Eisenstein Series and the Trace Formula

for GL(2) over a Function Field

Yuval Z. Flicker

Received: April 5, 2013 Revised: October 24, 2013

Communicated by Peter Schneider

Abstract. We write out and prove the trace formula for a convolution operator on the space of cusp forms on GL(2) over the function field F of a smooth projective absolutely irreducible curve over a finite field. The proof – which follows Drinfeld – is complete and all terms in the formula are explicitly computed. The structure of the homogeneous space GL(2, F)\ GL(2, 𝔸) is studied in section 2 by means of locally free sheaves of O_{X}-modules. Section 3 deals with the regularization and computation of the geometric terms, over conjugacy classes. Section 4 develops the theory of intertwining operators and Eisenstein Series, and the trace formula is proven in section 5.

2010 Mathematics Subject Classification: Primary 11F70, 11F72; Secondary 22E35, 22E55, 11G20, 11R39, 11R52, 11R58, 14H30, 11S37 Keywords and Phrases: Eisenstein series, intertwining operators, trace formula, automorphic representations, GL(2), function fields, orbital integrals

1 Introduction and statement of the Trace Formula

1.1 Introduction

The (non-invariant) trace formula for GL(2) over a number field was stated and its proof sketched in chapter 15 of the influential book of Jacquet and Langlands [JL70] of 1970. It was used there for comparison of automorphic representations of the multiplicative group of a quaternion algebra, with automorphic representations of GL(2). Drinfeld used the trace formula for GL(2) over a function field F to prove Langlands’ conjecture for GL(2, F), and to count in [D81] the number of two dimensional irreducible representations of the fundamental group of a smooth projective geometrically irreducible curve X over a finite field. To check the statement of the trace formula of [JL70] in the function field case, Drinfeld gave a detailed (but unpublished) proof. It differs from the one sketched in [JL70]. It is this proof of Drinfeld which is given in this paper. The main reason why this proof is still interesting is the elementary and unconventional treatment of Eisenstein series (see subsections 4.7-4.8 below), and the computation of traces in the spirit of Tate [T68], see subsection 5.2. In both cases it is based on a “baby model” (see Proposition 4.31, Corollary 4.32, Lemma 5.11), which cries out for generalization.

Let us describe the contents of this article. The trace formula itself is stated in subsection 1.2 with a few comments. More comments, including informal ones, are given in section 3. Section 2 contains a dictionary between the language of adèles and the language of vector bundles on the smooth projective curve X corresponding to F. In particular, the set of rank n vector bundles on X is identified with GL(n, F)\ GL(n, 𝔸)/ GL(n, O_{𝔸}), where O_{𝔸} ⊂ 𝔸 is the ring of integral adèles. This dictionary goes back to A. Weil [W38], although in an older language. It underlies Drinfeld’s Geometric Langlands program [BD]. The terms which appear in the geometric part of the trace formula – orbital integrals and weighted orbital integrals – are estimated and regularized in section 3. In section 4 intertwining operators, Eisenstein series, and L-functions are introduced. The rationality of the intertwining operator M(μ_{1}, μ_{2}, t) and the functional equation M^{2} = 1 are first proven using local computations: normalization of the intertwining operators by L-functions and ε-factors, and the functional equation of the L-functions. In subsections 4.7-4.8 these facts are proven using an alternative, global approach. The ideas might go back to Selberg. But technically the exposition is quite different and more elementary: in the case of function fields the analytic problems disappear. The trace formula is proven in section 5. The logarithmic derivative of the intertwining operator appears as a result of a computation of the trace of some operator in a power series space, see Lemma 5.11. This computation is probably related to Tate’s article [T68].

Here are some questions.

1. Could the methods of subsections 4.7-4.8 and section 5 be extended to prove the functional equation for Eisenstein series, and the trace formula, for an arbitrary reductive group over a function field?

2. Is there a modification of the technique from subsections 4.7-4.8 that would work in the case of number fields, e.g., for GL(2, ℚ)? One could try to replace the space of formal power series used in subsections 4.7-4.8 by some space of holomorphic functions.

3. What is the precise relationship between Lemma 5.11 and Tate’s [T68]?

4. What is the relationship between the approach to Eisenstein series of subsections 4.7-4.8, and the classical approaches: that of Selberg-Langlands-Arthur, and that of scattering theory (see [FP72] or [LP76])? This author’s initial motivation to write out Drinfeld’s expression and proof of the trace formula for GL(2) over a function field stems from his search for higher rank analogues of Drinfeld’s formula [D81]. This led us to count with Deligne [DF13] the number of rank n (≥ 2) local systems with principal unipotent local monodromy at least at two places. There we use the trace formula in the compact quotient case, and the transfer of automorphic representations from a compact form to GL(n). This explains the condition: “at least at two places”. The case of [D81] is rank n = 2, no monodromy. To complete the study of [D81] and of [DF13] in rank two one has to consider the case of principal unipotent local monodromy at a single place. This is done in [F], using the explicit computations of the trace formula for GL(2) over a function field of the present work. This was our initial motivation to write out this formula. Drinfeld’s proof in the case of rank two, no ramification, is also given in [F]. Of course there are numerous expositions of the trace formula of [JL70], e.g. [GJ79], geared to explain the lifting application of [JL70], mainly in the number field case. But none computes explicitly (and accurately, cf. [D81]) all the terms which appear in the trace formula. The latter is precisely what is needed for the counting applications of [D81] and [F]. An attempt at a complete exposition of the computations for GL(2) in the number field case is at [AFOO]. Of course the trace formula of [JL70] was generalized to the higher rank case by Arthur, see e.g. [A05], in the number field case, and by Lafforgue, see e.g. [Lf97], in the function field case. But the important applications of these works did not require explicit evaluation of all the terms which appear in the trace formula, so our results are not included in those of [Lf97], even in the case of GL(2) considered here. In the number field case, the Remark on p. 112 of [A05] states: “As a matter of fact, it is only in the case of GL(2) that the general coefficients have been evaluated. It would be very interesting to understand them better in other examples, although this does not seem to be necessary for presently conceived applications of the trace formula”. Indeed the applications of [D81], [DF13],

[F] – counting rather than comparing – are of different nature than those of [JL70], [A05], [Lf97], where most terms can be erased a-priori in the comparison so they need not be computed. To repeat what is explained above, we also think the approach of subsections 4.7-4.8 and section 5 is original, substantially different from the currently known methods (which are developed in [A05], [Lf97]), interesting and warrants further development. I am deeply grateful to V. Drinfeld for making available to me his unpublished notes, for teaching me lots of mathematics in the process, and for his permission to publish this paper; to A. Beilinson for telling me at IHES about Drinfeld’s notes; to the referee for the very careful reading. The author was a Schonbrunn visiting Professor at the Hebrew University, Jerusalem. Work partially supported by a grant (#267097) from the Simons Foundation.

1.2 Statement of the Trace Formula

Let us write the trace formula for GL(2) over a function field F of a smooth projective geometrically connected curve X over a finite field 𝔽_{q}, and a test function f in C_{c}^{∞}(GL(2, 𝔸)) (subscript c for “compactly supported”, superscript ∞ for “locally constant”, 𝔸 denotes the ring of adèles of F). Let r_{0} be the representation of GL(2, 𝔸) by right translation on the space A_{0,α} of cusp forms on α^{ℤ} · GL(2, F)\ GL(2, 𝔸), and r_{0}(f) = ∫ f(g)r_{0}(g)dg (g ∈ GL(2, 𝔸)) the convolution operator; dg = ⊗_{v}dg_{v} is a Haar measure. Here α is a fixed idèle of degree 1, whose components are almost all equal to 1. A cusp form is a function ϕ : GL(2, F)\ GL(2, 𝔸) → E (E is a fixed algebraically closed subfield of ℂ) which is invariant on the right by some open compact subgroup of GL(2, 𝔸), and ∫_{N(F)\N(𝔸)} ϕ(nx)dn = 0 for all x in GL(2, 𝔸). Here N denotes the unipotent upper triangular subgroup of GL(2). We also write A for the diagonal subgroup, and A′ = A − Z where Z is the center of GL(2). By a well known result of G. Harder, when F is a function field (but not a number field) a cusp form is compactly supported modulo Z(𝔸).

Theorem 1.1. For any f ∈ C_{c}^{∞}(GL(2, 𝔸)) we have tr r_{0}(f) = ∑_{1≤i≤8} S_{i}(f). The terms are:

S_{1}(f) = |α^{ℤ} · GL(2, F)\ GL(2, 𝔸)| ∑_{γ∈α^{ℤ}·F^{×}} f(γ).

S_{2}(f) = ∑_{F_{2}} S_{2,F_{2}}(f),

S_{2,F_{2}}(f) = | Aut_{F} F_{2}|^{−1} ∑_{γ∈α^{ℤ}(F_{2}−F)} ∫_{GL(2,𝔸)/α^{ℤ}·F_{2}^{×}} f(xγx^{−1})dx.

Here F_{2} ranges over the set of isomorphism classes of quadratic extensions of the field F. For each F_{2} we fix an embedding F_{2} ↪ M(2, F) into the ring of 2 × 2 matrices over F.

S_{3}(f) = ∑_{γ∈α^{ℤ}·A′(F)} ∫_{A(𝔸)\ GL(2,𝔸)} f(x^{−1}γx)v(x)dx.

Any x ∈ GL(2, 𝔸) can be written in the form ank, a ∈ A(𝔸), k ∈ GL(2, O_{𝔸}), n = ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ), b is determined uniquely by x up to b ↦ ub + w, u ∈ O_{𝔸}^{×}, w ∈ O_{𝔸}. Put v(x) = ∑_{v} log_{q}(max(1, |b_{v}|_{v})).

S_{4}(f) = ∑_{a∈F^{×}α^{ℤ}} θ̃_{a,f}(1),  θ̃_{a,f}(t) = \frac{1}{2}(θ_{a,f}(t) + θ_{a,f}(t^{−1})),

θ_{a,f}(t) = ∫_{F^{×}α^{ℤ}N(F)\ GL(2,𝔸)} f (x^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) x) t^{ht^{+}(x)}dx,

ht^{+} : GL(2, 𝔸) → ℤ is defined by ht^{+} (( \begin{matrix}a & c \\ 0 & b \end{matrix}) k) = deg a − deg b (k ∈ GL(2, O_{𝔸}); a, b ∈ 𝔸^{×}; c ∈ 𝔸).

S_{5}(f) = \frac{−1}{4πi} ∑_{μ_{1},μ_{2}} ∮_{|z|=1} tr I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}, f)\frac{m′(μ_{1}/μ_{2}, z)}{m(μ_{1}/μ_{2}, z)}2zdz.

Here m(μ, z) = L(μ, z)/L(μ, z/q). The μ_{1}, μ_{2} range over the set of characters of 𝔸^{×}/F^{×} · α^{ℤ}, ν_{z}(x) = z^{deg(x)}. Also I(μ_{1}, μ_{2}) is the space of right locally constant functions ϕ on GL(2, 𝔸) with

ϕ (( \begin{matrix}a & c \\ 0 & b \end{matrix}) x) = |a/b|^{1/2}μ_{1}(a)μ_{2}(b)ϕ(x) (x ∈ GL(2, 𝔸); a, b ∈ 𝔸^{×}; c ∈ 𝔸).

It is a GL(2, 𝔸)-module by right translation, and tr I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}, f) is the trace of the indicated convolution operator.

S_{6}(f) = \frac{−1}{4πi} ∑_{μ_{1},μ_{2}} ∮_{|z|=1} tr[I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}, f) · R(μ_{1}, μ_{2}, z)^{−1} \frac{d}{dz} R(μ_{1}, μ_{2}, z)]dz.

Notations are as in S_{5}(f), and R(μ_{1}, μ_{2}, z) : I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}) → I(μ_{2}ν_{z^{−1}}, μ_{1}ν_{z}) is an operator, rational in z, defined as a product ⊗_{v}R(μ_{1v}, μ_{2v}, z_{v}), z_{v} = z^{deg(v)}. The product is well defined as the local operator maps the function in the source whose restriction to GL(2, O_{v}) is 1 to such function in the target. Further, R(μ_{1v}, μ_{2v}, z) is defined to be

[L(μ_{1v}/μ_{2v}, z^{2}/q_{v})/L(μ_{1v}/μ_{2v}, z^{2})]M(μ_{1v}, μ_{2v}, z).

The operator M(μ_{1v}, μ_{2v}, z) = M(μ_{1v}ν_{z}, μ_{2v}ν_{z^{−1}}) is defined first by an integral

ϕ ↦ ∫ ϕ (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) x) dy if |(μ_{1v}/μ_{2v})(π π π_{v})z^{2}| < 1,

then by analytic continuation, as it is a rational function in z. The operators I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}, f) and R(μ_{1}, μ_{2}, z) are considered as operators on

I_{0}(μ_{1}, μ_{2}) = \{ϕ ∈ C^{∞}(GL(2, O_{𝔸})); ϕ (( \begin{matrix}a & c \\ 0 & b \end{matrix}) x) = μ_{1}(a)μ_{2}(b)ϕ(x);

x ∈ GL(2, O_{𝔸}), a, b ∈ O_{𝔸}^{×}; c ∈ O_{𝔸}}.

S_{7}(f) = \frac{1}{4} ∑_{μ} tr I(μ, μ, f), S_{8}(f) = − ∑_{μ} ∫_{GL(2,𝔸)} f(x)μ(det x)dx.

Both sums range over all characters μ of 𝔸^{×}/F^{×} · α^{2ℤ}. The sum of S_{8} is over all automorphic one dimensional representations (μ∘det) of α^{ℤ}\ GL(2, 𝔸). The integral there represents the trace of the convolution operator associated with f.

The terms S_{1}(f) and S_{2}(f) are finite by Proposition 3.5, 3.6, 3.9. The argument used in the proof of Proposition 3.9 shows that for any γ ∈ α^{ℤ}(A(F) − Z(F)) the function x ↦ f(x^{−1}γx) on A(𝔸)\ GL(2, 𝔸) has compact support, hence the integral in S_{3}(f) converges.

By Proposition 3.11 the function θ_{a,f}(t) is rational and may have at t = 1 a pole of order at most 1, for each a ∈ 𝔸^{×}. Hence θ̃_{a,f}(t) is regular at t = 1. From Proposition 3.5 it follows that the sums in S_{3}(f) and S_{4}(f) are finite, so these terms are well defined. For any f = ⊗f_{v} in C_{c}^{∞}(GL(2, 𝔸)), the operator I(μ_{1}, μ_{2}, f) is zero unless μ_{i} are unramified at each v where f_{v} is GL(2, O_{v}) biinvariant. This implies that the sums in S_{i}(f) (5 ≤ i ≤ 8) are finite, for a given f. To see that S_{5}(f) and S_{6}(f) are well defined, note that the rational functions m(μ, t), R(μ_{1}, μ_{2}, t), R(μ_{1}, μ_{2}, t)^{−1} are regular on |t| = 1 for all characters μ, μ_{1}, μ_{2} of 𝔸^{×}/F^{×} · α^{ℤ}. For m(μ, t) this follows from Proposition 4.11, for R and R^{−1} from Corollary 4.28. The distributions [linear forms on C_{c}^{∞}(GL(2, 𝔸))] f ↦ tr r_{0}(f), S_{i}(f) (i = 1, 2, 5, 7, 8) are invariant, namely take the same value at f and f^{h}(x) = f(h^{−1}xh), h ∈ GL(2, 𝔸). For i = 3, 4, 6 we have S_{i}(f^{h}) = S_{i}(f) if h ∈ GL(2, O_{𝔸}), but S_{i} is not invariant. If f ∈ C_{c}^{∞}(GL(2, 𝔸)) takes values in ℚ then tr r_{0}(f) ∈ ℚ, since the representation r_{0} is defined over ℚ. For i = 1, 2, 3, 4, 8 it is clear that S_{i}(f) ∈ ℚ. For i = 7 the integrand contains the factor μ(ab)|a/b|^{1/2} which involves \sqrt{q}. However the sum includes with μ also με, ε(α) = −1, and so the sum of the terms indexed by μ and με can be written as an integral over the domain where |a/b| is in q^{2ℤ}. To see that S_{5}(f) is rational, we put a(μ_{1}, μ_{2}) = \frac{1}{2πi} ∮_{|t|=1} f(μ_{1}, μ_{2}, t)dt where

f(μ_{1}, μ_{2}, t) = tr I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}, f) · \frac{d}{dt} ln m(μ_{1}/μ_{2}, t^{2}),

and claim that for any σ ∈ Gal(\overline{ℚ}/ℚ) one has σ(a(μ_{1}, μ_{2})) = a(^{σ}μ_{1},^{σ}μ_{2}). Note that Gal(\overline{ℚ}/ℚ) acts on the group of characters on 𝔸^{×}/F^{×} · α^{ℤ} as they are all ℚ-valued. Now a(μ_{1}, μ_{2}) is the sum of the residues of f(μ_{1}, μ_{2}, t) at the points of the unit disc. We have that σ(f(μ_{1}, μ_{2}, t)) = f(^{σ}μ_{1},^{σ}μ_{2}, ε(σ) · ^{σ}t) with ε(σ) = σ(\sqrt{q})/\sqrt{q}. However, if f(μ_{1}, μ_{2}, t) has a pole at t = t_{0} and |t_{0}| < 1, then by Proposition 4.11, |σ(t_{0})| < 1 for any σ ∈ Gal(\overline{ℚ}/ℚ). Hence S_{5}(f) ∈ ℚ. To see that S_{6}(f) ∈ ℚ one proceeds similarly, using the results of Corollary 4.28 on the poles of R(μ_{1}, μ_{2}, t) and R(μ_{1}, μ_{2}, t)^{−1}.

2 Locally free sheaves of O_{X}-modules

2.1 Stable bundles

Let X be a smooth geometrically connected projective curve over 𝔽_{q} (we take minimal q). Denote by O_{X} the structure sheaf of X. Denote by Bun_{n} the set of isomorphism classes of rank n locally free sheaves of O_{X}-modules. By a (vector) bundle we mean here simply a locally free sheaf. In particular, Bun_{1} = Pic X. The Picard group Pic X of invertible, or rank 1, locally free sheaves L of O_{X}- modules, is naturally isomorphic to the group of classes \overline{D} of (Weil) divisors D = ∑_{v} n_{v}v (n_{v} ∈ ℤ, v ∈ |X|). Here |X| is the set of closed points of X, and the divisors D, D′ lie in the same class (are linearly equivalent) if their difference is the (principal) divisor (f) = ∑_{v} ord_{v}(f)v where f is a nonzero rational function on X and ord_{v}(f) is the order of f at v ∈ |X| (ord_{v}(f) > 0 if v is a zero, ord_{v}(f) < 0 if v is a pole, ord_{v}(f) = 0 otherwise). If L, M ∈ Pic X correspond to the divisors D, D′ then L ⊗ M corresponds to D + D′. There is a degree map deg on Pic X: deg(∑_{v} n_{v}v) = ∑_{v} n_{v} deg(v) defines deg(L) = deg(D), where deg(v) = [k_{v} : 𝔽_{q}]. Here k_{v} is the residue field of the function field F = 𝔽_{q}(X) of X over 𝔽_{q} at v; assume 𝔽_{q} is algebraically closed in F. We write F_{v} for the completion of F at v, O_{v} for its ring of integers. The cardinality of the residue field k_{v} = 𝔽_{q_{v}} at v is denoted by q_{v}, thus q_{v} = q^{deg(v)}. We also write deg(\overline{D}) for deg(D), as the degree of a principal divisor is 0; recall that \overline{D} denotes the class of D. Denote by χ(L) = dim_{𝔽_{q}} H^{0}(X, L) − dim_{𝔽_{q}} H^{1}(X, L) the Euler-Poincaré characteristic of L ∈ Pic X. Here H^{i}(X, L) are finite dimensional vector spaces over 𝔽_{q}. Then χ(O_{X}) = 1 − g where g = dim_{𝔽_{q}} H^{1}(X, O_{X}) is named the genus of X. The Riemann-Roch theorem asserts that χ(L) − deg(L) = χ(O_{X}) is independent of L ∈ Pic X. Define the degree of a locally free sheaf E of O_{X}-modules of rank n to be deg E = χ(E) − nχ(O_{X}). The determinant of E is det E = ⋀^{n} E ∈ Pic X. We have deg E = deg det E. This gives an alternative definition of the degree. A proof of this equality is as follows. If E is a line bundle, then there is nothing to prove. In the general case, use the fact that both deg E and deg det E are additive (if E′ ⊂ E is a subbundle, then deg E = deg E′+deg(E/E′) and similarly for deg det E), and that each vector bundle has a flag, E_{i}, such that E_{i}/E_{i−1} are line bundles. The height of a rank two locally free sheaf E of O_{X}-modules is the integer ht(E) = max_{L}(2 deg L − deg E), L ranges over all invertible subsheaves of E.

Proposition 2.1. We have −2g ≤ ht(E) < ∞.

Proof. Let L be an invertible subsheaf of E. From the Riemann-Roch theorem χ(L) = deg L + 1 − g we obtain dim_{𝔽_{q}} H^{0}(X, L) ≥ deg L + 1 − g, whence deg L ≤ dim_{𝔽_{q}} H^{0}(X, L) + g − 1 ≤ dim_{𝔽_{q}} H^{0}(X, E) + g − 1, so ht(E) is finite. Let L be an invertible subsheaf of E of maximal degree. Let M be an invertible sheaf with deg M = deg L + 1. Then Hom(M, E) = 0. Also, by Riemann-Roch for the rank 2 sheaf E, dim_{𝔽_{q}} Hom(M, E) = dim_{𝔽_{q}} H^{0}(X, M^{−1}E) ≥ deg(M^{−1}E) + 2 − 2g = deg E − 2 deg M + 2 − 2g = deg E − 2 deg L − 2g, so 2 deg L − deg E ≥ −2g.

A rank two locally free sheaf E of O_{X}-modules is called stable if ht(E) < 0 and semistable if ht(E) ≤ 0. In general, the slope μ(E) of a locally free sheaf E over an algebraic curve is defined to be deg E/ rk E, and E is called stable if μ(F) < μ(E) for all proper nonzero subbundles F of E (semistable if ≤). A locally free sheaf E of rank two is called almost stable if ht(E) < 2g − 1, and very unstable if ht(E) ≥ 2g − 1. If g = 0, every E is very unstable.

Remark 1. A very unstable vector bundle E of rank 2 splits into the direct sum of two line bundles. We give here a relatively elementary treatment. An extension can be found in the work of Harder and Narasimhan. If E is very unstable, L is an invertible subsheaf of E of maximal degree, and M = E/L, then M is invertible and Ext(M, L) = H^{1}(X, M^{−1}L) is 0 since

deg M^{−1}L = deg L − deg M = 2 deg L − deg E = ht E ≥ 2g − 1.

Indeed, by Serre duality H^{1}(X, M^{−1}L) = H^{0}(X, L^{−1}Mω) where ω denotes the canonical bundle. But deg L^{−1}Mω ≤ 2g − 2 − (2g − 1) < 0, and H^{0}(X, F) = 0 for an invertible sheaf F with negative degree.

Proposition 2.2. The number of isomorphism classes of almost stable rank two locally free sheaves E of O_{X}-modules with a fixed degree is finite.

Proof. The height of an almost stable sheaf lies in [−2g, 2g − 2]. Hence it suffices to show the finiteness for E with a fixed degree n and height h. Every such sheaf lies in an exact sequence 0 → L → E → M → 0, where L and M are invertible sheaves and 2 deg L − deg E = h. Then deg L = (n + h)/2, deg M = (n − h)/2. Since the degrees of L and M are fixed, there are only finitely many possibilities for L and M (set of cardinality of the 𝔽_{q}-points on the abelian variety Pic^{0}(X)). With L and M fixed there are only finitely many choices for E as Ext(L, M) is finite.

The group Pic X acts on Bun_{2} : (L ∈ Pic X, E ∈ Bun_{2}) ↦ L ⊗ E. As

deg(L ⊗ E) = 2 deg(L) + deg(E),

the set of almost stable sheaves is invariant under this action. In a Pic X-orbit we may choose E to have deg(E) in {0, 1}. Hence we deduce

Corollary 2.3. The number of Pic X-orbits on the set of isomorphism classes of almost stable rank two locally free sheaves of O_{X}-modules is finite.

2.2 Bundles and lattices

Let E be a rank n locally free sheaf of O_{X}-modules. Denote by E_{η} the fiber (= stalk) of E over the generic point η of X. Let E_{(v)} be the stalk of E at the closed point v ∈ |X|. Let O_{(v)} be the local ring of X at v. Then E_{η} is an n-dimensional vector space over F, and E_{(v)} is an O_{(v)}-lattice in E_{η}, namely a rank n free O_{(v)}-submodule of E_{η}. A set M of O_{(v)}-lattices M_{(v)} in a finite dimensional vector space V over F, v ranges over the set |X| of closed points in X, is called adelic if there exists a basis {e_{1}, …, e_{n}} in V such that M_{(v)} = O_{(v)}e_{1} + ⋯ + O_{(v)}e_{n} for almost all v in |X|. “Almost all” means “with at most finitely many exceptions”. If M is adelic then it is adelic with respect to any basis {e_{1}, …, e_{n}} of V. The set of stalks {E_{(v)}; v ∈ |X|} of a locally free sheaf E of O_{X}-modules is adelic. Conversely, an adelic set of lattices M = {M_{(v)}; v ∈ |X|} in a finite dimensional vector space V over F is the set of stalks of the locally free sheaf E of O_{X}-modules defined by

H^{0}(U, E) = {s ∈ V; ∀v ∈ U, s ∈ M_{(v)}}

for any open subset U of X. Obtained is an equivalence of the category of finite rank locally free sheaves of O_{X}-modules, with the category of finite dimensional vector spaces over F with adelic sets of O_{(v)}-lattices. Let O_{v} be the completion of O_{(v)}. The completion of F at v is denoted F_{v}. Let V be a finite dimensional vector space over F. Put V_{v} = V ⊗_{F} F_{v}. There is a natural bijection between the set of O_{(v)}-lattices in V, and O_{v}-lattices in V_{v}: an O_{(v)}-lattice M ⊂ V corresponds to the lattice M ⊗_{O_{(v)}} O_{v} in V_{v}; an O_{v}-lattice N ⊂ V_{v} corresponds to the O_{(v)}-lattice N ∩ V. The category C whose objects are finite dimensional F -vector spaces V with adelic sets {M_{v}; v ∈ |X|} of O_{v}-lattices M_{v} in V_{v} is equivalent to the category of finite rank locally free sheaves of O_{X}-modules E, by E ↦ (E_{η}, {E_{v}}), where E_{η} is the generic fiber of E and E_{v} is the completion of the stalk of E at the closed point v ∈ |X|. Let R_{n} be the set of isomorphism classes of pairs (E, i) where E is a rank n locally free sheaf of O_{X}-modules, and i is an isomorphism from the generic fiber of E to F^{n}. The pairs (E, i) and (E′, i′) are isomorphic if there is an isomorphism E ˜→E′ which induces a commutative diagram when restricted to the generic fiber with sides i and i′ and the identity F^{n} → F^{n}. The group GL(n, F) acts on R_{n} by g : (E, i) ↦ (E, g ∘ i). Then GL(n, F)\R_{n} = Bun_{n} is the set of isomorphism classes of rank n locally free sheaves of O_{X}-modules. The set R_{n} is the set of adelic collections of O_{v}-lattices M_{v} ⊂ F_{v}^{n}, v ∈ |X|. The group GL(n, F_{v}) acts transitively on the set of O_{v}-lattices in F_{v}^{n}. The stabilizer of the standard lattice O_{v}^{n} in F_{v}^{n} is GL(n, O_{v}). Thus the set of O_{v}-lattices in F_{v}^{n} is GL(n, F_{v})/ GL(n, O_{v}), and R_{n} is GL(n, 𝔸)/ GL(n, O_{𝔸}), where 𝔸 is the ring of adèles in F and O_{𝔸} = ∏_{v∈|X|} O_{v}. Thus

Bun_{n} = GL(n, F)\ GL(n, 𝔸)/ GL(n, O_{𝔸}).

The elements of GL(n, 𝔸)/ GL(n, O_{𝔸}) are called matrix divisors, and the elements of GL(n, F)\ GL(n, 𝔸)/ GL(n, O_{𝔸}) classes of matrix divisors. For n = 1, the identification of GL(n, F)\ GL(n, 𝔸)/ GL(n, O_{𝔸}) with Bun_{n} is the identification of classes of divisors with invertible sheaves. The group GL(n, 𝔸) can be identified with the set of triples

(E, i_{η} : E_{η} →^{∼} F^{n}, (i_{v} : E_{v} →^{∼} O_{v}^{n})).

Given a rank n locally free sheaf E, an isomorphism i_{η} : E_{η} →^{∼} F^{n}, and for each closed point v in |X| an isomorphism i_{v} : E_{v} →^{∼} O_{v}^{n} of the completion E_{v} of the stalk E_{(v)} at v with O_{v}^{n}, let us define the corresponding g = (g_{v}) in GL(n, 𝔸). Each g_{v} has to be an automorphism F_{v}^{n} → F_{v}^{n}, with g_{v}(O_{v}^{n}) = O_{v}^{n} for almost all v. Construct g_{v} as the composition i_{v} ∘ i_{η}^{−1}:

F_{v}^{n} = F^{n} ⊗_{F} F_{v} ←^{∼}^{i_{η}} E_{η} ⊗_{F} F_{v} = E_{F_{v}} = E_{v} ⊗_{O_{v}} F_{v} →^{∼}^{i_{v}} O_{v}^{n} ⊗_{O_{v}} F_{v} = F_{v}^{n}.

Note that since E is locally free, for almost all v the map g_{v} = i_{v} ∘ i_{η}^{−1} takes O_{v}^{n} ⊂ F_{v}^{n} to E_{v} ⊂ E_{η}⊗_{F} F_{v} via i_{η}^{−1}, and then to O_{v}^{n} via i_{v}. To show that the map {(E, i_{η}, (i_{v}))} → GL(n, 𝔸) is bijective one shows that GL(n, 𝔸) acts on the set of triples, simply transitively. Viewing the trivial locally free sheaf as O_{𝔸}^{n} (space of columns), g(E, i_{η}, (i_{v})) is defined to be (gE, i_{η}, (i_{v} ∘g_{v}^{−1})), where i_{v} ∘g_{v}^{−1} maps the stalk g_{v}E_{v} of gE at v to O_{v}^{n}. The set of pairs {(E, i_{η})} then corresponds to GL(n, 𝔸)/ GL(n, O_{𝔸}), the set of pairs {(E, (i_{v}))} to GL(n, F)\ GL(n, 𝔸), and the set {E} to GL(n, F)\ GL(n, 𝔸)/ GL(n, O_{𝔸}). To an idèle a = (π π π_{v}^{−n_{v}}u_{v}; v ∈ |X|), where π π π_{v} denotes a generator of the maximal ideal in the ring O_{v} of integers in F_{v}, u_{v} ∈ O_{v}^{×} and n_{v} ∈ ℤ, we associate the divisor D = ∑_{v} n_{v}v, and the degree

deg(a) = deg(D) = ∑_{v} n_{v} deg(v), deg(v) = [𝔽_{v} : 𝔽_{q}],

where 𝔽_{v} is the residue field of F at v, a finite field of q_{v} = q^{deg(v)} elements. For g ∈ GL(2, 𝔸) write deg g for deg det g. Recall that O_{𝔸} = ∏_{v} O_{v} (v ∈ |X|). For t ∈ ℂ^{×} we write

ν_{t}(a) = t^{− deg(a)} = ∏_{v} t_{v}^{−n_{v}}

where t_{v} = t^{deg(v)}. Then ν_{q^{−1}}(a) = ∏_{v} q_{v}^{n_{v}} = |a| is equal to ν(a) = q^{deg(a)}. Also ν_{t}(π π π_{v}) = t_{v}, ν_{q^{−1}}(π π π_{v}) = |π π π_{v}|. Let L and M be invertible sheaves. Fix isomorphisms i_{L}, i_{M} of their generic fibers with F. Each of (L, i_{L}) and (M, i_{M}) defines an element of 𝔸^{×}/O_{𝔸}^{×}, namely a divisor on X. Choose representatives a, b in 𝔸^{×}, for example ∑_{v} n_{v}v is represented by (π π π_{v}^{−n_{v}}). Given an exact sequence 0 → L → E → M → 0 of locally free sheaves, choose an isomorphism φ between the generic fiber of E and F^{2} so that the induced exact sequence of generic fibers 0 → F → F^{2} → F → 0 is standard (x ↦ ( \begin{matrix}x \\ 0 \end{matrix} ) , ( \begin{matrix}x \\ y \end{matrix}) ↦ y). The isomorphism φ is defined uniquely up to left multiplication by an automorphism of F^{2} of the form ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ), t ∈ F. The pair (E, φ) determines an element of GL(2, 𝔸)/ GL(2, O_{𝔸}), of the form u = ( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}), with z in 𝔸. Since u is defined up to right multiplication by an element of GL(2, O), z is uniquely defined up to addition of an element of \frac{a}{b}O_{𝔸}. Replacing φ by ( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) φ with t ∈ F replaces z by z + t. Thus we get a bijection

Ext(M, L) → 𝔸/(F + \frac{a}{b}O_{𝔸}).

This is an isomorphism of 𝔽_{q}-vector spaces. In summary, if the invertible sheaves L and M correspond to idèles a and b, then Ext(M, L) ≃ 𝔸/(F + \frac{a}{b}O_{𝔸}), and the map Ext(M, L) → Bun_{2} which associates to the exact sequence 0 → L → E → M → 0 its middle term, coincides with the map 𝔸/(F + \frac{a}{b}O_{𝔸}) ≃ H^{1}(X, M^{−1}L), see [S97], II. 5. The isomorphism 𝔸/(F + \frac{a}{b}O_{𝔸}) →^{∼} Ext(M, L) is H^{1}(X, M^{−1}L) ˜→ Ext(M, L).

2.3 The space GL(2, F)\ GL(2, 𝔸)

Proposition 2.4. Given a ∈ 𝔸^{×}, deg a ≥ 2g − 1, then aO_{𝔸} + F = 𝔸.

Proof. If L is an invertible sheaf on X associated with a, then 𝔸/(F + aO_{𝔸}) = H^{1}(X, L). By Serre duality H^{1}(X, L) ≃ H^{0}(X, L^{−1}ω), where ω is the canonical bundle of degree 2g − 2. Then deg(L^{−1}ω) ≤ (2g − 2) − (2g − 1) = −1 < 0, hence H^{0}(X, L^{−1}ω) = {0}.

Define a function

ht^{+} : GL(2, 𝔸) → ℤ  by  ht^{+} (( \begin{matrix}a & c \\ 0 & b \end{matrix}) k) = deg a − deg b

for all a, b ∈ 𝔸^{×}, c ∈ 𝔸, k ∈ GL(2, O_{𝔸}). It is clearly a well defined function on B(F)\ GL(2, 𝔸). For x ∈ GL(2, 𝔸), put

ht(x) = max_{γ∈GL(2,F)} ht^{+}(γx).

On GL(2, F)\ GL(2, 𝔸) it is well defined.

Proposition 2.5. For any x ∈ GL(2, 𝔸) we have −2g ≤ ht(x) < ∞.

Proof. This follows from Proposition 2.1 as if E is a rank two locally free sheaf of O_{X}-modules associated to the image of x in GL(2, F)\ GL(2, 𝔸)/ GL(2, O_{𝔸}), then ht(x) = ht(E).

Put H_{B} = {x ∈ B(F)\ GL(2, 𝔸); ht^{+}(x) > 0} and

H = {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) > 0}.

Proposition 2.6. (1) The restriction p to H_{B} of the natural projection p′ : B(F)\ GL(2, 𝔸) → GL(2, F)\ GL(2, 𝔸) is a homeomorphism H_{B} → H.

(2) The set {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) ≤ n} is compact modulo the center Z(𝔸) of GL(2, 𝔸) for every integer n.

Proof. (1) The map p is clearly onto. To show that p is injective it suffices to show for any x in GL(2, 𝔸), γ ∈ GL(2, F), that ht^{+}(x) > 0 and ht^{+}(γx) > 0 implies γ ∈ B(F). This is a typical application of the Harder-Narasimhan filtration. In simple, explicit terms, this follows from

Lemma 2.7. If g ∈ GL(2, F) − B(F) then ht^{+}(x) + ht^{+}(gx) ≤ 0.

Proof. Write g as g_{1}wg_{2} with g_{1}, g_{2} in B(F), w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Put x′ = g_{2}x. Then ht^{+}(x) = ht^{+}(x′), ht^{+}(gx) = ht^{+}(wx′). Thus we need to show that

ht^{+}(x′) + ht^{+}(wx′) ≤ 0. Suppose x′ = ( \begin{matrix}a_{1} & c_{1} \\ 0 & b_{1} \end{matrix} ) k_{1}, wx′ = ( \begin{matrix}a_{2} & c_{2} \\ 0 & b_{2} \end{matrix} ) k_{2} with

k_{1}, k_{2} ∈ GL(2, O_{𝔸}). Put k_{2}k_{1}^{−1} = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ). Then ( \begin{matrix}a_{2} & c_{2} \\ 0 & b_{2} \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = w ( \begin{matrix}a_{1} & c_{1} \\ 0 & b_{1} \end{matrix} ) =

( \begin{matrix}0 & b_{1} \\ a_{1} & c_{1} \end{matrix} ). Hence b_{2}γ = a_{1}. Thus deg a_{1} ≤ deg b_{2} (as deg γ ≤ 0, since γ ∈ O_{𝔸}). But deg a_{2}b_{2} = deg a_{1}b_{1}. Hence deg a_{2} ≤ deg b_{1}. Then ht^{+}(x′) + ht^{+}(wx′) = deg a_{1} − deg b_{1} + deg a_{2} − deg b_{2} ≤ 0.

Now the natural projection p′ : B(F)\ GL(2, 𝔸) → GL(2, F)\ GL(2, 𝔸) is open and H_{B} is an open subset of B(F)\ GL(2, 𝔸). Hence the bijection p = p′|H_{B} : H_{B} → H is open. Since it is also continuous, p is a homeomorphism.

(2) The image of the set S = {x ∈ B(F)\ GL(2, 𝔸); −2g ≤ ht^{+}(x) ≤ n} in GL(2, F)\ GL(2, 𝔸) under p′ contains the set {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) ≤ n}. So it suffices to show that S is compact mod Z(𝔸). Choose a compact C in 𝔸^{×} with

CF^{×} = {t ∈ 𝔸^{×}; −2g ≤ deg t ≤ n}.

Choose an idèle d with deg d ≥ 2g − 1. Put

Y = \{( \begin{matrix}1 & c \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) k; k ∈ GL(2, O_{𝔸}), a, b ∈ 𝔸^{×}, \frac{a}{b} ∈ C, c ∈ dO_{𝔸}\} .

Lemma 2.8. The map Y → S is surjective.

Proof. Let x ∈ GL(2, 𝔸), −2g ≤ ht^{+}(x) ≤ n. We need to show that x can be written as hy with y ∈ Y and h ∈ B(F). Write x as ( \begin{matrix}r & s \\ 0 & t \end{matrix}) K with k ∈ GL(2, O_{𝔸}), r, t ∈ 𝔸^{×}, s ∈ 𝔸. It remains to show that ( \begin{matrix}r & s \\ 0 & t \end{matrix}) can be expressed as

( \begin{matrix}α & γ \\ 0 & β \end{matrix} ) ( \begin{matrix}1 & c \\ 0 & 1 \end{matrix} ) ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) with a, b ∈ 𝔸^{×}, \frac{a}{b} ∈ C, c ∈ dO_{𝔸}, α, β ∈ F^{×}, γ ∈ F. Thus we

need to show the existence of a, b, c, α, β, γ such that (*) aα = r, βb = t, a, b ∈ 𝔸^{×}, α, β ∈ F^{×}, \frac{a}{b} ∈ C, (**) b(αc + γ) = s, c ∈ dO_{𝔸}, γ ∈ F. By definition of x, deg r − deg t lies in [−2g, n], so the existence of a, b, α, β satisfying (*) follows from the definition of C. The existence of c ∈ dO_{𝔸} and γ ∈ F satisfying αc + γ = s/b follows from: cO_{𝔸} + F = 𝔸 if deg c ≥ 2g − 1.

Since Y is compact mod Z(𝔸), so is S, and (2) follows. □

In summary, the homogeneous space GL(2, F)\ GL(2, 𝔸) is the union of the compact mod Z(𝔸) set {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) ≤ 0}, and the set H = {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) > 0}, whose structure is simpler. The set H_{B}, hence also the sets H and GL(2, F)\ GL(2, 𝔸), are noncompact modulo Z(𝔸). Indeed the function ht^{+} takes arbitrary large values. The image of H in Bun_{2} = GL(2, F)\ GL(2, 𝔸)/ GL(2, O_{𝔸}) is the set of nonsemistable locally free sheaves. The set GL(2, F)\ GL(2, 𝔸)/ GL(2, O_{𝔸}) is analogous to the set

SL(2, ℤ)\ SL(2, ℝ)/ SO(2) = SL(2, ℤ)\𝔥,

where 𝔥 = {z ∈ ℂ; Im z > 0}, the upper half plane, is isomorphic to SL(2, ℝ)/ SO(2), by

g ↦ g(i) = (ai + b)/(ci + d).

The set B(F)\ GL(2, 𝔸)/ GL(2, O_{𝔸}) is analogous to N\𝔥 where N is the group of transformations z ↦ z+n (n ∈ ℤ) on 𝔥. The function ht^{+} is analogous to the function z ↦ ln Im z on N\𝔥. The statement −2g ≤ ht(x) < ∞ corresponds to the statement that the natural map from the half plane \{z ∈ ℂ; Im z ≥ \sqrt{3}/2\} to SL(2, ℤ)\𝔥 is onto. The statement that p : H_{B} → H is homeomorphism corresponds to the statement that the map N\{z ∈ ℂ; Im z > 1} → SL(2, ℤ)\𝔥 is injective, and the compactness of {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) ≤ n} corresponds to the statement that the complement in SL(2, ℤ)\𝔥 of the image of the half plane {z ∈ ℂ; Im z > h} is compact.

2.4 ℓ-groups

An ℓ-space is a Hausdorff topological space such that each of its points has a fundamental system of open compact neighborhoods. We shall consider on ℓ-spaces only measures for which every open compact subset is measurable, and its volume is a rational number. If dx is such a measure on an ℓ-space Y, and f is a locally constant compactly supported function on Y with values in a field E of characteristic zero, then ∫_{Y} f(x)dx reduces to a finite sum, and it is well defined. On topological groups we consider only left- or right-invariant measures. An ℓ-group is a topological group with an ℓ-space structure.

Proposition 2.9. Let G be an ℓ-group. Then (1) there exists a fundamental system of neighborhoods of the identity in G consisting of open compact subgroups;

(2) there exists a left Haar measure on G such that the volume of each open compact set is a rational number.

Proof. (1) Let U be a neighborhood of the identity in G. We shall show that U contains an open compact subgroup. Since G is ℓ-space, we may assume that U is open and compact. Put V = {x ∈ G; xU ⊂ U}. Then V = ∩_{u∈U}Uu^{−1}, hence it is compact. Now for each v in V and u in U, by continuity of multiplication m there exists an open subset W_{u} containing v, and U_{u} in U containing u, such that m(W_{u}, U_{u}) ⊂ U. As U is compact and U = ∪_{u∈U}U_{u}, there are finitely many u_{1}, …, u_{n} in U with U = ∪_{1≤i≤n}U_{u_{i}}. Then W = ∩_{1≤i≤n}W_{u_{i}} is open in V and it contains v. Thus V is an open neighborhood of the identity, and V · V = V. Then V ∩ V^{−1} is an open compact subgroup in U.

(2) Fix some left Haar measure on G. Denote the volume of an open compact subgroup U by |U|. For two such groups, U_{1} and U_{2} we have

\frac{|U_{1}|}{|U_{2}|} = \frac{|U_{1}|}{|U_{1} ∩ U_{2}|}/\frac{|U_{2}|}{|U_{1} ∩ U_{2}|} = \frac{[U_{1} : U_{1} ∩ U_{2}]}{[U_{2} : U_{1} ∩ U_{2}]} ∈ ℚ.

Consequently the Haar measure on G can be chosen to assign rational volume to every open compact subgroup of G. But then the volume of every open compact subset K in G is rational, since as in (1) for such K there is a compact open subgroup U of G with KU ⊂ K, and then |K| = [K : U]|U| is rational, where K is a disjoint union of [K : U] translates of U.

Fix an ℓ-group G and a left Haar measure on G such that the volume of any open compact set is a rational number. Fix a field E of characteristic zero. The Evector space H_{G} of compactly supported locally constant functions f : G → E is an algebra under the convolultion (f_{1} ∗ f_{2})(g) = ∫_{G} f_{1}(h)f_{2}(h^{−1}g)dh. For an open compact subgroup U in G the set of U-biinvariant functions in H_{G} is a subalgebra H_{G}^{U}, called the Hecke algebra of (G, U). Although H_{G} has no unit (unless G is discrete, when the δ-function is in H_{G}), H_{G}^{U} does: it is δ_{U} : G → ℚ, the characteristic function of U divided by |U|. A representation π of the group G on a vector space V is called smooth if the stabilizer of any vector of V is open, and admissible if it is smooth and for any open subgroup U of G the space V^{U} of U-fixed vectors in V is finite dimensional. If π is a smooth representation of an ℓ-group G on a vector space V over E, for each f ∈ H_{G} define the operator π(f) : V → V by π(f)v = ∫_{G} f(g)π(g)vdg. This integral reduces to a finite sum since π is smooth, and π(f_{1} ∗ f_{2}) = π(f_{1}) ∘ π(f_{2}). Then V is naturally an H_{G}-module, and for any open compact subgroup U of G, the space V^{U} is a unital module over H_{G}^{U}.

Proposition 2.10. (1) A smooth G-module V ≠ {0} is irreducible iff for every open compact subgroup U of G either V^{U} = 0 or V^{U} is an irreducible H_{G}^{U}-module.

(2) Given an open compact subgroup U of G and an irreducible unital H_{G}^{U}- module M, there exists a smooth irreducible G-module V such that V^{U} is isomorphic to M as an H_{G}^{U}-module, and V is determined by this property up to isomorphism.

For a proof see [BZ76], 2.10. See [BZ76], 2.11 for Schur’s Lemma. Let π be an irreducible admissible representation of G in a vector space V over an algebraically closed field E. Then any nonzero G-module morphism (intertwining operator) V → V is a scalar.

Proposition 2.11. Let π be an irreducible admissible representation of G in a vector space V over an algebraically closed field E. For any field extension E′ of E, the representation of G in V ⊗_{E} E′ is also irreducible.

Proof. By Proposition 2.10, the statement reduces to a similar statement for finite dimensional algebras, since π is assumed to be admissible.

Let E be a subfield of ℂ invariant with respect to complex conjugation. A representation of G on a vector space V over E is unitary if there is a Ginvariant scalar product on V (thus a bilinear function (·, ·) : V × V → E with \overline{(v, w)} = (w, v) and (v, v) = 0 iff v = 0, and (gv, gw) = (v, w) for all v, w in V and g in G).

Note that we do not require V to be complete with respect to the scalar product, even in the case E = ℂ. If E is algebraically closed and the representation of G in E is irreducible and admissible, then the G-invariant inner product on V is unique up to a scalar multiple, if it exists.

Proposition 2.12. Let π be an admissible unitary representation of G in the E-space V. Fix a G-invariant scalar product on V. Let L be an invariant subspace of V, and L^{⊥} its orthogonal complement. Then V = L ⊕ L^{⊥}.

Proof. Given x ∈ V, we need to express it as x_{1} + x_{2} with x_{1} ∈ L and x_{2} ∈ L^{⊥}. Since π is smooth there exists a compact open subgroup U of G with x ∈ V^{U}. Since π is admissible, dim_{E} V^{U} is finite. Thus x = x_{1} + x_{2} for some x_{1} ∈ L^{U}, x_{2} ∈ V^{U}, x_{2} orthogonal to L^{U}. It remains to show that x_{2} is orthogonal to the entire space L. Let δ_{U} be the unit in H_{G}^{U}. Then π(δ_{U}) is the orthogonal projector V ↦ V^{U}. Hence for every y in L, (x_{2}, y) = (π(δ_{U})x_{2}, y) = (x_{2}, π(δ_{U})y) = 0 since π(δ_{U})y ∈ L^{U}.

It follows that every admissible unitary representation of G is a direct sum of irreducible representations. This sum is not necessarily finite. However, given an open compact subgroup U of G, only finitely many summands contain nonzero U-invariant vectors.

2.5 Automorphic forms

Let E be an algebraically closed field of characteristic zero. An automorphic form is a smooth function ϕ : GL(2, F)\ GL(2, 𝔸) → E, where by smooth we mean that there is an open subgroup U_{ϕ} of GL(2, 𝔸) such that ϕ(xu) = ϕ(x) for all u ∈ U_{ϕ} and x ∈ GL(2, 𝔸). A cusp form is an automorphic form ϕ with ∫_{𝔸/F} ϕ (( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x) dz = 0 for all x ∈ GL(2, 𝔸). Since ϕ is right locally constant (= smooth) and 𝔸/F is compact, the integral here is well defined and reduces to a finite sum. Let A_{0}^{E} be the space of cusp forms ϕ : GL(2, F)\ GL(2, 𝔸) → E. The group GL(2, 𝔸) acts on A_{0}^{E} by right translation: (r(h)ϕ)(g) = ϕ(gh). By a character of an ℓ-group G with values in E we mean a locally constant homomorphism χ : G → E^{×}. If E ⊂ ℂ such χ is called a unitary character if |χ(g)| = 1 for all g in G. Denote by A_{0}^{E}(χ) the space of ϕ ∈ A_{0}^{E} with ϕ(ax) = χ(a)ϕ(x), a ∈ 𝔸^{×} (identified with the center of GL(2, 𝔸)), x ∈ GL(2, F)\ GL(2, 𝔸). The space A_{0}^{E}(χ) is invariant under the GL(2, 𝔸)-action. Let π be an irreducible representation of GL(2, 𝔸) over E. By Schur’s lemma, there is a character χ : 𝔸^{×} → E^{×} such that for every a in 𝔸^{×}, π(a) is multiplication by χ(a). This χ is called the central character of π. If V ⊂ A_{0}^{E} is an irreducible admissible representation π of GL(2, 𝔸) and χ is the central character of V, then V ⊂ A_{0}^{E}(χ). Since the center of GL(2, F) acts trivially on A_{0}^{E}, χ is trivial on F^{×}. Thus every irreducible admissible π ⊂ A_{0}^{E} lies in A_{0}^{E}(χ), where χ is the central character of π, which is a character of 𝔸^{×}/F^{×}. The following is known also e.g. for GL(n).

Proposition 2.13. Fix an open subgroup U of GL(2, 𝔸). There exists a compact mod Z(𝔸) subset K of GL(2, F)\ GL(2, 𝔸) such that the support of any U-invariant cusp form is contained in K.

Proof. We first show that there is an integer n such that given z ∈ 𝔸 and x ∈ GL(2, 𝔸) with ht^{+}(x) ≥ n, there exist u ∈ U and β ∈ F with ( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x =

( \begin{matrix}1 & β \\ 0 & 1 \end{matrix} ) xu.

To see this, fix an effective divisor −D = ∑_{v∈|X|} n_{v}v on X, put d = (π π π_{v}^{n_{v}}) and let J_{D} = dO_{𝔸} be the corresponding ideal in O_{𝔸}. The groups Γ(D) = {γ ∈ GL(2, O_{𝔸}); γ ≡ I mod J_{D}} make a basis of neighborhoods of the identity in GL(2, 𝔸). Thus we may assume in this proof that U = Γ(D). In this case we shall show that n = 2g − 1 − deg(d). Indeed, fix z ∈ 𝔸 and x = ( \begin{matrix}a & c \\ 0 & b \end{matrix}) k with k ∈ GL(2, O_{𝔸}) and ht^{+}(x) = deg a − deg b ≥ 2g − 1 − deg(d) (note: deg(d) = − deg D = ∑_{v} n_{v} deg v). Then \frac{ad}{b}O_{𝔸} + F = 𝔸 and z = \frac{ad}{b}t + β for some β ∈ F and t ∈ O_{𝔸}. Put u = k^{−1} ( \begin{matrix}1 & td \\ 0 & 1 \end{matrix} ) k. Then u ∈ Γ(D) and

( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x = ( \begin{matrix}1 & β \\ 0 & 1 \end{matrix} ) xu.

We claim the proposition holds with K = {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) < n}. This K is compact modulo Z(𝔸). Let ϕ be a U-invariant cusp form, x ∈ GL(2, 𝔸), ht(x) ≥ n. We shall show that ϕ(x) = 0. Replacing x by γx for suitable γ ∈ GL(2, F), we assume that ht^{+}(x) ≥ n. By our choice of n, ϕ (( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x) = ϕ(x) for all z in 𝔸. Since ϕ is a cusp form, ϕ(x) = 0.

Corollary 2.14. The representation of GL(2, 𝔸) in A_{0}^{E}(χ) is admissible.

Proposition 2.15. Let E′ be an extension of E, and χ : 𝔸^{×}/F^{×} → E^{×} a character. Then A_{0}^{E′}(χ) = A_{0}^{E}(χ) ⊗_{E} E′.

Proof. The space A_{0}^{E}(χ) ⊗_{E} E′ consists of the functions ϕ in A_{0}^{E′}(χ) whose values span a finite dimensional space over E, since ϕ ∈ A_{0}^{E}(χ) takes finite number of values times the set Γ of values of χ. But every ϕ in A_{0}^{E′}(χ) has this property, since the set of its values lies in finitely many cosets of Γ.

Given a representation π of GL(2, 𝔸) over E and a character ω : 𝔸^{×} → E^{×}, write ωπ or πω or ω ⊗π or π ⊗ω for the representation (πω)(x) = ω(det x)π(x) in the space of π.

Proposition 2.16. For any characters χ, ω : 𝔸^{×}/F^{×} → E^{×}, we have A_{0}^{E}(χ) ⊗ ω = A_{0}^{E}(χω^{2}).

Proof. We need to construct an invertible linear map L : A_{0}^{E}(χ) → A_{0}^{E}(χω^{2}) such that for every ϕ ∈ A_{0}^{E}(χ) and h ∈ GL(2, 𝔸) we have r(h)L(ϕ) = ω(det h)L(r(h)ϕ), where (r(h)ϕ)(x) = ϕ(xh). Such L is (Lϕ)(x) = ϕ(x)ω(det x).

Proposition 2.17. Given a character χ : 𝔸^{×}/F^{×} → E^{×} there exists a character ω : 𝔸^{×}/F^{×} → E^{×} such that χ(x)ω(x)^{2} is a root of unity for every x in 𝔸^{×}/F^{×}.

Proof. Fix α ∈ 𝔸^{×}/F^{×} with deg α = 1. Such α exists since in the finite field extension F/𝔽_{q}(t), where t ∈ F is transcendental over 𝔽_{q}, there are always primes which split completely. Fix c in the algebraically closed field E with c^{2} = χ(α). Define ω : 𝔸^{×}/F^{×} → E^{×} by ω(x) = c^{− deg(x)}, put χ_{1}(x) = χ(x)ω^{2}(x), put α^{ℤ} = {α^{n}; n ∈ ℤ}. Then χ_{1} is a character of the profinite group 𝔸^{×}/F^{×} · α^{ℤ}, hence the values of χ_{1} are roots of 1.

Proposition 2.18. Let E be a subfield of ℂ invariant under complex conjugation, χ an E^{×}-valued unitary character of 𝔸^{×}/F^{×}. Then the representation of GL(2, 𝔸) in A_{0}^{E}(χ) is unitary.

Proof. The function x ↦ ϕ_{1}(x)\overline{ϕ}_{2}(x) on GL(2, F)\ GL(2, 𝔸), where ϕ_{1}, ϕ_{2} ∈ A_{0}^{E}(χ), is invariant under Z(𝔸) and is compactly supported as a function on PGL(2, F)\ PGL(2, 𝔸). Let dx be an invariant measure on PGL(2, F)\ PGL(2, 𝔸). It exists since PGL(2, F) is a discrete subgroup of PGL(2, 𝔸), a group with a two-sided invariant measure. Then

(ϕ_{1}, ϕ_{2}) = ∫ ϕ_{1}(x)\overline{ϕ}_{2}(x)dx (x ∈ PGL(2, F)\ PGL(2, 𝔸))

is an invariant scalar product on A_{0}^{E}(χ).

Corollary 2.19. The representation of GL(2, 𝔸) in A_{0}^{E}(χ) is a direct sum of irreducible subrepresentations.

Note that we may assume that all values of χ are roots of unity, and that E = \overline{ℚ}. The multiplicity one theorem asserts that in A_{0}^{E}(χ) any irreducible representation of GL(2, 𝔸) occurs with multiplicity one. An irreducible representation of GL(2, 𝔸) over an algebraically closed field E is called cuspidal if it is isomorphic to a subrepresentation of A_{0}^{E}.

2.6 Factorizability

Irreducible admissible representations of GL(2, 𝔸) are factorizable, as we proceed to show. Let E denote an algebraically closed subfield of ℂ. An irreducible representation of GL(2, F_{v}) in an E-space V is unramified if V contains a nonzero GL(2, O_{v})-invariant vector.

Proposition 2.20. The space of GL(2, O_{v})-invariant vectors V^{GL(2,O_{v})} in an unramified representation (π, V) of GL(2, F_{v}) is one dimensional.

Proof. Denote by H_{v} = C_{c}(GL(2, O_{v})\ GL(2, F_{v})/ GL(2, O_{v})) the Hecke convolution algebra of compactly supported GL(2, O_{v})-biinvariant E-valued functions on GL(2, F_{v}). We claim it is a commutative algebra. Indeed, for any f ∈ H_{v}, the function^{t}f(x) = f(^{t}x), where^{t}x is the transpose of x, is also in H_{v}. Since ^{t}(xy) = ^{t}y^{t}x, we have ^{t}(f_{1} ∗ f_{2}) = ^{t}f_{2} ∗^{t}f_{1} for all f_{1}, f_{2} ∈ H_{v}. By Cartan decomposition every GL(2, O_{v})-double coset in GL(2, F_{v}) contains a diagonal matrix. Hence ^{t}f = f for all f ∈ H_{v}, and f_{1} ∗f_{2} = ^{t}(f_{1} ∗f_{2}) = ^{t}f_{2} ∗^{t}f_{1} = f_{2} ∗f_{1} for all f_{1}, f_{2} ∈ H_{v}. If V is unramified, V^{GL(2,O_{v})} is a nonzero irreducible H_{v}- module. But H_{v} is commutative, so dim_{E} V^{GL(2,O_{v})} is 1.

Given an irreducible admissible representation π_{v} of GL(2, F_{v}) in a space V_{v} for every closed point v ∈ |X| such that π_{v} is unramified for all v ∈ S, S ⊂ |X| finite, construct a representation π = ⊗π_{v} of GL(2, 𝔸) as follows. For each v ∈ |X| − S choose a nonzero vector ξ_{v}^{0} ∈ V_{v}^{GL(2,O_{v})}. For any finite set S′ ⊃ S of closed points of X put V_{S′} = ⊗_{v∈S′}V_{v}. If S′′ ⊃ S′ ⊃ S, define an inclusion

V_{S′} ↪ V_{S′′} by x ↦ (⊗_{v∈S′′−S′}ξ_{v}^{0}) ⊗ x. Put V = lim_{S′⊃S^{→}} V_{S′}. It is the span of

the vectors ⊗_{v∈|X|}ξ_{v}, ξ_{v} = ξ_{v}^{0} for almost all v, and ξ_{v} ∈ V_{v} for all v ∈ |X|. Then V is a GL(2, 𝔸)-module in a natural way; denote by π the corresponding representation of GL(2, 𝔸). The vectors ξ_{v}^{0} are determined uniquely up to a scalar multiple, hence π is uniquely determined by the π_{v} for all v ∈ |X|. Reducing to irreducible finite dimensional representations of tensor products of algebras, we have

Proposition 2.21. Given an irreducible admissible representation π_{v} of GL(2, F_{v}) for every v in |X| which is unramified for almost all v, π = ⊗_{v}π_{v} is an irreducible admissible representation of GL(2, 𝔸). Every irreducible admissible representation π of GL(2, 𝔸) equals ⊗_{v}π_{v} for some irreducible admissible representations π_{v} of GL(2, F_{v}) which are almost all unramified. The representations π_{v} are determined by π uniquely up to isomorphism.

3 Looking for a trace formula

3.1 Trace formula in the compact case

Let X be an ℓ-space. Denote by C^{∞}(X) the space of E-valued locally constant (= smooth) functions on X. Here E is a fixed algebraically closed subfield of ℂ. Let C_{c}^{∞}(X) be the space of smooth compactly supported E-valued functions on X. Let r be an admissible representation of an ℓ-group G in an E-space V. Fix a Haar measure dx on G. Given f ∈ C_{c}^{∞}(G), define r(f) = ∫_{G} f(x)r(x)dx, an endomorphism of V. Since f is C^{∞}, that is smooth, it is right invariant under an open subgroup U of G. Then Im r(f) ⊂ V^{U}, so Im r(f) is finite dimensional, and the trace tr r(f) is well defined. Let r be now the representation of G on C^{∞}(Γ\G) by right translation, where Γ is a discrete cocompact subgroup of G. Since r is admissible, tr r(f) is defined.

Proposition 3.1. Let G be an ℓ-group. Let Γ be a discrete cocompact sugroup of G. Then G has a two sided invariant measure and Γ\G has a G-invariant measure.

Proof. Since (see [BZ76]) Γ\G admits a measure which when translated by x in G is multiplied by Δ(x), where Δ is the modulus of G, we have |Γ\G| = Δ(x)|Γ\G|, thus Δ = 1.

Proposition 3.2. Let X be an ℓ-space, dx a measure on X, K ∈ C_{c}^{∞}(X ×X). Define a linear endomorphism A of C^{∞}(X) by (Aϕ)(y) = ∫_{X} K(x, y)ϕ(x)dx. Then the image of A is finite dimensional and tr A = ∫_{X} K(x, x)dx.

Proof. We may assume that K(x, y) is of the form φ(x)ψ(y), as such functions span C_{c}^{∞}(X × X). In this case the claim is clear.

Proposition 3.3. Let G be an ℓ-group, Γ a discrete cocompact subgroup, r the representation of G in C^{∞}(Γ\G) by right translation, dx a Haar measure on G, f ∈ C_{c}^{∞}(G), S a set of representatives of the conjugacy classes in Γ, Z_{Γ}(γ) the centralizer of γ in Γ. Then tr r(f) = ∑_{γ∈S} ∫_{G/Z_{Γ}(γ)} f(xγx^{−1})dx.

Proof. We first show that for each γ ∈ Γ the function x ↦ f(xγx^{−1}) on G/Z_{Γ}(γ) is compactly supported, and that there are at most finitely many γ ∈ S for which x ↦ f(xγx^{−1}) is not identically zero. For this, fix a compact subset K in G with KΓ = G. Given x ∈ G there are k ∈ K, δ ∈ Γ, with x = kδ. Fix γ ∈ Γ. If f(xγx^{−1}) ≠ 0 then kδγδ^{−1}k^{−1} lies in suppf, thus δγδ^{−1} ∈ K_{f} = K·suppf · K. Since K_{f} is compact K_{f} ∩ Γ is finite, and there are only finite number of possibilities for δγδ^{−1}. Hence there are only a finite number of possibilities δ_{1}, …, δ_{n} for δ modulo Z_{Γ}(γ). Then f(xγx^{−1}) ≠ 0 implies that x ∈ K′Z_{Γ}(γ), where K′ = ∪_{1≤i≤n}Kδ_{i} is compact. If f(xγx^{−1}) ≠ 0, the conjugacy class of γ in Γ intersects the finite set K_{f} ∩ Γ. The number of such classes is finite. Thus the sum is finite and the integrals converge. Now given ϕ in C^{∞}(Γ\G), for any y in G we have

(r(f)ϕ)(y) = ∫_{G} f(x)ϕ(yx)dx = ∫_{G} f(y^{−1}x)ϕ(x)dx = ∫_{Γ\G} K_{f}(x, y)ϕ(x)dx

where K_{f}(x, y) = ∑_{γ∈Γ} f(y^{−1}γx). Then

tr r(f) = ∫_{Γ\G} K_{f}(x, x)dx = ∫_{Γ\G} ∑_{γ∈Γ} f(x^{−1}γx)dx

= ∫_{Γ\G} ∑_{γ∈S} ∑_{δ∈Z_{Γ}(γ)\Γ} f(x^{−1}δ^{−1}γδx)dx = ∑_{γ∈S} ∫_{Γ\G} ∑_{δ∈Z_{Γ}(γ)\Γ} f(x^{−1}δ^{−1}γδx)dx

= ∑_{γ∈S} ∫_{Z_{Γ}(γ)\G} f(x^{−1}γx)dx.

3.2 Case of GL(2), oversimplified

Let now A_{0}^{E} denote the space of E-valued cusp forms on GL(2, F)\ GL(2, 𝔸). The right-shifts representation of GL(2, 𝔸) on A_{0}^{E} is not admissible since the center Z(𝔸) of GL(2, 𝔸) is not compact. Fix a degree-one idèle α and put α^{ℤ} = {α^{n}; n ∈ ℤ}. It is a cyclic subgroup of 𝔸^{×}, and we view 𝔸^{×} as the center of GL(2, 𝔸). Denote by A_{0,α}^{E} the space of cusp forms in A_{0}^{E} invariant under α, and by r_{0} the representation of GL(2, 𝔸) on A_{0,α}^{E} by right translation. Since 𝔸^{×}/F^{×}α^{ℤ} is compact and every U-invariant cusp form – where U is an open subgroup of GL(2, 𝔸) – is supported on some compact modulo Z(𝔸) set K ⊂ GL(2, F)\ GL(2, 𝔸), the representation r_{0} is admissible. Hence tr r_{0}(f) is defined for every f ∈ C_{c}^{∞}(GL(2, 𝔸)). Put A_{c,α} = C_{c}^{∞}(α^{ℤ} · GL(2, F)\ GL(2, 𝔸)). Fix f ∈ C_{c}^{∞}(GL(2, 𝔸)). Let r be the right representation of GL(2, 𝔸) on A_{c,α}. We proceed to compute tr r(f) as if the space α^{ℤ} · GL(2, F)\ GL(2, 𝔸) were compact, to see what needs to be corrected. This space is not compact and r is not admissible, so that in fact tr r(f) makes no sense. For any ring R define A(R) = {diag(a, b); a, b ∈ R^{×}}, A′(R) = {diag(a, b); a, b ∈ R^{×}, a ≠ b\}, N(R) = \{( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) ; a ∈ R\}. Let Q be the set of quadratic extensions of the field F. For each L ∈ Q choose an embedding L ↪ M(2, F); it exists and is unique up to an automorphism of M(2, F); all automorphisms of M(2, F) are inner. Given γ ∈ α^{ℤ} · GL(2, F), denote by Z(γ) the centralizer of γ in α^{ℤ} GL(2, F).

Proposition 3.4. Every conjugacy class of α^{ℤ} · GL(2, F) intersects precisely

one of : F^{×} · α^{ℤ}; a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ), a ∈ F^{×} · α^{ℤ}; α^{ℤ} · A′(F); α^{ℤ} · (L^{×} − F^{×}) for some

L ∈ Q. In the first two cases the number of intersection points is 1, in the 3rd case 2, in the 4th case: the number of automorphisms of L over F. The centralizers Z(γ) are α^{ℤ} ·GL(2, F), α^{ℤ}F^{×}N(F), α^{ℤ} ·A(F), α^{ℤ}L^{×}, respectively.

Immitating the trace formula in the compact case, one may expect

tr r(f) = S_{1}(f) + ∑_{L∈Q} S_{2,L}(f) + S_{3}(f) + S_{4}(f)

with

S_{1}(f) = |α^{ℤ} · GL(2, F)\ GL(2, 𝔸)|,

S_{2,L}(f) = | Aut_{F} (L)|^{−1} ∑_{γ∈α^{ℤ}·(L^{×}−F^{×})} ∫_{α^{ℤ}·L^{×}\ GL(2,𝔸)} f(x^{−1}γx)dx,

S_{3}(f) = \frac{1}{2} ∑_{γ∈α^{ℤ}A′(F)} ∫_{α^{ℤ}A(F)\ GL(2,𝔸)} f(x^{−1}γx)dx,

S_{4}(f) = ∑_{a∈α^{ℤ}·F^{×}} ∫_{α^{ℤ}F^{×}N(F)\ GL(2,𝔸)} f(x^{−1}a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x)dx.

The left side of this wrong trace formula is divergent. So is S_{3}(f), since the homogeneous space A(𝔸)/α^{ℤ} · A(F) is not compact. We shall show that S_{1}(f) and ∑_{L∈Q} S_{2,L}(f) converge, and although S_{4}(f) diverges, we shall show in which way it does.

Proposition 3.5. Given f ∈ C_{c}^{∞}(GL(2, 𝔸)), the number of conjugacy classes of γ ∈ α^{ℤ} · GL(2, F) with x ∈ GL(2, 𝔸) and f(xγx^{−1}) ≠ 0 is finite.

Proof. The sets K_{1} = {tr h; h ∈ suppf} ⊂ 𝔸, K_{2} = {det h; h ∈ suppf} ⊂ 𝔸^{×} are compact. It suffices to show that the set {γ ∈ α^{ℤ} · GL(2, F); tr γ ∈ K_{1}, det γ ∈ K_{2}} is a union of finitely many conjugacy classes. Put γ = α^{n}x for some x ∈ GL(2, F). Then 2n = deg γ, so n lies in a finite set. Fix n. Then tr x ∈ α^{−n}K_{1}, det x ∈ α^{−2n}K_{2}. But the sets F ∩ α^{−n}K_{1} and F^{×} ∩ α^{−2n}K_{2} are finite. Hence the trace and determinant of x can take only finitely many values. As the number of conjugacy classes of elements in GL(2, F) with fixed trace and determinant is at most two, we are done.

3.3 Central elements

Proposition 3.6. The volume | GL(2, F) · α^{ℤ}\ GL(2, 𝔸)| is finite.

Proof. This volume is equal to (below x ∈ α^{ℤ} GL(2, F)\ GL(2, 𝔸)/ GL(2, O_{𝔸}))

∑_{x} |α^{ℤ} GL(2, F) ∩ x GL(2, O_{𝔸})x^{−1}\x GL(2, O_{𝔸})|
= | GL(2, O_{𝔸})| ∑_{x} |α^{ℤ} GL(2, F) ∩ x GL(2, O_{𝔸})x^{−1}|^{−1}.

For x in GL(2, 𝔸)/ GL(2, O_{𝔸}), let E = xO_{𝔸}^{2} be the associated rank 2 locally free sheaf on X. Then Aut(E) consists of the g ∈ GL(2, 𝔸) which map (E =)xO_{𝔸}^{2} to xO_{𝔸}^{2} and the generic fiber F^{2} to itself, thus Aut E is

GL(2, F) ∩ x GL(2, O_{𝔸})x^{−1} = α^{ℤ} GL(2, F) ∩ x GL(2, O_{𝔸})x^{−1}.

We then need to show the convergence of

∑_{E∈Bun_{2} /J} | Aut E|^{−1},

J being the image of α^{ℤ} under the natural homomorphism 𝔸^{×} → Pic X. The number of J-orbits on the set of stable rank two locally free sheaves on X is finite, so it remains to show that the sum of | Aut E|^{−1} over the set Bun_{2}^{un} of J-orbits of unstable rank two locally free sheaves on X is convergent.

Lemma 3.7. (1) A rank two locally free sheaf E on X is very unstable (ht(E) ≥ 2g −1) iff E ≃ L⊕M where L, M are invertible sheaves with deg L−deg M ≥ 2g − 1.

(2) If L, M ∈ Pic X and deg L − deg M ≥ max(2g − 1, 1) then

| Aut(L ⊕ M)| = (q − 1)^{2}q^{deg L−deg M+1−g}.

(3) If L ⊕ M ≃ L′ ⊕ M′ with deg L > deg M, deg L′ > deg M′ then L ≃ L′, M ≃ M′.

Proof. (1) If L is an invertible sheaf of E of maximal degree and M = E/L, then M is invertible, and Ext(M, L) = H^{1}(X, M^{−1}L) is 0 (by Serre duality) as

deg M^{−1}L = deg L − deg M = 2 deg L − deg E = ht(E) ≥ 2g − 1.

The exact sequence

0 → Hom(M, L) → Aut(L ⊕ M) → Aut L × Aut M → 0

implies (2) since Hom(M, L) = H^{0}(X, M^{−1}L) and H^{1}(X, M^{−1}L) = {0}, so the Riemann-Roch theorem implies that dim H^{0}(X, M^{−1}L) = deg(M^{−1}L) + 1−g. If the invertible sheaf L corrsponds to aO_{𝔸}, then Aut L consists of g ∈ 𝔸^{×} which map the generic fiber F onto itself (thus g ∈ F^{×}) and map aO_{𝔸} onto itself (thus g ∈ O_{𝔸}^{×}). Then Aut L = F^{×} ∩ O_{𝔸}^{×} = 𝔽_{q}^{×} has cardinality q − 1. For (3), put E = L ⊕ M →^{∼} L′ ⊕ M′. Since deg L > (deg E)/2 > deg M′, we have Hom(L, M′) = {0}. Hence the image of L under the isomorphism

L ⊕ M →^{∼} L′ ⊕ M′ lies in L′. Hence L ≃ L′ and M ≃ E/L ≃ E/L′ ≃ M′.

Assume g ≥ 1, so that 2g − 1 ≥ 1 (the case g = 0 is similar). The lemma implies

∑_{E∈Bun_{2}^{un} /J} | Aut E|^{−1} = (q − 1)^{−2}| Pic^{0}(X)| ∑_{n≥2g−1} q^{g−1−n} < ∞.

Corollary 3.8. If the Haar measure on GL(2, 𝔸) is normalized so that | GL(2, O_{𝔸})| is a rational number, then |α^{ℤ} · GL(2, F)\ GL(2, 𝔸)| ∈ ℚ.

This follows from the proof of the last proposition.

3.4 Elliptic elements

Proposition 3.9. Let L be a quadratic extension of F, γ ∈ α^{ℤ} · (L^{×} − F^{×}) ⊂ GL(2, 𝔸), and f ∈ C_{c}^{∞}(GL(2, 𝔸)). Then the function x ↦ f(xγx^{−1}) on GL(2, 𝔸)/α^{ℤ} · L^{×} has compact support.

Proof. We need to show that the map x ↦ xγx^{−1} on GL(2, 𝔸)/α^{ℤ}·L^{×} is proper (the preimage of a compact is compact). Since (L ⊗_{F} 𝔸)^{×}/α^{ℤ} · L^{×} is compact, it suffices to show that the map ψ(x) = xγx^{−1}, ψ : GL(2, 𝔸)/𝔸_{L}^{×} → GL(2, 𝔸), is proper (𝔸_{L} = L ⊗_{F} 𝔸 is the ring of adèles of L).

Lemma 3.10. Let F be a local field in this lemma. Suppose γ ∈ M(2, F) is regular, i.e. the subalgebra E = F [γ] generated by γ is a field or is F × F. Then the map ψ : GL(2, F)/E^{×} → GL(2, F), x ↦ xγx^{−1}, is proper. Moreover, if γ ∈ GL(2, O) and the ring O[γ] is integrally closed, then ψ^{−1}(GL(2, O)) = GL(2, O)/E^{×} ∩ GL(2, O).

Proof. The conjugacy class C of γ is a closed subset of GL(2, F), since γ is regular. So it suffices to show that ψ maps GL(2, F)/E^{×} homeomorphically onto C. It is clear that ψ is continuous, injective and Im ψ = C. It remains to show that the map ψ′ : GL(2, F) → C, x ↦ xγx^{−1}, is open. For this, it suffices to show that C is the set of F -points of a smooth variety C over F, and that ψ′ is smooth, that is its differential is everywhere onto. Since C is a homogeneous space under a connected group G is suffices to show that the tangent map dψ′ of ψ′ at the identity is onto. When verifying these properties of C and ψ′, we may replace F with an extension, thus we may assume that γ is of the form diag(a, b) with a ≠ b, or ( \begin{matrix}a & 1 \\ 0 & a \end{matrix}) (if E is nonseparable over F). To compute the tangent map dψ′ : Lie G → T_{γ}(C) of ψ′(x) = xγx^{−1} near the identity x = 1, let Y be in Lie G, and put x = 1 + ϵY, where ϵ^{2} = 0. Then x^{−1} = 1 − ϵY and ψ′(x) = (1 + ϵY)γ(1 − ϵY) = 1 + ϵ(Y γ − γY), so dψ′(Y) = Y γ − γY is onto the tangent space T_{γ}(C) of C at γ, and ψ is proper. If x ∈ GL(2, F) and xγx^{−1} ∈ GL(2, O), put M = x^{−1}O^{2}. Then γM ⊂ M. In addition, γ ∈ GL(2, O), so γO^{2} ⊂ O^{2}. Thus M and O^{2} are O[γ]-submodules in F^{2}. Both modules are of finite type. As F^{2} is a rank one free E = F [γ]-module, and we assume that O[γ] is integrally closed, namely it is the ring of integers in E = F [γ], both M and O^{2} are rank one torsion free over the discrete valuation ring O[γ] (being rank two over O). Hence there exists a ∈ E^{×} with M = aO^{2}. Thus xaO^{2} = O^{2}, that is xa ∈ GL(2, O).

Now for γ as in the proposition, for almost all closed points in X the component of α at v is 1, γ ∈ GL(2, O_{v}), and the ring O_{v}[γ] is integrally closed. This and the lemma imply the proposition. □

3.5 Regularization of the unipotent terms

To study the integral which occurs in S_{4}(f), we regularize it as

θ_{a,f}(t) = ∫_{α^{ℤ}·F^{×}N(F)\ GL(2,F)} f(ax^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x)t^{ht^{+}(x)}dx.

Proposition 3.11. (1) For every f ∈ C_{c}^{∞}(GL(2, 𝔸)) and a ∈ 𝔸^{×}, the integral θ_{a,f}(t) converges as an element of ℂ((t)), and ζ_{F} (q^{−1}t)^{−1}θ_{a,f}(t) ∈ ℂ[t, t^{−1}], where ζ_{F} (t) = ∏_{v∈|X|}(1 − t_{v})^{−1}, t_{v} = t^{deg v}.

(2) If f is the characteristic function of GL(2, O_{𝔸}) in GL(2, 𝔸), then

θ_{1,f}(t) = | GL(2, O_{𝔸})| · (q − 1)^{−1}q^{g−1} · | Pic^{0}(X)|ζ_{F} (q^{−1}t).

Proof. (1) It suffices to consider f(x) = ∏_{v} f_{v}(x_{v}), x = (x_{v}) ∈ GL(2, 𝔸), where f_{v} ∈ C_{c}^{∞}(GL(2, F_{v})) for all v ∈ |X| and f_{v} is the characteristic function f_{v}^{0} of GL(2, O_{v}) at almost all v, since such functions span C_{c}^{∞}(GL(2, 𝔸)). Normalize the measures on F_{v}^{×} and F_{v} so that |O_{v}^{×}| = 1 = |O_{v}|. Denote by val_{v}(x_{v}) the valuation of x_{v} ∈ F_{v}^{×}, normalized by val_{v}(π π π_{v}) = 1. Define a function

h_{v}^{+} : GL(2, F_{v}) → ℤ by h_{v}^{+}(( \begin{matrix}a & b \\ 0 & c \end{matrix}) k) = val_{v}(a) − val_{v}(c), k ∈ GL(2, O_{v}).

Then h_{v}^{+} is well-defined and ht^{+}(x) = ∑_{v∈|X|} h_{v}^{+}(x_{v}) deg(v). We have

θ_{a,f}(t) = |𝔸^{×}/α^{ℤ} · F^{×}| · |𝔸/F | ∏_{v} θ_{a_{v},f_{v}}(t_{v})

where

θ_{a_{v},f_{v}}(t_{v}) = ∫_{F_{v}^{×}N(F_{v})\ GL(2,F_{v})} f_{v}(a_{v}x^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x)t^{h_{v}^{+}(x) deg v}dx

and t_{v} = t^{deg(v)}. To compute it, note that p_{n,v} = diag(π π π_{v}^{n}, 1) (n ∈ ℤ) make a set of representatives of the two sided coset space

F_{v}^{×}N(F_{v})\ GL(2, F_{v})/ GL(2, O_{v}).

Then

θ_{a_{v},f_{v}}(t_{v}) = ∑_{n∈ℤ} t_{v}^{n} ∫_{F_{v}^{×}N(F_{v})∩p_{n,v}^{−1} GL(2,O_{v})p_{n,v}\p_{n,v}^{−1} GL(2,O_{v})} f_{v}(a_{v}x^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x)dx

= ∑_{n∈ℤ} t_{v}^{n}|F_{v}^{×}N(F_{v}) ∩ p_{n,v}^{−1} GL(2, O_{v})p_{n,v}|^{−1} ∫_{p_{n,v}^{−1} GL(2,O_{v})} f_{v}(a_{v}x^{−1} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) x)dx

= ∑_{n∈ℤ} q_{v}^{−n}t_{v}^{n} ∫_{GL(2,O_{v})} f_{v}(a_{v}yp_{n,v} ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) p_{n,v}^{−1}y^{−1})dy = ∑_{n∈ℤ} τ_{n}(f_{v})q_{v}^{−n}t_{v}^{n},

where τ_{n}(f_{v}) = ∫_{GL(2,O_{v})} f_{v}(a_{v}y ( \begin{matrix}1 & π π π_{v}^{n} \\ 0 & 1 \end{matrix} ) y^{−1})dy is 0 if n << 0 and τ_{n}(f_{v}) =

f_{v}(a_{v}) for n >> 0. If a_{v} ∈ O_{v}^{×} and f_{v} is the characteristic function of GL(2, O_{v}), then τ_{n}(f_{v}) = | GL(2, O_{v})| for n ≥ 0 and u_{n,v} = 0 for n < 0, so

θ_{a_{v},f_{v}}(t_{v}) = | GL(2, O_{v})|(1 − t_{v}/q_{v})^{−1}.

(2) It remains to compute (note that |O_{𝔸}^{×}| = 1 and |O_{𝔸}| = 1) :

|𝔸^{×}N(𝔸)/α^{ℤ}F^{×}N(F)| = (|𝔸^{×}/α^{ℤ}F^{×}|/|O_{𝔸}^{×}|)(|𝔸/F |/|O_{𝔸}|).

The exact sequence 1 → 𝔽_{q}^{×} → O_{𝔸}^{×} → 𝔸^{×}/α^{ℤ}F^{×} → Pic X/α^{ℤ}(= Pic^{0}(X)) → 1 implies that the first factor on the right is | Pic^{0}(X)|/(q−1). The exact sequence

0 → 𝔽_{q} → O_{𝔸} → 𝔸/F → H^{1}(X, O_{X}) → 0

implies that the second factor on the right is q^{g−1}.

4 Intertwining operators and Eisenstein series

4.1 Intertwining operators

Let E be an algebraically closed field of characteristic zero, and v ∈ |X| a closed point of X. Denote by |a|_{v} the absolute value of a ∈ F_{v}^{×} normalized by |π π π_{v}| = q_{v}^{−1}. It is an E^{×}-valued character of F_{v}^{×}. Fix a square root \sqrt{q} = q^{1/2} of q in E. If E ⊂ ℂ we choose q^{1/2} > 0. For E-valued characters μ_{1}, μ_{2} of F_{v}^{×} denote by I(μ_{1}, μ_{2}) both the space of right locally constant func-

tions ϕ : GL(2, F_{v}) → E with ϕ(( \begin{matrix}a_{1} & b \\ 0 & a_{2} \end{matrix} ) x) = |a_{1}/a_{2}|_{v}^{1/2}μ_{1}(a_{1})μ_{2}(a_{2})ϕ(x) (x ∈

GL(2, F_{v}); a_{1}, a_{2} ∈ F_{v}^{×}; b ∈ F_{v}), and the action of the group GL(2, F_{v}) by right translation on I(μ_{1}, μ_{2}). The induced representation I(μ_{1}, μ_{2}) is admissible by the Iwasawa decomposition G = BK. It is unitarizable when μ_{1}, μ_{2} are unitary. It is possible to work with I(| · |_{v}^{1/2}μ_{1}, | · |_{v}^{1/2}μ_{2}), in whose definition the factor |a_{1}/a_{2}|_{v}^{1/2}μ_{1}(a_{1})μ_{2}(a_{2}) becomes |a_{1}|_{v}μ_{1}(a_{1})μ_{2}(a_{2}), but later we shall need to multiply back by | · |_{v}^{−1/2}. The following is a standard basic result.

Proposition 4.1. If μ_{1}/μ_{2} ≠ |·|_{v}, |·|_{v}^{−1}, then the representations of GL(2, F_{v}) in I(μ_{1}, μ_{2}) and I(μ_{2}, μ_{1}) are irreducible and isomorphic. If μ_{1}/μ_{2} = | · |_{v} or | · |_{v}^{−1} then I(μ_{1}, μ_{2}) contains a unique proper invariant subspace I′(μ_{1}, μ_{2}) and there is a GL(2, F_{v})-isomorphism I′(μ_{1}, μ_{2}) ≃ I(μ_{2}, μ_{1})/I′(μ_{2}, μ_{1}). If μ_{2}/μ_{1} = | · |_{v}, the subspace I′(μ_{1}| · |_{v}^{−1/2}, μ_{1}| · |_{v}^{1/2}) is one dimensional; x ∈ GL(2, F_{v}) acts on I′(μ_{1}| · |_{v}^{−1/2}, μ_{1}| · |_{v}^{1/2}) via multiplication by μ_{1}(x). The subspace

I′(μ_{2}| · |_{v}^{1/2}, μ_{2}| · |_{v}^{−1/2}) is denoted by St(μ_{2}) = St(μ_{2}| · |_{v}^{1/2}, μ_{2}| · |_{v}^{−1/2}).

It is isomorphic to I(μ_{2}| · |_{v}^{−1/2}, μ_{2}| · |_{v}^{1/2})/I′(μ_{2}| · |_{v}^{−1/2}, μ_{2}| · |_{v}^{1/2}). It consists of

ϕ ∈ I(μ_{2}| · |_{v}^{1/2}, μ_{2}| · |_{v}^{−1/2}) with ∫_{GL(2,O_{v})} μ_{2}(det x)^{−1}ϕ(x)dx = 0.

If I(μ_{1}, μ_{2}) ≃ I(μ′_{1}, μ′_{2}) then {μ_{1}, μ_{2}} = {μ′_{1}, μ′_{2}}, the representations I(μ_{1}, μ_{2}) (μ_{1}/μ_{2} ≠ | · |_{v} or | · |_{v}^{−1}) and St(μ′_{2}) are infinite dimensional and inequivalent, and St(μ_{1}) ≃ St(μ_{2}) implies μ_{1} = μ_{2}.

We proceed to describe the operator intertwining I(μ_{1}, μ_{2}) and I(μ_{2}, μ_{1}).

Proposition 4.2. If |μ_{1}(π π π_{v})/μ_{2}(π π π_{v})| < 1 the integral

(Mϕ)(x) = ∫_{F_{v}} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) x)dy

converges for each ϕ ∈ I(μ_{1}, μ_{2}) and x ∈ GL(2, F_{v}), and Mϕ ∈ I(μ_{2}, μ_{1}).

Proof. As ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}y^{−1} & −1 \\ 0 & y \end{matrix} ) ( \begin{matrix}1 & 0 \\ y^{−1} & 1 \end{matrix} ), the integrand is

μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}^{−1}ϕ (( \begin{matrix}1 & 0 \\ y^{−1} & 1 \end{matrix} ) x) ,

which is 0 if |y|_{v} is small, and μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}^{−1}ϕ(x) if |y|_{v} is big enough. For sufficiently large n then the part of the integral over |y|_{v} ≥ q_{v}^{n} is bounded by ϕ(x) times

∫_{|y|_{v}≥q_{v}^{n}} |μ_{2}(y)/μ_{1}(y)| · |y|_{v}^{−1}dy = |O_{v}^{×}| ∑_{k≥n} |μ_{1}(π π π_{v})/μ_{2}(π π π_{v})|^{k} < ∞.

It is clear that (Mϕ)(( \begin{matrix}1 & c \\ 0 & 1 \end{matrix} ) x) = (Mϕ)(x) (c ∈ F_{v}) and (Mϕ)(( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) x) equals

∫_{F_{v}} ϕ(( \begin{matrix}b & 0 \\ 0 & a \end{matrix}) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & yb/a \\ 0 & 1 \end{matrix} ) x)dy = μ_{1}(b)μ_{2}(a) |\frac{b}{a}|_{v}^{1/2} |\frac{a}{b}|_{v} (Mϕ)(x).

We obtained, if |μ_{1}(π π π_{v})/μ_{2}(π π π_{v})| < 1, a GL(2, F_{v})-equivariant map

M = M(μ_{1}, μ_{2}) : I(μ_{1}, μ_{2}) → I(μ_{2}, μ_{1}).

Let ν_{t} be the unramified character of F_{v}^{×} with ν_{t}(π π π_{v}) = t. Put M(μ_{1}, μ_{2}, t) = M(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}). It converges for any μ_{1}, μ_{2}, provided t ∈ ℂ is small enough in absolute value. To define M(μ_{1}, μ_{2}) as the value at t = 1 of the analytic continuation of M(μ_{1}, μ_{2}, t), we need these operators to be defined on the same space, which we will take to be

I_{0}(μ_{1}, μ_{2}) = \{ϕ ∈ C^{∞}(GL(2, O_{v})); ϕ(( \begin{matrix}a_{1} & b \\ 0 & a_{2} \end{matrix} ) x) = μ_{1}(a_{1})μ_{2}(a_{2})ϕ(x),
a_{1}, a_{2} ∈ O_{v}^{×}, b ∈ O_{v}, x ∈ GL(2, O_{v})}.

By the Iwasawa decomposition G = BK, the restriction map I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) → I_{0}(μ_{1}, μ_{2}) is bijective for any t. Identifying these spaces, the operator M(μ_{1}, μ_{2}, t) becomes a map I_{0}(μ_{1}, μ_{2}) → I_{0}(μ_{2}, μ_{1}). Write L(μ, t) for (1 − μ(π π π_{v})t)^{−1} if μ is unramified, and L(μ, t) = 1 if μ is a ramified character of F_{v}^{×}.

Proposition 4.3. The operator valued function M(μ_{1}, μ_{2}, t) is rational in t ∈ ℂ^{×}. In fact the function t ↦ L(μ_{1}/μ_{2}, t^{2})^{−1}(M(μ_{1}, μ_{2}, t)ϕ)(x) is a polynomial in t for all ϕ ∈ I_{0}(μ_{1}, μ_{2}), x ∈ GL(2, O_{v}). If μ_{1}, μ_{2} are unramified and the restrictions of ϕ ∈ I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) and ψ ∈ I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}) to GL(2, O_{v}) are 1, then M(μ_{1}, μ_{2}, t)ϕ = \frac{L(μ_{1}/μ_{2},t^{2})}{L(μ_{1}/μ_{2},q_{v}^{−1}t^{2})}ψ.

Proof. Put ϕ_{t} = M(μ_{1}, μ_{2}, t)ϕ and a_{1} = ∫_{|y|_{v}≤1} ϕ(( \begin{matrix}0 & −1 \\ 1 & y \end{matrix} ) x)dy where x ∈

GL(2, O_{v}). Then

ϕ_{t}(x) = a_{1} + ∫_{|y|_{v}>1} μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}^{−1}ν_{t}(y)^{−2}ϕ (( \begin{matrix}1 & 0 \\ y^{−1} & 1 \end{matrix} ) x) dy.

We shall show that this is the Taylor series of a rational function. If n is large enough, ϕ (( \begin{matrix}1 & 0 \\ y^{−1} & 1 \end{matrix} ) x) = ϕ(x) for |y|_{v} ≥ q_{v}^{n}. Then ϕ_{t}(x) = a_{1} + a_{2}(t) + a_{3}(t) with

a_{2}(t) = ∫_{1<|y|_{v}<q_{v}^{n}} μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}^{−1}ν_{t}(y)^{−2}ϕ (( \begin{matrix}1 & 0 \\ y^{−1} & 1 \end{matrix} ) x) dy,
a_{3}(t) = ϕ(x) ∫_{|y|_{v}≥q_{v}^{n}} μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}ν_{t}(y)^{−2}dy.

Clearly a_{2}(t) is a polynomial in t (since ν_{t}(π π π_{v}^{−1})^{−1} = t) and a_{3}(t) = ct^{2n}L(μ_{1}/μ_{2}, t^{2}). If μ_{1}, μ_{2} are unramified and x ∈ GL(2, O_{v}), a_{1} = 1 and the expression for ϕ_{t}(x) is

ϕ_{t}(x) = 1 + ∫_{|y|_{v}>1} μ_{2}(y)μ_{1}(y)^{−1}|y|_{v}^{−1}ν_{t}(y)^{−2}dy
= 1 − (1 − q_{v}^{−1}) ∑_{k≥1}(μ_{1}(π π π_{v})/μ_{2}(π π π_{v}))^{k}t^{2k}

= 1 + \frac{(1 − q_{v}^{−1})(μ_{1}(π π π_{v})/μ_{2}(π π π_{v}))t^{2}}{1 − (μ_{1}(π π π_{v})/μ_{2}(π π π_{v}))t^{2}} = \frac{L(μ_{1}/μ_{2}, t^{2})}{L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2})}.

The operator M(μ_{1}, μ_{2}, t) : I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) → I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}) intertwines the GL(2, F_{v})-modules for every t where it is defined. It can be regarded as a rational function of t (in fact, of t^{2}) with values in the set of operators I_{0}(μ_{1}, μ_{2}) → I_{0}(μ_{2}, μ_{1}). Indeed,

M(μ_{1}, μ_{2}, t) = M(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) = M(μ_{1}ν_{t^{2}}, μ_{2}).

Define

R(μ_{1}, μ_{2}, t) = \frac{L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2})}{L(μ_{1}/μ_{2}, t^{2})}M(μ_{1}, μ_{2}, t).

Corollary 4.4. Suppose μ_{1} and μ_{2} are unramified and φ ∈ I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}), ψ ∈ I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}) are the functions whose restrictions to GL(2, O_{v}) are one, then R(μ_{1}, μ_{2}, t)φ = ψ. □

Given characters μ_{1}, μ_{2} of 𝔸^{×}, write I(μ_{1}, μ_{2}) for the space of right locally constant functions ϕ on GL(2, 𝔸) which satisfy

ϕ (( \begin{matrix}a_{1} & b \\ 0 & a_{2} \end{matrix} ) x) = μ_{1}(a_{1})μ_{2}(a_{2})|a_{1}/a_{2}|^{1/2}ϕ(x). Put ν(a) = q^{deg(a)}.

Then I(μ_{1}, μ_{2}) is the restricted tensor product of the spaces I(μ_{1v}, μ_{2v}) where μ_{iv} is the component of μ_{i} at v (the restriction of μ_{i} to F_{v}^{×} ↪ 𝔸^{×}); it is spanned by ⊗_{v}ϕ_{v} with ϕ_{v} ∈ I(μ_{1v}, μ_{2v}) for all v and ϕ_{v}| GL(2, O_{v}) = 1 for almost all v, where μ_{iv}|O_{v}^{×} = 1, i.e. μ_{iv} are unramified. Define the character ν_{t} of 𝔸^{×} by ν_{t}(a) = t^{deg(a)}. Then the restriction of ν_{t} to F_{v}^{×} is ν_{t_{v}}, the unramified character of F_{v}^{×} with ν_{t_{v}}(π π π_{v}) = t_{v}(= t^{deg(v)}). As in the local case, we identify the spaces I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) with I_{0}(μ_{1}, μ_{2}) for all t. The operator R(μ_{1}, μ_{2}, t) from I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) to I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}) defined by R(μ_{1}, μ_{2}, t) = ⊗_{v}R(μ_{1v}, μ_{2v}, t_{v}) is rational in t. On any element in I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) at most finitely many components R(μ_{1v}, μ_{2v}, t_{v}) do not act as the identity. Also write m(μ, t) for L(μ, t)/L(μ, t/q).

4.2 Eisenstein series

Write A_{α} = C^{∞}(α^{ℤ} · GL(2, F)\ GL(2, 𝔸)),

A_{c,α} = C_{c}^{∞}(α^{ℤ} · GL(2, F)\ GL(2, 𝔸)), Y = A(F)N(𝔸)\ GL(2, 𝔸)

and Y_{α} = Y/α^{ℤ}. Normalize the Haar measure on N(𝔸) ≃ 𝔸 by |N(𝔸)/N(F)| = |𝔸/F | = 1. The Haar measure on N(𝔸) is invariant with respect to conjugation by the elements of A(F) by the product formula. So it extends to a two-sided invariant measure on the space α^{ℤ} · A(F)N(𝔸). This, and the two-sided Haar measure on GL(2, 𝔸) induce an invariant measure on Y_{α}. Let φ and ψ be locally constant functions on Y_{α}, at least one of which is compactly supported. Put (φ, ψ) = ∫_{Y_{α}} φ(x)\overline{ψ}(x)dx. On α^{ℤ} · GL(2, F)\ GL(2, 𝔸) a scalar product is similarly defined. Define the map E^{∗} : A_{α} → C^{∞}(Y_{α}) by

ϕ ↦ ϕ_{N}, ϕ_{N}(x) = ∫_{N(F)\N(𝔸)} ϕ(nx)dn, x ∈ GL(2, 𝔸).

Note that N(F)\N(𝔸) is compact, so the integral converges. Note that ker E^{∗} is the space A_{0,α} of cusp forms invariant under α. For any f ∈ C_{c}^{∞}(Y_{α}) define a function Ef on α^{ℤ} · GL(2, F)\ GL(2, 𝔸) by

(Ef)(x) = ∑_{γ∈A(F)N(F)\ GL(2,F)} f(γx), x ∈ GL(2, 𝔸).

Proposition 4.5. The sum defining (Ef)(x) converges. For f ∈ C_{c}^{∞}(Y_{α}) and ϕ ∈ A_{α} we have (Ef, ϕ) = (f, E^{∗}ϕ).

Proof. Consider the diagram

Y_{α} ←^{r} α^{ℤ} · A(F)N(F)\ GL(2, 𝔸) →^{s} α^{ℤ} · GL(2, F)\ GL(2, 𝔸).

Since N(F)\N(𝔸) is compact, the map r is proper. Hence the natural embedding r^{∗} maps C_{c}^{∞}(Y_{α}) to C_{c}^{∞}(α^{ℤ} · A(F)N(F)\ GL(2, 𝔸)). Given

ψ ∈ C_{c}^{∞}(α^{ℤ}A(F)N(F)\ GL(2, 𝔸)),

define a function s_{∗}ψ on α^{ℤ} GL(2, F)\ GL(2, 𝔸) by

(s_{∗}ψ)(x) = ∑_{γ∈A(F)N(F)\ GL(2,F)} ψ(γx), x ∈ GL(2, 𝔸).

The sum is finite since ψ is compactly supported, and

s_{∗}ψ ∈ C_{c}^{∞}(α^{ℤ} GL(2, F)\ GL(2, 𝔸)).

The sum which defines (Ef)(x) converges since E = s_{∗}r^{∗}. Now define E^{∗} = r_{∗}s^{∗}, where s^{∗} is the natural embedding, and

r_{∗} : C^{∞}(α^{ℤ}A(F)N(F)\ GL(2, 𝔸)) → C^{∞}(Y_{α})

is defined by (r_{∗}h)(x) = ∫_{N(F)\N(𝔸)} h(nx)dn, x ∈ GL(2, 𝔸). Since (r^{∗}, r_{∗}) and (s_{∗}, s^{∗}) are adjoint pairs, so is (E = s_{∗}r^{∗}, E^{∗} = r_{∗}s^{∗}).

The image A_{E,α} of the Eisenstein map E = s_{∗}r^{∗} : C_{c}^{∞}(Y_{α}) → A_{c,α} is called the Eisenstein part of A_{c,α}. The maps E and E^{∗} intertwine the GL(2, 𝔸)-action; A_{E,α} is an invariant subspace of A_{c,α}.

Proposition 4.6. The space A_{c,α} is an orthogonal direct sum of the space A_{0,α} of cusp forms and of A_{E,α}.

Proof. Cusp forms are compactly supported. Since A_{0,α} = ker E^{∗} and A_{E,α} = im E, we have A_{0,α} ⊥ A_{E,α}. Given a compact open subgroup U in GL(2, 𝔸), put A_{α}^{U} for the space of U-invariant functions in A_{α}, and

A_{c,α}^{U} = A_{c,α} ∩ A_{α}^{U}, A_{0,α}^{U} = A_{0,α} ∩ A_{α}^{U}, A_{E,α}^{U} = A_{E,α} ∩ A_{α}^{U}.

It remains to show that A_{0,α}^{U} +A_{E,α}^{U} = A_{c,α}^{U}. If not there exists a nonzero linear form ℓ : A_{c,α}^{U} → ℂ which is zero on A_{0,α}^{U} +A_{E,α}^{U}. There exists f ∈ A_{α}^{U} such that ℓ(ϕ) = (ϕ, f) for every ϕ ∈ A_{c,α}^{U}. For any U-invariant function ψ ∈ C_{c}^{∞}(Y_{α}) we have (ψ, E^{∗}f) = (Eψ, f) = ℓ(Eψ) = 0. Hence E^{∗}f = 0, thus f ∈ A_{0,α}^{U}. This however is impossible since f is orthogonal to the space A_{0,α}^{U} of U-invariant cusp forms.

Given ϕ ∈ C_{c}^{∞}(Y_{α}) and x ∈ GL(2, 𝔸), put (Mϕ)(x) = ∫_{N(𝔸)} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn.

The integral converges, by

Proposition 4.7. The map N(𝔸) → Y_{α}, n ↦ α^{ℤ}A(F)N(𝔸) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx, is proper.

Proof. It suffices to consider the case of x = 1. The function

ht^{+} : Y_{α} → ℤ, ( \begin{matrix}a & c \\ 0 & b \end{matrix}) k ↦ deg a − deg b,

is continuous. Thus it suffices to show that the map φ(a) = ht^{+}(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} )), φ : 𝔸 → ℤ, is proper. But ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & a_{v} \\ 0 & 1 \end{matrix} ) is in GL(2, O_{v}) if |a_{v}|_{v} ≤ 1; otherwise

it is = ( \begin{matrix}a_{v}^{−1} & −1 \\ 0 & a_{v} \end{matrix} ) ( \begin{matrix}1 & 0 \\ a_{v}^{−1} & 1 \end{matrix} ). If a = (a_{v}), then φ(a) = −2 ∑_{v} max(0, log_{q} |a_{v}|_{v}),

as log_{q} |a_{v}|_{v} = − val_{v}(a_{v}) deg(v). Hence φ is proper.

By definition, x ↦ (Mϕ)(x) is invariant under left translation by N(𝔸), and also by α^{ℤ} · A(F). Indeed,

(Mϕ)(( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) x) = ∫_{𝔸} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) n ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) x)dy = |\frac{a}{b}| ∫_{N(ℤ)} ϕ(( \begin{matrix}b & 0 \\ 0 & a \end{matrix}) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn

and |a/b| = q^{deg(a/b)}. Thus M maps C_{c}^{∞}(Y_{α}) to C^{∞}(Y_{α}).

Proposition 4.8. Denote by I the natural embedding of C_{c}^{∞}(Y_{α}) in C^{∞}(Y_{α}). Then

E^{∗}E = I + M.

Proof. By the Bruhat decomposition, an element of GL(2, F) which is not in A(F)N(F) has a unique decomposition n_{1}a ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) n_{2} with n_{i} ∈ N(F), a ∈ A(F). Thus, for any ϕ ∈ C_{c}^{∞}(Y_{α}), x ∈ GL(2, 𝔸), we have

(Eϕ)(x) = ∑_{γ∈A(F)N(F)\ GL(2,F)} ϕ(γx) = ϕ(x) + ∑_{ν∈N(F)} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) νx).

Hence

(E^{∗}Eϕ)(x) = |N(𝔸)/N(F)|ϕ(x) + ∫_{N(F)\N(𝔸)} ∑_{ν∈N(F)} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) νnx)dn

= ϕ(x) + ∫_{N(𝔸)} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn = ϕ(x) + (Mϕ)(x).

Proposition 4.9. Let μ_{1}, μ_{2} be characters of 𝔸^{×}/F^{×}. If t is sufficiently small, for all ϕ ∈ I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) and x ∈ GL(2, 𝔸), the integral (M(μ_{1}, μ_{2}, t)ϕ)(x) = ∫_{N(𝔸)} ϕ(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn converges and defines a function in I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}). Moreover, M(μ_{1}, μ_{2}, t) = q^{1−g}m(μ_{1}/μ_{2}, t^{2})R(μ_{1}, μ_{2}, t).

Proof. Recall that |a| = q^{deg(a)} and that I(μ_{1}, μ_{2}) consists of the ϕ in C^{∞}(GL(2, 𝔸)) with

ϕ(( \begin{matrix}a_{1} & 0 \\ 0 & a_{2} \end{matrix} ) x) = |a_{1}/a_{2}|^{1/2}μ_{1}(a_{1})μ_{2}(a_{2})ϕ(x),

while ν_{t}(a) = t^{deg a}. We put t_{v} = t^{deg(v)}. We may assume that ϕ(x) = ∏_{v} ϕ_{v}(x_{v}) with ϕ_{v} ∈ I(μ_{1v}ν_{t_{v}}, μ_{2v}ν_{t_{v}^{−1}}). For almost all v, the restriction of ϕ_{v} to GL(2, O_{v}) is 1. We may replace ϕ_{v}, μ_{i}, t by their complex absolute values to assume t > 0 and ϕ_{v}, μ_{i} take real nonnegative values.

Then (M(μ_{1}, μ_{2}, t)ϕ)(x) = c ∏_{v} τ_{v}, with τ_{v} = ∫_{N(F_{v})} ϕ_{v}(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx_{v})dn =

∫_{F_{v}} ϕ_{v}(( \begin{matrix}0 & −1 \\ 1 & z \end{matrix} ) x_{v})dz. The measure dn_{v} on N(F_{v}) is normalized by |N(O_{v})| = 1, and c = |N(𝔸)/N(F)| in the measure ⊗_{v}dn_{v} on N(𝔸). We saw that for small enough t the integral which defines τ_{v} converges for all v. For almost all v we have τ_{v} = L(μ_{1v}/μ_{2v}, t_{v}^{2})/L(μ_{1v}/μ_{2v}, q_{v}^{−1}t_{v}^{2}), so the product ∏_{v} τ_{v} converges for small t. Now M(μ_{1}, μ_{2}, t) = c ∏_{v} M(μ_{1v}, μ_{2v}, t_{v}). Each factor here is \frac{L(μ_{1v}/μ_{2v},t_{v}^{2})}{L(μ_{1v}/μ_{2v},q_{v}^{−1}t_{v}^{2})}R(μ_{1v}, μ_{2v}, t_{v}). Put R(μ_{1}, μ_{2}, t) = ⊗_{v}R(μ_{1v}, μ_{2v}, t_{v}),

and m(μ, t) = \frac{L(μ,t)}{L(q^{−1}t,μ)}, where L(μ, t) = ∏_{v} L(μ_{v}, t_{v}). Note that c is |O| = q^{1−g}, using 0 → 𝔽_{q} → O → 𝔸/F → H^{1}(X, O_{X}) → 0.

It follows (since L(μ, t) is a rational function of t) that after identifying the spaces I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) for all t, the operator

M(μ_{1}, μ_{2}, t) : I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) → I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t})

(defined for small t) depends on t rationally. Hence M(μ_{1}, μ_{2}, t) is defined for almost all t, and it commutes with the action of GL(2, 𝔸).

4.3 L-functions

Let us review the theory of L-functions for GL(2). Let E be an algebraically closed field of characteristic zero. The valuation val_{v}(a) of a ∈ F_{v}^{×} is the largest integer n with a ∈ π π π_{v}^{n}O_{v}. For any character ψ : F_{v} → E^{×}, ψ ≠ 1, let r(ψ) be the largest n such that ψ(π π π_{v}^{−n}O_{v}) = 1. Normalize the Haar measure on F_{v} by |O_{v}| = 1. The conductor of a character χ : F_{v}^{×} → E^{×} is n = 0 if χ(O_{v}^{×}) = 1, i.e., χ is unramified; otherwise it is the smallest n ≥ 1 such that χ(1 + π π π_{v}^{n}O_{v}) = 1. Given χ, put L(t, χ) = (1 − χ(π π π_{v})t)^{−1} if χ is unramified, L(t, χ) = 1 is χ is ramified. Given ψ ≠ 1, put

Γ(χ, ψ, t) = ∫_{F_{v}^{×}} χ(x)^{−1}ψ(x)t^{− val_{v}(x)}dx, ψ : F_{v} → E^{×}.

This Γ(χ, ψ, t) is a formal power series in t which contains positive and negative powers of t. Tate’s thesis (see [Lg94], VII, section 3-4) establishes

Proposition 4.10. The formal series Γ(χ, ψ, t) has finitely many positive powers of t. It is a rational function of t, namely a Laurent series of a rational function of t at t = ∞. Put ε(χ, ψ, t) = \frac{L(χ,t)Γ(χ,ψ,t)}{L(χ^{−1},q_{v}^{−1}t^{−1})}. It has the form

c(χ, ψ)t^{n(χ,ψ)}. If r(ψ) = 0 then n(χ, ψ) is the conductor of χ. If in addition χ is unramified then ε(χ, ψ, t) is 1. If a ∈ F_{v}^{×}, ψ_{a}(x) = ψ(ax), then ε(χ, ψ_{a}, t) = χ(a)(q_{v}t)^{val_{v}(a)}ε(χ, ψ, t).

Note that L and ε are usually considered, in the case where E = ℂ, as functions of s, where t = q_{v}^{−s}, rather than of t. The Haar measure on F_{v} is usually normalized by |O_{v}| = q_{v}^{−r(ψ)/2}, as this measure is self-dual with respect to the pairing F_{v}×F_{v} → E^{×}, (x, y) ↦ ψ(xy). This choice of measure is not convenient if E ≠ ℂ since E has no distinguished square root of q. Given a character χ of 𝔸^{×}, denote its restriction to F_{v}^{×} by χ_{v}. The restriction to F_{v} of a character ψ of 𝔸 is denoted ψ_{v}. For a closed point v of X, we write deg(v) for the dimension of the residue field at v over 𝔽_{q}, and q_{v} = q^{deg(v)}. Given a character χ : 𝔸^{×}/F^{×} → E^{×}, put L(χ, t) = ∏_{v} L(χ_{v}, t_{v}), where t_{v} = t^{deg(v)}; the product converges in E[[t]]. Let ψ : 𝔸/F → E^{×} be a character ≠ 1. Then ε(χ, t) = q^{1−g} ∏_{v} ε(χ_{v}, ψ_{v}, t_{v}) converges as almost all factors are 1, and ε(χ, t) is independent of ψ by Proposition 4.10.

Proposition 4.11. For any character χ : 𝔸^{×}/F^{×} → E^{×} the formal series L(χ, t) is rational in t, and L(χ, t) = ε(χ, t)L(χ^{−1}, q^{−1}t^{−1}). If the restriction of χ to the group of x ∈ 𝔸^{×}/F^{×} with deg(x) = 0 is nontrivial, then L(χ, t) is a polynomial. If the restriction is trivial, χ is given by χ(x) = u^{deg(x)}, and then L(χ, t) has precisely two poles: t = u^{−1} and t = q^{−1}u^{−1}, both poles are simple. If χ : 𝔸^{×}/F^{×} → ℂ^{×} is a unitary character (|χ(x)| = 1 for all x) then the zeroes of L(χ, t) lie in the doughnut {t ∈ ℂ; q^{−1} < |t| < 1}.

The proof of this is also in [Lg94], Chapter VII, sections 7-8. The following is due to [W45].

Theorem 4.12. (A. Weil). For any unitary character χ : 𝔸^{×}/F^{×} → ℂ^{×}, all zeroes of L(χ, t) lie on the circle |t| = q^{−1/2}.

Given a character ψ : 𝔸/F → E^{×}, ψ ≠ 1, let W (ψ) be the space of locally constant functions ϕ : GL(2, F_{v}) → E with ϕ(( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x) = ψ(z)ϕ(x) for all z ∈ F_{v}, x ∈ GL(2, F_{v}). The group GL(2, F_{v}) acts on W (ψ) by right translation. Fix a Haar measure d^{×}x on F_{v}^{×}. For any ϕ ∈ W (ψ) put

Λ_{ϕ}(t) = ∫ ϕ(( \begin{matrix}a & 0 \\ 0 & 1 \end{matrix} ))(q_{v}t)^{val_{v}(a)}d^{×}a, Λ̃_{ϕ}(t) = ∫ ϕ(( \begin{matrix}0 & 1 \\ a & 0 \end{matrix} ))(q_{v}t)^{val_{v}(a)}d^{×}a.

Both Λ_{ϕ}(t) and Λ̃_{π}(t) are formal power series in t, containing positive and negative powers of t. Let π be an irreducible admissible representation of GL(2, F_{v}) over E. Then π(( \begin{matrix}a & 0 \\ 0 & a \end{matrix})) is the operator of multiplication by a scalar η(a) ∈ E^{×}. The character η : F_{v}^{×} → E^{×} is called the central character of π.

Proposition 4.13. Let π be an irreducible admissible infinite dimensional representation over E of GL(2, F_{v}). Let η be the central character of π. (1) There exists a unique GL(2, F_{v})-invariant subspace W (π, ψ) of W (ψ) equivalent to π. (2) If ϕ ∈ W (π, ψ) then Λ_{ϕ}(t) is the Laurent series at t = 0 of a rational function, and Λ̃_{ϕ}(t) is the Laurent series at t = ∞ of a rational function. (3) There exists a nonzero polynomial P ∈ E[t] such that for any ϕ ∈ W (π, ψ) we have P (t)Λ_{ϕ}(t) ∈ E[t, t^{−1}]. There exists ϕ ∈ W (π, ψ) with Λ_{ϕ}(t) ≠ 0. (4) The quotient Λ̃_{ϕ}(t)/Λ_{ϕ}(t) of rational functions in t does not depend on the choice of ϕ in W (π, ψ) with Λ_{ϕ}(t) ≠ 0. (5) The lowest degree polynomial P ∈ E[t] which satisfies (3) and P (0) = 1 is independent of ψ. (6) Put Γ(π, ψ, t) = Λ̃_{ϕ}(t)/Λ_{ϕ}(t) and ε(π, ψ, t) = \frac{Γ(π,ψ,t)L(π,t)}{L(π⊗η^{−1},q_{v}^{−2}t^{−1})} where L(π, t) = P (t)^{−1} with P of (5). Then

ε(π, ψ, t) has the form c(π, ψ)t^{n(π,ψ)}, c(π, ψ) in E^{×} and n(π, ψ) in ℤ. (7) If ψ_{a}(x) is ψ(ax) for a ∈ F_{v}^{×}, then ε(π, ψ_{a}, t) = η(a)(q_{v}t)^{2 val_{v}(a)}ε(π, ψ, t).

This is [JL70], Theorem 2.18. Our L and ε relate to those L_{JL}, ε_{JL} of Jacquet-Langlands by L_{JL}(π, s) = L(π, t_{v}), t_{v} = q_{v}^{−s}, ε_{JL}(π, ψ, s) = ε(π, ψ, t_{v}). Note that the proof of [JL70], which claims that Λ_{ϕ}(t) is a Laurent series of a meromorphic function in ℂ − {0}, shows that Λ_{ϕ}(t) is rational. In general, the meromorpic functions of s over p-adic and global function fields are rational functions of q^{s}. Every smooth finite dimensional irreducible representation of GL(2, F_{v}) is one dimensional, of the form x ↦ χ(det x), where χ : F_{v}^{×} → E^{×} is a character ([JL70], Proposition 2.7).

Proposition 4.14. Let π, π′ be irreducible admissible infinite dimensional representations of GL(2, F_{v}) with equal central characters. If there is a character ψ : F_{v} → E^{×} such that for every character ω : F_{v}^{×} → E^{×} we have Γ(πω, ψ, t) = Γ(π′ω, ψ, t), then π ≃ π′.

For a proof see [JL70], Corollary 2.19.

The conductor of an irreducible admissible infinite dimensional representation π of GL(2, F_{v}) is the integer n(π, ψ), with ψ normalized by r(ψ) = 0. It is well defined, as from (7) above, the integer n(π, ψ) of (6) is not changed if ψ is replaced by ψ_{a} : x ↦ ψ(ax).

Proposition 4.15. The conductor of π is the least integer n such that the representation space of π contains a nonzero vector invariant under the group

H_{n} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL(2, O_{v}); c ∈ π π π_{v}^{n}O_{v}, d ∈ 1 + π π π_{v}^{n}O_{v}\}. For this n, dim_{E} π π π^{H_{n}} =

1.

For a proof see Casselman, Math. Ann. 201 (1973), 301-314.

Proposition 4.16. Let π be an irreducible admissible infinite dimensional representation, with central character η, of GL(2, F_{v}). Let ψ : F_{v} → E^{×} be a nontrivial character. Then there exists an integer m_{π} such that if χ : F_{v}^{×} → E^{×} is any character with conductor > m_{π}, then L(πχ, t) = 1 and

ε(πχ, ψ, t) = ε(χ, ψ, t)ε(χη, ψ, q_{v}t)q_{v}^{−r(ψ)}.

For a proof see [JL70], Proposition 3.8. See [JL70], Proposition 3.5, 3.6, for a proof of:

Proposition 4.17. Let μ_{1}, μ_{2} be characters of F_{v}^{×}, and ψ ≠ 1 a character of F_{v}. If μ_{1}/μ_{2} ≠ | · |_{v}^{±1} then L(I(μ_{1}, μ_{2}), t) = L(μ_{1}, t)L(μ_{2}, t) and

ε(I(μ_{1}, μ_{2}), ψ, t) = ε(μ_{1}, ψ, t)ε(μ_{2}, ψ, t)q_{v}^{−r(ψ)}.

If μ_{2}/μ_{1} = | · |_{v}, then

L(St(μ_{1}| · |_{v}^{−1/2}, μ_{1}| · |_{v}^{1/2}), t) = L(μ_{1}| · |_{v}^{1/2}, t),

ε(St(μ_{1}| · |_{v}^{−1/2}, μ_{1}| · |_{v}^{1/2}), ψ, t) = \frac{L(μ_{1}^{−1}, t^{−1})}{L(μ_{1}, t)}ε(μ_{1}, ψ, t)ε(μ_{1}| · |_{v}, ψ, t)q_{v}^{−r(ψ)}.

If π is a cuspidal representation of GL(2, F_{v}) then L(π, t) is 1.

Recall that an irreducible admissible infinite dimensional representation π of GL(2, F_{v}) on a vector space V is called unramified if its space V^{K} of K = GL(2, O_{v})-fixed vectors is nonzero. In this case V^{K} is one dimensional, and π = I(μ_{1}, μ_{2}) with unramified μ_{1}, μ_{2} and μ_{1}/μ_{2} ≠ | · |^{±1}.

Corollary 4.18. Let π be an unramified irreducible admissible infinite dimensional representation of GL(2, F_{v}) and ψ ≠ 1 with r(ψ) = 0. Then ε(π, ψ, t) = 1.

Proof. Here π = I(μ_{1}, μ_{2}) with unramified μ_{1}, μ_{2}, so the claim follows from the last proposition and Tate’s Thesis.

Let π be an admissible irreducible representation of GL(2, 𝔸) whose local components are all infinite dimensional. Put L(π, t) = ∏_{v} L(π_{v}, t_{v}), t_{v} = t^{deg(v)}; the infinite product converges in E[[t]]. For any character ψ : 𝔸/F → E^{×}, ψ ≠ 1, put ε(π, ψ, t) = ∏_{v} ε(π_{v}, ψ_{v}, t_{v}); almost all factors here are 1. From (7) it follows that if the central character of π is trivial on F^{×}, then ε(π, ψ, t) is independent of the choice of ψ : 𝔸/F → E^{×}. We denote it in this case by ε(π, t). Theorems 11.1, 11.3 of [JL70] assert:

Theorem 4.19. Let π be an irreducible admissible representation of GL(2, 𝔸) over E. Denote by η : 𝔸^{×} → E^{×} its central character. Then π is cuspidal iff (1) η is trivial on F^{×}; (2) all local components of π are infinite dimensional; (3) for any character ω : 𝔸^{×}/F^{×} → E^{×}, the formal series L(πω, t) is a polynomial in t, and (4) L(πω, t) = ε(πω, t)L(πη^{−1}ω^{−1}, q^{−2}t^{−1}).

Note that (4) makes sense due to (3). In [JL70], (3) is formulated as stating that the product ∏_{v} L(π_{v}ω_{v}, t_{v}) converges absolutely for sufficiently small t, and its value has an analytic continuation to a holomorphic function in ℂ−{0}. But the argument of [JL70] can be modified to lead to (3) in our case of E which is not ℂ, over a function field F. Note that (4) is not ∏_{v} Γ(π_{v}ω_{v}, ψ_{v}, t_{v}) = 1; indeed the product here does not converge.

Proposition 4.20. If π, π′ are cuspidal representations of GL(2, 𝔸) and π_{v} ≃ π′_{v} for almost all v, then π ≃ π′.

Proof. Let S be a finite set of closed points of X with π_{v} ≃ π′_{v} at v ∉ S. Let η, η′ be the central characters of π, π′, and η_{v}, η′_{v} their components at v (restrictions to F_{v}^{×}). By our assumption, η′_{v} = η_{v} for all v ∉ S. But the groups F_{v}^{×}, v ∉ S, generate a dense subgroup of 𝔸^{×}/F^{×}. Hence η′ = η. By the Theorem 4.19, of [JL70], above, fixing a character ψ : 𝔸/F → E^{×}, ψ ≠ 1, for any character ω : 𝔸^{×}/F^{×} → E^{×} one has

∏_{v} L(π_{v}ω_{v}, t_{v}) = ∏_{v} ε(π_{v}ω_{v}, ψ_{v}, t_{v})L(π_{v}η_{v}^{−1}ω_{v}^{−1}, q_{v}^{−2}t_{v}^{−1}),
∏_{v} L(π′_{v}ω_{v}, t_{v}) = ∏_{v} ε(π′_{v}ω_{v}, ψ_{v}, t_{v})L(π′_{v}η′_{v}^{−1}ω_{v}^{−1}, q_{v}^{−2}t_{v}^{−1}).

Since π_{v} ≃ π′_{v} at all v ∉ S, we conclude

∏_{v∈S} Γ(π_{v}ω_{v}, ψ_{v}, t_{v}) = ∏_{v∈S} \frac{ε(π_{v}ω_{v}, ψ_{v}, t_{v})L(π_{v}η_{v}^{−1}ω_{v}^{−1}, q_{v}^{−2}t_{v}^{−1})}{L(π_{v}ω_{v}, t_{v})}

= ∏_{v∈S} \frac{ε(π′_{v}ω_{v}, ψ_{v}, t_{v})L(π′_{v}η′_{v}^{−1}ω_{v}^{−1}, q_{v}^{−2}t_{v}^{−1})}{L(π′_{v}ω_{v}, t_{v})} = ∏_{v∈S} Γ(π′_{v}ω_{v}, ψ_{v}, t_{v}).

Since η = η′, it follows from Proposition 4.16 that for each v ∈ S there exists m_{v} > 0 such that if χ : F_{v}^{×} → E^{×} is any character whose conductor is ≥ m_{v}, then Γ(π_{v}χ, ψ_{v}, t) = Γ(π′_{v}χ, ψ_{v}, t). Fix v ∈ S and a character χ of F_{v}^{×}. By Proposition 4.14, it suffices to show Γ(π_{v}χ, ψ_{v}, t) = Γ(π′_{v}χ, ψ_{v}, t). For this, it suffices to choose a character ω : 𝔸^{×}/F^{×} → E^{×} in the last displayed equation with ω_{v} = χ and such that for each u ∈ S − {v}, the conductor of ω_{u} is bigger than m_{u}. But the group H = F_{v}^{×} ∏_{u∈S−{v}} O_{u}^{×} maps isomorphically and homeomorphically onto its image in 𝔸^{×}/F^{×}. Hence any character of H extends to a character of 𝔸^{×}/F^{×}.

Proposition 4.21. Let η be a character of 𝔸^{×}/F^{×}, S a finite set of closed points of X, ψ ≠ 1 a character of 𝔸/F with r(ψ_{u}) = 0 for all u in S. Suppose that for any closed point v ∈ |X| − S, π_{v} is an irreducible admissible infinite dimensional representation of GL(2, F_{v}) with central character η_{v} such that almost all π_{v} are unramified, there is no pair μ_{1}, μ_{2} of characters of 𝔸^{×}/F^{×} with π_{v} = π(μ_{1v}, μ_{2v}) for almost all v ∈ |X|−S, and for any character ω of 𝔸^{×}/F^{×} which is unramified at all points of S, the formal series ∏_{v∉S} L(π_{v}ω_{v}, t_{v}) and ∏_{v∉S} L(π_{v}η_{v}^{−1}ω_{v}^{−1}, t_{v}) are polynomials, and there exists a number c ∈ E^{×} and integers n_{u} > 0 (u ∈ S) such that

∏_{v∉S} L(π_{v}ω_{v}, t_{v}) = c ∏_{u∈S}(ω(π π π_{u})t_{u})^{n_{u}} ∏_{v∉S} ε(π_{v}ω_{v}, ψ_{v}, t_{v})L(π_{v}η_{v}^{−1}ω_{v}^{−1}, q_{v}^{−2}t_{v}^{−1}).

Then there exists a cuspidal representation π of GL(2, 𝔸) with central character η such that for every v ∈ |X| − S the local component of π at v is π_{v}.

A proof is in [JL70], Theorem 11, Corollary 11.6, proof of Theorem 12.2. The representation π is unique by Proposition 4.20.

4.4 Intertwining again

We can now return to the study of the intertwining operators.

Proposition 4.22. Let μ_{1}, μ_{2} be characters of F_{v}^{×}. Let ψ ≠ 1 be a character of F_{v}. Then

R(μ_{1}, μ_{2}, t)R(μ_{2}, μ_{1}, t^{−1}) = ε (\frac{μ_{1}}{μ_{2}} , ψ, q_{v}^{−1}t^{2}) ε (\frac{μ_{2}}{μ_{1}} , ψ, q_{v}^{−1}t^{−2}) .

Proof. By the transformation formula for the ε-factors, the right hand side does not depend on ψ. We then choose ψ with ker ψ ⊃ O_{v} and ker ψ ⊅ π π π_{v}^{−1}O_{v}. We can rewrite the asserted equality as

M(μ_{1}, μ_{2}, t)M(μ_{2}, μ_{1}, t^{−1}) = Γ (\frac{μ_{2}}{μ_{1}} , ψ, q_{v}^{−1}t^{2}) Γ (\frac{μ_{2}}{μ_{1}} , ψ, q_{v}^{−1}t^{−2}) .

The restriction map I(μ_{1}, μ_{2}) → I(μ_{1}/μ_{2}), where

I(μ) = \{f ∈ C^{∞}(SL(2, F_{v})); f (( \begin{matrix}a & b \\ 0 & 1/a \end{matrix} ) x) = μ(a)|a|_{v}f(x)\},

is an isomorphism (μ : F_{v}^{×} → E^{×} is a character). The group SL(2, F_{v}) acts transitively on F_{v}^{2} − {(0, 0)} on the right. The stabilizer of the vector (0, 1) is N(F_{v}). Then N(F_{v})\ SL(2, F_{v}) can be identified with F_{v}^{2} −\{(0, 0)\} by ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ (c, d) ∈ F_{v}^{2} − {(0, 0)}. Using this we identify I(μ) with

V (μ) = {f ∈ C^{∞}(F_{v}^{2} − {(0, 0)});
f(ax) = μ(a)^{−1}|a|_{v}^{−1}f(x), a ∈ F_{v}^{×}, x ∈ F_{v}^{2} − {(0, 0)}},

so I(μ_{1}, μ_{2}) with V (μ_{1}/μ_{2}). The operator M(μ_{1}, μ_{2}, t) corresponds to the operator \overline{M}(μ_{1}/μ_{2}, t^{2}) where

\overline{M}(μ, s) : V (μν_{s}) → V (μ^{−1}ν_{s^{−1}}), (\overline{M}(μ, s)f)(x) = ∫_{\{y;x∧y=1\}} f(y)dy.

Here ∧ denotes the symplectic form (a, b)∧(c, d) = ad−bc on F_{v}^{2}. The measure on the line ℓ_{x} = {y ∈ F_{v}^{2}; x ∧ y = 1} is transferred from the Haar measure on F_{v} via the map F_{v} → ℓ_{x} given by a ↦ y_{0} + ax where y_{0} is a fixed point on ℓ_{x}. So we need to show:

\overline{M}(μ, s)\overline{M}(μ^{−1}, s^{−1}) = Γ(μ, ψ, q_{v}^{−1}s)Γ(μ^{−1}, ψ, q_{v}^{−1}s^{−1}).

For sufficiently small s ∈ ℂ^{×} define operators A_{s} : C_{c}^{∞}(F_{v}^{2}) → V (μν_{s}) and B_{s} : C_{c}^{∞}(F_{v}^{2}) → V (μ^{−1}ν_{s}) by

(A_{s}f)(x) = ∫_{F_{v}} f(ax)μ(a)ν_{s}(a)da, (B_{s}f)(x) = ∫_{F_{v}} f(ax)μ(a)^{−1}ν_{s}(a)da.

Restriction defines an isomorphism V (μν_{s}) → V_{0}(μ), where

V_{0}(μ) = {f ∈ C^{∞}(O_{v}^{2} − {(0, 0)});
f(ax) = μ(a)^{−1}f(x), x ∈ O_{v}^{2} − {(0, 0)}, a ∈ O_{v}^{×}},

so we can identify the spaces V (μν_{s}) as s varies. The operators A_{s} and B_{s}, defined above for small s, depend rationally on s. Hence they can be extended to all s. Consider the Fourier transform

F : C_{c}^{∞}(F_{v}^{2}) → C_{c}^{∞}(F_{v}^{2}), (F f)(y) = ∫_{F_{v}^{2}} f(x)ψ(x ∧ y)dx.

Lemma 4.23. We have \overline{M}(μ, s)A_{s} = Γ(μ^{−1}, ψ, q_{v}^{−1}s^{−1})B_{s^{−1}}F,

\overline{M}(μ^{−1}, s^{−1})B_{s^{−1}} = Γ(μ, ψ, q_{v}^{−1}s)A_{s}F.

Proof. Given f ∈ C_{c}^{∞}(F_{v}^{2}), x ∈ F_{v}^{2} − {(0, 0)}, we first show

Γ(μ^{−1}, ψ, q_{v}^{−1}s^{−1})(B_{s^{−1}}F f)(x) = (\overline{M}(μ, s)A_{s}f)(x).

The operators F, A_{s}, B_{s} commute with the action of SL(2, F_{v}). This action is transitive on F_{v}^{2} − {(0, 0)}, so we may assume x = (0, 1). We compute

(B_{s^{−1}}F f)((0, 1)) = ∫_{F_{v}}(F f)((0, a))μ(a)^{−1}ν_{s^{−1}}(a)da,
(F f)((0, a)) = ∫_{F_{v}^{2}} f(y, z)ψ(ya)dydz = φ̂(−a),
φ̂(a) = ∫ φ(y)ψ(−ya)dy,  φ(y) = ∫ f(y, z)dz.

Tate’s functional equation (see [L], VII, section 3-4) is

Γ(μ^{−1}, ψ, q_{v}^{−1}s^{−1}) ∫ φ̂(a)μ^{−1}(a)ν_{s^{−1}}(a)da = ∫ φ(y)μ(y)ν_{s}(y)\frac{dy}{|y|}.

(Formally this can be deduced from the definition of the Γ-function and the inversion formula φ(y) = ∫ φ̂(a)ψ(ay)da. However the left side converges for large |s|, while the right for small |s|, so one has to show both sides are rational in s). We conclude that the left side of the equation to be shown is

∫ φ(y)μ(−y)ν_{s}(y)|y|^{−1}dy = ∫ ∫ f(y, z)μ(−y)ν_{s}(y)|y|^{−1}dydz

while the right side is (recall: x = (0, 1), so (0, 1) ∧ (y, z) = −y)

∫ (A_{s}f)(−1, z)dz = ∫ ∫ f(−y, yz)μ(y)ν_{s}(y)dydz.

The proof of the second identity of the lemma is similar. □

The inverse Fourier transform coincides with F since the form (x, y) ↦ x ∧ y in the definition of F is skew-symmetric. Hence F^{2} = 1, and it follows from the Lemma that

\overline{M}(μ, s)\overline{M}(μ^{−1}, s^{−1})B_{s^{−1}} = Γ(μ, ψ, q_{v}^{−1}s)Γ(μ^{−1}, ψ, q_{v}^{−1}s^{−1})B_{s^{−1}}.

However, the operator B_{s^{−1}} is onto for those s where it is defined (even its restriction to C_{c}^{∞}(F_{v}^{2} −{(0, 0)}) is onto), as V (μν_{s}) is irreducible, so the proposition follows.

Proposition 4.24. For any characters μ_{1}, μ_{2} of 𝔸^{×}/F^{×} we have

M(μ_{1}, μ_{2}, t)M(μ_{2}, μ_{1}, t^{−1}) = 1.

Proof. From Proposition 4.21, M(μ_{1}, μ_{2}, t)M(μ_{2}, μ_{2}, t^{−1}) is equal to

q^{2−2g}m(μ_{1}/μ_{2}, t^{2})m(μ_{2}/μ_{1}, t^{−2})R(μ_{1}, μ_{2}, t)R(μ_{2}, μ_{1}, t^{−1}),

while Proposition 4.22 implies, for any character ψ ≠ 1 of 𝔸/F, that

R(μ_{1}, μ_{2}, t)R(μ_{2}, μ_{1}, t^{−1})

is

∏_{v}[ε(μ_{1v}/μ_{2v}, ψ_{v}, q_{v}^{−1}t_{v}^{2})ε(μ_{2v}/μ_{1v}, ψ_{v}, q_{v}^{−1}t_{v}^{−2})]

= q^{2g−2}ε(μ_{1}/μ_{2}, q^{−1}t^{2})ε(μ_{2}/μ_{1}, q^{−1}t^{−2}).

As ε(χ, t) = q^{1−g} ∏_{v} ε(χ_{v}, ψ_{v}, t_{v}) satisfies the functional equation L(χ, t) = ε(χ, t)L(χ^{−1}, q^{−1}t^{−1}), we have that

ε(μ_{1}/μ_{2}, q^{−1}t^{2})ε(μ_{2}/μ_{1}, q^{−1}t^{−2})m(μ_{1}/μ_{2}, t^{2})m(μ_{2}/μ_{1}, t^{−2}),

which is equal to

\frac{ε(μ_{1}/μ_{2}, q^{−1}t^{2})L(μ_{2}/μ_{1}, t^{2})}{L(μ_{1}/μ_{2}, q^{−1}t^{2})} · \frac{ε(μ_{1}/μ_{2}, q^{−1}t^{−2})L(μ_{1}/μ_{2}, t^{2})}{L(μ_{2}/μ_{1}, q^{−1}t^{−2})}

is equal to 1.

4.5 M^{2} = 1 via Mellin transform

We shall next study the relationship between M : C_{c}^{∞}(Y_{α}) → C^{∞}(Y_{α}) and M(μ_{1}, μ_{2}, t) : I(μ_{1}ν^{t}, μ_{2}ν^{−t}) → I(μ_{2}ν^{−t}, μ_{1}ν^{t}), and conclude that M^{2} = 1. Both are defined by the same integral formula. Here μ_{1}, μ_{2} are characters of 𝔸^{×}/F^{×} · α^{ℤ}. Put

η(( \begin{matrix}a & 0 \\ 0 & b \end{matrix})) = μ_{1}(a)μ_{2}(b)|a/b|^{1/2}ν_{t}(a/b), η : A(𝔸)/A(F) · α^{ℤ} → E^{×}.

It is a character. Recall that Y_{α} = α^{ℤ}N(𝔸)A(F)\ GL(2, 𝔸) and (Mf)(x) =

∫_{N(𝔸)} f(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn. Suppose that f ∈ C_{c}^{∞}(Y_{α}), and t ∈ E^{×}. Define a

function T (f, μ_{1}, μ_{2}, t) : GL(2, 𝔸) → ℂ by

(T (f, μ_{1}, μ_{2}, t))(x) = ∫_{α^{ℤ}A(F)\A(𝔸)} f(a^{−1}x)η(a)d^{×}a.

Then T (f, μ_{1}, μ_{2}, t) ∈ I(μ_{1}ν_{t}, μ_{2}ν_{−t}) is called the Mellin transform of f. The notation T can be used also when f ∈ C^{∞}(Y_{α}) is not compactly supported, whenever the integral converges.

Proposition 4.25. For φ ∈ C_{c}^{∞}(Y_{α}), characters μ_{1}, μ_{2} : 𝔸^{×}/F^{×} · α^{ℤ} → E^{×} and large enough t ∈ ℂ^{×}, the integral defining T converges, and T (Mφ, μ_{1}, μ_{2}, t) = M(μ_{2}, μ_{1}, t^{−1})T (φ, μ_{2}, μ_{1}, t^{−1}).

Proof. By definition,

(T (f, μ_{1}, μ_{2}, t))(x) = ∫∫ f(( \begin{matrix}a & 0 \\ 0 & b \end{matrix})^{−1} x)μ_{1}(a)μ_{2}(b)|a/b|^{1/2}ν_{t}(a/b)d^{×}ad^{×}b.

Put f = Mφ, so f(( \begin{matrix}a & 0 \\ 0 & b \end{matrix})^{−1} x) = |b/a| ∫_{N(𝔸)} φ(( \begin{matrix}b & 0 \\ 0 & a \end{matrix})^{−1} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn. Hence

(T (f, μ_{1}, μ_{2}, t))(x) equals

∫ ∫ ∫ φ(( \begin{matrix}b & 0 \\ 0 & a \end{matrix})^{−1} ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)μ_{1}(a)μ_{2}(b)|b/a|^{1/2}ν_{t}(a/b)d^{×}ad^{×}bdn

= ∫_{N(𝔸)}(T (φ, μ_{2}, μ_{1}, t^{−1}))(( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx)dn
= (M(μ_{2}, μ_{1}, t^{−1})T (φ, μ_{2}, μ_{1}, t^{−1}))(x).

If t is large enough, the integral which defines M(μ_{2}, μ_{1}, t^{−1}) converges, and so is the integral which defines T (f, μ_{1}, μ_{2}, t), which justifies the computation.

Proposition 4.26. If φ ∈ C_{c}^{∞}(Y_{α}) then Mφ ∈ C^{∞}(Y_{α}). If Mφ ∈ C_{c}^{∞}(Y_{α}) then M^{2}φ = φ.

Proof. Put f = Mφ and h = Mf = M^{2}φ (h is defined if f ∈ C_{c}^{∞}(Y_{α})). By Proposition 4.25,

T (h, μ_{1}, μ_{2}, t) = M(μ_{2}, μ_{1}, t^{−1})T (f, μ_{2}, μ_{1}, t^{−1}),
T (f, μ_{2}, μ_{1}, t^{−1}) = M(μ_{1}, μ_{2}, t)T (φ, μ_{1}, μ_{2}, t).

The first equation holds only for large enough t, and the second only for small enough t. However, both sides of the second equality depend rationally on t (for the left side, this is true since f = Mφ is compactly supported), hence it holds for all t in ℂ^{×}. Hence for large enough t, by Proposition 4.24 T (h, μ_{1}, μ_{2}, t) = T (φ, μ_{1}, μ_{2}, t) for all μ_{1}, μ_{2}. This implies h = φ.

4.6 Poles, zeroes and values of R and M

Recall that ν_{t}(x) = t^{deg(x)} is a character of 𝔸^{×}/F^{×} with ν_{t}(π π π_{v}) = t_{v} (= t^{deg(v)}), and locally we write ν_{t} for the unramified character of F_{v}^{×} with ν_{t}(π π π_{v}) = t. Let μ_{1}, μ_{2} be characters of F_{v}^{×}. Recall:

R(μ_{1}, μ_{2}, t) = \frac{L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2})}{L(μ_{1}/μ_{2}, t^{2})}M(μ_{1}, μ_{2}, t).

Proposition 4.27. (1) The function R(μ_{1}, μ_{2}, t) is regular at t = 0. It has a pole at τ ∈ ℂ^{×} iff μ_{2}ν_{τ^{−1}}/μ_{1}ν_{τ} = ν (with ν(π π π_{v}) = q_{v}^{−1}). This pole has order 1. The function R(μ_{1}, μ_{2}, t)^{−1} has a pole at τ ∈ ℂ^{×} iff μ_{1}ν_{τ} /μ_{2}ν_{τ^{−1}} = ν. This pole has order 1.

(2) Suppose R(μ_{1}, μ_{2}, t)^{−1} has a pole at τ ∈ ℂ^{×}. Then the function R(μ_{1}, μ_{2}, t) is regular at t = τ. Put L = lim_{t→τ} (t − τ)R(μ_{1}, μ_{2}, t)^{−1} and Q = R(μ_{1}, μ_{2}, τ). The operators Q : I(μ_{1}ν_{τ} , μ_{2}ν_{τ^{−1}}) → I(μ_{2}ν_{τ^{−1}}, μ_{1}ν_{τ}) and L : I(μ_{2}ν_{τ^{−1}}, μ_{1}ν_{τ}) → I(μ_{1}ν_{τ} , μ_{2}ν_{τ^{−1}}) intertwine the GL(2, F_{v})-action. The representations of GL(2, F_{v}) in the spaces ker Q, coker Q, im L are isomorphic to the square integrable St(μ_{1}ν_{τ} , μ_{2}ν_{τ^{−1}}). The representations of GL(2, F_{v}) in the spaces ker L, coker L, im Q are isomorphic to the one dimensional x ↦ μ_{2}(x)(νν_{τ^{−1}})(x) = μ_{1}(x)ν_{τ} (x).

(3) The statement (2) remains true with R(μ_{1}, μ_{2}, t) replaced by R(μ_{1}, μ_{2}, t)^{−1}.

Proof. From the first part of the proof of Proposition 4.3 it follows that

M(μ_{1}, μ_{2}, t)/L(μ_{1}/μ_{2}, t^{2}) = R(μ_{1}, μ_{2}, t)/L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2})

is regular. So R(μ_{1}, μ_{2}, t) could have a pole at t ∈ ℂ^{×} only if L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2}) is ∞, that is μ_{2}ν_{τ^{−1}}/μ_{1}ν_{τ} = ν (recall: ν(x) = |x|), and the order of the pole is at most 1. A similar statement holds for R(μ_{1}, μ_{2}, t)^{−1} = c(μ_{1}, μ_{2})t^{n(μ_{1},μ_{2})}R(μ_{2}, μ_{1}, t^{−1}). (The last equality follows from Proposition 4.22. In fact n(μ_{1}, μ_{2}) = 0, but we do not need this.) Namely R(μ_{1}, μ_{2}, t)^{−1} has a pole at τ ∈ ℂ^{×} iff μ_{1}ν_{τ} /μ_{2}ν_{τ^{−1}} = ν. This pole has order 1. Suppose μ_{1}ν_{τ} /μ_{2}ν_{τ^{−1}} = ν. Then μ_{2}ν_{τ^{−1}}/μ_{1}ν_{τ} ≠ ν so that R(μ_{1}, μ_{2}, t)^{−1} is regular at t = τ. With L, Q defined as in the proposition, it is clear they commute with the GL(2, F_{v})-action. If L = 0 then Q = R(μ_{1}, μ_{2}, τ) has no pole, in fact it is an isomorphism. If Q = 0 then L would be an isomorphism, as the operator lim_{t→τ} R(μ_{1}, μ_{2}, t)/(t − τ) would be the inverse of L. However, the representations of GL(2, F_{v}) in I(μ_{1}ν_{τ} , μ_{2}ν_{τ^{−1}}) and I(μ_{2}ν_{τ^{−1}}, μ_{1}ν_{τ}) are not equivalent, hence L ≠ 0, Q ≠ 0. As L ≠ 0, the function R(μ_{1}, μ_{2}, t)^{−1} does have a pole at t = τ. From the description of the invariant subspaces of I(μ_{1}ν_{τ} , μ_{2}ν_{τ^{−1}}) and I(μ_{2}ν_{τ^{−1}}, μ_{1}ν_{τ}) the claims in the proposition on the description of the action of GL(2, F_{v}) follow. The regularity of R(μ_{1}, μ_{2}, t) at t = 0 follows from that of L(μ_{1}/μ_{2}, q_{v}^{−1}t^{2})^{−1}R(μ_{1}, μ_{2}, t).

In conclusion, the representation of GL(2, F_{v}) in I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) is reducible iff R(μ_{1}, μ_{2}, t) or R(μ_{1}, μ_{2}, t)^{−1} has a pole at t = τ. These last operators are regular at t ∈ ℂ^{×} if μ_{1}/μ_{2} is ramified. If μ_{1}/μ_{2} is unramified and (μ_{1}/μ_{2})(π π π_{v}) = a, then the poles of R(μ_{1}, μ_{2}, t) are at ±\sqrt{q_{v}/a}, and those of R(μ_{1}, μ_{2}, t)^{−1} are at ±\sqrt{a/q_{v}}.

Corollary 4.28. Let μ_{1}, μ_{2} be characters of 𝔸^{×}/F^{×} · α^{ℤ}. If R(μ_{1}, μ_{2}, t) has a pole at t = τ ∈ ℂ^{×}, then |τ| = \sqrt{q}. If R(μ_{1}, μ_{2}, t)^{−1} has a pole at t = τ ∈ ℂ^{×} then |τ| = q^{−1/2}.

Indeed, a character of 𝔸^{×}/F^{×} which takes the value 1 at α is unitary, thus |a| = 1.

Proposition 4.29. Let μ_{1}, μ_{2} be characters of 𝔸^{×}/F^{×} · α^{ℤ} and τ ∈ ℂ^{×}, |τ| ≤ 1. If M(μ_{1}, μ_{2}, t) has a pole at t = τ then μ_{1} = μ_{2} and τ = ±q^{−1/2}. If μ_{1} = μ_{2} is denoted μ and τ = ±q^{−1/2} then M(μ, μ, t) has an order 1 pole at τ. The image of the operator C = lim_{t→τ} (t − τ)M(μ, μ, t) in this case is one dimensional and is spanned by the function f(x) = μ(det x)ν_{τ} (det x) in I(μν_{τ^{−1}}, μν_{τ}). Further, M(μ_{1}, μ_{2}, t) is regular at t = 0.

Proof. Recall that M(μ_{1}, μ_{2}, t) = q^{1−g}m(μ_{1}/μ_{2}, t^{2})R(μ_{1}, μ_{2}, t) where m(μ, t) = L(μ, t)/L(μ, t/q). Let τ ∈ ℂ^{×}, |τ| ≤ 1. By Corollary 4.28, the function R(μ_{1}, μ_{2}, t) is regular at τ. By Proposition 4.11, the function m(μ_{1}/μ_{2}, t^{2}) is not regular at τ only if μ_{1} = μ_{2} and τ = ±q^{−1/2}. In these cases it has a simple pole. Hence M(μ_{1}, μ_{2}, t) is regular at t = τ (0 < |τ| ≤ 1) unless μ_{1} = μ_{2} and τ = ±q^{−1/2} where the order of the pole is at most 1. When μ_{1} = μ_{2} = μ and τ = ±q^{−1/2}, the operator C = lim_{t→τ} (t − τ)M(μ, μ, t) is a scalar multiple of R(μ, μ, t) = ⊗_{v}R(μ_{v}, μ_{v}, τ_{v}), τ_{v} = τ^{deg(v)}. From (1) in Proposition 4.27, the function R(μ_{v}, μ_{v}, τ_{v})^{−1} has a pole at t = τ (t_{v} = τ_{v}). Its statement (2) implies that the image of R(μ_{v}, μ_{v}, τ_{v}) is one dimensional and GL(2, F_{v}) acts on it via the character x ↦ μ_{v}(det x)ν_{τ} (det x)^{deg v}. This implies the proposition, except the final claim, which follows from the regularily of R(μ_{1}, μ_{2}, t) at t = 0, and that of m(μ_{1}/μ_{2}, t^{2}) at t = 0.

Let μ_{1}, μ_{2} be characters of 𝔸^{×}/F^{×}. The operator M(μ_{1}, μ_{2}, t) maps I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) into the space I(μ_{2}ν_{t^{−1}}, μ_{1}ν_{t}), which in general is different from I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}). However, when μ_{1} = μ_{2} = μ and t = ±1, then M(μ_{1}, μ_{2}, t) maps I(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}) to itself; M(μ, μ, t) is regular at t = ±1. The representation of GL(2, 𝔸) in I(μν_{τ} , μν_{τ^{−1}}), τ = ±1, is irreducible, and hence M(μ, μ, τ) is a scalar operator. Moreover, from Proposition 4.26, M(μ, μ, τ)^{2} = 1 at τ = ±1.

Proposition 4.30. If μ is a character of 𝔸^{×}/F^{×} and τ = ±1, then M(μ, μ, τ) = −1.

Proof. In view of the relation between M and R, it suffices to verify that

lim_{t→1} \frac{L(1, t)}{L(1, t/q)} = −q^{g−1}  and  R(μ, μ, τ) = 1.

In fact, for any character ω of F_{v}^{×}, R(ω, ω, τ) is 1 at τ = ±1. Indeed, suppose first ω is unramified. Then there exists a function f in I(ων_{τ} , ων_{τ}) whose restriction to GL(2, O_{v}) is 1. By the normalization of the intertwining operator (Proposition 4.3(2)), R(ω, ω, τ)f = f. However, the representation of GL(2, F_{v}) on I(ων_{τ} , ων_{τ}) is irreducible, so R(ω, ω, τ) = 1 if ω is unramified. The general case reduces to the case where ω is unramified, or even ω = 1, by the commutativity of the diagram

I(ων_{τ} ,ων_{τ}) ⟶^{R(ω,ω,τ)} I(ων_{τ} ,ων_{τ})
↑  ↑
I(ν_{τ} ,ν_{τ})⊗ω ⟶^{R(1,1,τ)} I(ν_{τ} ,ν_{τ})⊗ω

To compute the limit of the ratio of L-functions, we use the functional equation L(1, t/q) = ε(1, t/q)L(1, t^{−1}). Then

lim_{t→1} L(1, t)/L(1, t/q) = ε(1, 1/q)^{−1} lim_{t→1} L(1, t)/L(1, t^{−1}).

By the definition of the global ε-function and its properties (Proposition 6.1, 6.3), ε(1, 1/q) = q^{1−g}. Since L(1, t) has a pole of order one at t = 1, by L’Hôpital rule lim_{t→1} L(1, t)/L(1, t^{−1}) is −1.

4.7 Global Eisenstein approach

These proofs of M^{2} = 1 and rationality of M(μ_{1}, μ_{2}, t) are based on local computations (normalization of the intertwining operators by L-functions and ε-factors), and the functional equation of the L-function. The following alternative proof of these results is based on properties of the Eisenstein map. The alternative approach of this subsection, the following subsection 4.8, and the computation of traces in subsection 5.2 are motivated by Tate [T68]. They are the newest part of this paper, which – as noted in the introduction – cries out for generalization from our context of GL(2), and for further study. We shall use the maps ht^{+} : Y_{α} → ℤ and ht : α^{ℤ} GL(2, F)\ GL(2, 𝔸) → ℤ. Both maps are proper. However, ht^{+} is onto while the image of ht contains the positive integers but only finitely many negatives. So in some sense Y_{α} is less compact than α^{ℤ} GL(2, F)\ GL(2, 𝔸), so the map E : C_{c}^{∞}(Y_{α}) → C_{c}^{∞}(α^{ℤ} GL(2, F)\ GL(2, 𝔸)) should have a big kernel. For φ in ker E we have (1 + M)φ = E^{∗}Eφ = 0. Hence M^{2}φ = φ. Unlike M, the operator M^{2} commutes with the action of A(𝔸) on C_{c}^{∞}(Y_{α}) by left translation. Hence M^{2}φ = φ not only for φ ∈ ker E but also for φ in the span of A(𝔸)-translates of φ in ker E. The number of such linear combinations is already sufficiently large to imply M^{2} = 1. We now turn to rigorous proofs.

Proposition 4.31. Let M : ℂ[z, z^{−1}]^{n} → ℂ((z))^{n} be a ℂ-linear map with M(zu) = z^{−1}M(u) for all u ∈ ℂ[z, z^{−1}]^{n}. Let I denote the natural embedding ℂ[z, z^{−1}]^{n} ↪ ℂ((z))^{n}. Put B = I + M. Suppose there is some k ∈ ℤ for which the vector space (Im B)/B(z^{k}ℂ[z^{−1}]^{n}) is finite dimensional. Then there is some

P (z) ∈ GL(n, ℂ(z)) ⊂ GL(n, ℂ((z)))

with P (z^{−1}) = P (z)^{−1} and (Mu)(z) = P (z)u(z^{−1}) for all u(z) ∈ ℂ[z, z^{−1}]^{n}.

Proof. Denote by e_{i} the column in ℂ^{n} with nonzero entry only at the ith row, where it is 1. From M(∑_{i}(∑_{j} c_{ij}z^{j})e_{i}) = ∑_{i}(∑_{j} c_{ij}z^{−j})Me_{i}, we see that (Mu)(z) = P (z)u(z^{−1}) where P (z) is the n × n matrix with columns Me_{1}, …, Me_{n} whose entries are in ℂ((z)). If u is in the kernel of B = I + M, then P (z)u(z^{−1}) = −u(z). Since Im B = ∪_{m≥1}B(z^{m}ℂ[z^{−1}]^{n}) and there is some k ≥ 0 such that B(z^{k}ℂ[z^{−1}]^{n}) has finite codimension in Im B, there is some ℓ with B(z^{ℓ}ℂ[z^{−1}]^{n}) = Im B. Then ker B + z^{ℓ}ℂ[z^{−1}]^{n} = ℂ[z, z^{−1}]^{n}. For each i (1 ≤ i ≤ n), z^{ℓ+1}e_{i} ∈ ker B + z^{ℓ}ℂ[z^{−1}]^{n}. Hence there is a matrix W ∈ M(n, ℂ[z, z^{−1}]) whose columnes are in ker B and W −z^{ℓ+1} Id ∈ z^{ℓ}M(n, ℂ[z^{−1}]), where Id is the identity matrix. But then W ∈ GL(n, ℂ(z)), and since the columns of W are in ker B, we have P (z)W (z^{−1}) = −W (z). Then P (z) = −W (z)W (z^{−1})^{−1}, and P (z^{−1}) = −W (z^{−1})W (z)^{−1} = P (z)^{−1}.

Corollary 4.32. A ℂ-linear map M : ℂ[z, z^{−1}] → ℂ[z, z^{−1}] which satisfies the conditions of Proposition 4.31 has M^{2} = Id.

Recall that Y_{α} = α^{ℤ}A(F)N(𝔸)\ GL(2, 𝔸). Write C_{+}^{∞}(Y_{α}) for the space of the E-valued functions f on Y_{α} with (1) f(x) = 0 if ht^{+}(x) is large enough, and (2) f is invariant under right translation by some open subgroup U of GL(2, 𝔸). Note that C_{c}^{∞}(Y_{α}) ⊂ C_{+}^{∞}(Y_{α}) ⊂ C^{∞}(Y_{α}).

Proposition 4.33. The image of C_{c}^{∞}(Y_{α}) under M lies in C_{+}^{∞}(Y_{α}).

Proof. For f ∈ C_{c}^{∞}(Y_{α}) there exists an integer m such that f(x) = 0 if ht^{+}(x) < −m. We shall show that for such f, (Mf)(x) = ∫_{N(𝔸)} f (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dx is zero if ht^{+}(x) > m. It suffices to show then that for x ∈ GL(2, 𝔸) with ht^{+}(x) > m, and any n ∈ N(𝔸), we have ht^{+} (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) < −m. But by Lemma 2.7 we have

ht^{+}(x) + ht^{+} (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) = ht^{+}(nx) + ht^{+} (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) ≤ 0.

Proposition 4.34. Let U be an open subgroup of GL(2, O). For every integer m ≥ 1 define

W_{m}^{U} = {φ ∈ C_{c}^{∞}(Y_{α})^{U}; φ(x) = 0 if ht^{+}(x) < m},
Y_{m}^{U} = {φ ∈ C_{c}^{∞}(α^{ℤ} · GL(2, F)\ GL(2, 𝔸))^{U}; φ(x) = 0 if ht^{+}(x) < m}.

Then E(W_{m}^{U}) = Y_{m}^{U} for large enough m.

Proof. Put

Z_{m}^{U} = {φ ∈ C_{c}^{∞}(α^{ℤ} · A(F)N(F)\ GL(2, 𝔸))^{U}; φ(x) = 0 if ht^{+}(x) < m}.

Recall that

E = s_{∗}r^{∗},  s_{∗}(x) = ∑_{γ} ψ(γx),  γ ∈ A(F)N(F)\ GL(2, F).

It is clear that s_{∗}(Z_{m}^{U}) = Y_{m}^{U}. It suffices to show that r^{∗}(W_{m}^{U}) = Z_{m}^{U} for sufficiently large m. In fact, we showed, as the first claim in the proof of Proposition 2.13, that for an open subgroup U of GL(2, 𝔸), that there is an integer m with the property that if z ∈ 𝔸, x ∈ GL(2, 𝔸), ht^{+}(x) ≥ m, then there is u ∈ U, β ∈ F, with ( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) x = ( \begin{matrix}1 & β \\ 0 & 1 \end{matrix} ) xu. In other words, if x ∈ GL(2, 𝔸) and ht^{+}(x) is large enough, then N(𝔸)x ⊂ N(F)xU.

We shall now give a different proof of Proposition 4.26.

Proposition 4.35. If φ ∈ C_{c}^{∞}(Y_{α}) and Mφ ∈ C_{c}^{∞}(Y_{α}) then M^{2}φ = φ.

Proof. Let us introduce a structure of ℂ[z, z^{−1}]-module on C^{∞}(Y_{α}) by

(zf)(x) = \frac{1}{\sqrt{q}} f (( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) x) , f ∈ C^{∞}(Y_{α}), x ∈ GL(2, 𝔸).

From

(Mϕ) (( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) x) = |\frac{a}{b}| ∫_{N(𝔸)} ϕ (( \begin{matrix}b & 0 \\ 0 & a \end{matrix}) ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dn

it follows that M(zf) = z^{−1}M(f); recall that |α| = q, and f is invariant under α. This is the reason for introducing the factor \sqrt{q}. Let U be an open subgroup of GL(2, O). Put

W_{c}^{U} = C_{c}^{∞}(Y_{α})^{U}, W_{+}^{U} = C_{+}^{∞}(Y_{α})^{U}.

Both are ℂ[z, z^{−1}]-submodules in C^{∞}(Y_{α}). Denote by W_{0}^{U} the set of functions f ∈ C^{∞}(Y_{α})^{U} such that f(x) = 0 if ht^{+}(x) ≠ 0. Then the natural map W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}] → W_{c}^{U} is an isomorphism. In the same way we have a canonical isomorphism W_{0}^{U} ⊗_{ℂ} ℂ((z)) → W_{+}^{U}. The operator

M : W_{c} = C_{c}^{∞}(Y_{α}) → W_{+} = C_{+}^{∞}(Y_{α})

maps W_{c}^{U} into W_{+}^{U}. Hence it defines a map

M : W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}] → W_{0}^{U} ⊗_{ℂ} ℂ((z))

satisfying the first condition of Proposition 4.31. It remains to check the second condition of that Proposition. The space W_{m}^{U} can be identified with W_{0}^{U} ⊗_{ℂ} z^{−m}ℂ[z^{−1}], and then the operator B = I + M is just E^{∗}E. Thus it suffices to show that for some m ∈ ℤ, the space E^{∗}E(W_{c}^{U})/E^{∗}E(W_{m}^{U}) is finite dimensional. Since E(W_{m}^{U}) = Y_{m}^{U} for large m, and {x ∈ GL(2, F)\ GL(2, 𝔸); ht(x) ≤ m} is compact mod Z(𝔸), it follows that the subspace

E(W_{m}^{U}) ⊂ C_{c}^{∞}(α^{ℤ} GL(2, F)\ GL(2, 𝔸))^{U}

has finite codimension. Thus M satisfies both conditions of Proposition 4.31, and our claim follows from Corollary 4.32.

To use Proposition 4.31 to give another proof of the rationality of M(μ_{1}, μ_{2}, t), we take a different view of the Mellin transform and the relationship between the operators M and M(μ_{1}, μ_{2}, t). Let I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) be the space of locally constant functions f : GL(2, 𝔸) → ℂ[z, z^{−1}] with

f (( \begin{matrix}a & c \\ 0 & b \end{matrix}) x) = μ_{1}(a)μ_{2}(b)ν_{z}(b/a)|a/b|^{1/2}f(x).

Let I_{+}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) be

I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) ⊗_{ℂ[z,z^{−1}]} ℂ((z)).

The group α^{ℤ} ⊂ GL(2, 𝔸) acts trivially on these I_{c} and I_{+}. We put

I_{c} = ⊕I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}), I_{+} = ⊕I_{+}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}),

where the sums range over all characters μ_{1}, μ_{2} of 𝔸^{×}/F^{×} · α^{ℤ}.

Proposition 4.36. There exists an isomorphism of ℂ((z))-modules I_{+} →^{∼} C_{+}^{∞}(Y_{α}) which is GL(2, 𝔸)-equivariant and maps I_{c} to C_{c}^{∞}(Y_{α}).

Proof. Define a map F : I_{+} → C_{+}^{∞}(Y_{α}) by mapping

φ = {φ_{μ_{1},μ_{2}}} ∈ I_{+}, φ_{μ_{1},μ_{2}} ∈ I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}),

to

(F φ)(x) = constant term of the formal series ∑_{μ_{1},μ_{2}} φ_{μ_{1},μ_{2}}(x) ∈ ℂ((z)),

for any x ∈ GL(2, 𝔸). The map F is well defined, commutes with the actions of ℂ((z)) and GL(2, 𝔸). The inverse of F exists, as follows. If ψ ∈ C_{+}^{∞}(Y_{α}) then F^{−1}(ψ) = {φ_{μ_{1},μ_{2}}} with φ_{μ_{1},μ_{2}} ∈ I_{+}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) given by

φ_{μ_{1},μ_{2}}(x) = ∫_{A(𝔸)/α^{ℤ}·A(F)} ψ(h^{−1}x)η(h)dh,

where

η : A(𝔸) → ℂ((z))^{×}, η(diag(a, b)) = μ_{1}(a)μ_{2}(b)ν_{z}(a/b).

The last integral converges in the field ℂ((z)). A base of the topology is given by z^{n}ℂ[[z]], n > 0. The map F maps I_{c} to C_{c}^{∞}(Y_{α}).

Put I_{0} = ⊕_{μ_{1},μ_{2}}I_{0}(μ_{1}, μ_{2}), with

I_{0}(μ_{1}, μ_{2}) = \{f ∈ C^{∞}(GL(2, O)); f (( \begin{matrix}a & c \\ 0 & b \end{matrix}) x) = μ_{1}(a)μ_{2}(b)f(x)\}.

Denote by M(z) the map I_{0} → I_{0} which takes I_{0}(μ_{1}, μ_{2}) to I_{0}(μ_{2}, μ_{1}) via M(μ_{1}, μ_{2}, z). We use the isomorphism F to identify the spaces I_{+} and C_{+}^{∞}(Y_{α}), as well as I_{c} and C_{c}^{∞}(Y_{α}). The natural isomorphism

I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) →^{∼} I_{0}(μ_{1}, μ_{2}) ⊗_{ℂ} ℂ[z, z^{−1}]

and

I_{+}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}) →^{∼} I_{0}(μ_{1}, μ_{2}) ⊗_{ℂ} ℂ((z))

permit us to identify I_{c} and I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] as well as I_{+} and I_{0} ⊗_{ℂ} ℂ((z)). Thus the map M : C_{c}^{∞}(Y_{α}) → C_{+}^{∞}(Y_{α}) induces an operator

M_{0} : I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] → I_{0} ⊗_{ℂ} ℂ((z)).

Proposition 4.37. Regard the elements of I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] as functions of z with values in I_{0} and the elements of I_{0} ⊗_{ℂ} ℂ((z)) as formal series in z with coefficients in I_{0}. Then for any u ∈ I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] one has (M_{0}u)(z) = M(z)u(z^{−1}), M(z) is viewed as a formal series in z.

Proof. Write ι for the automorphism of ℂ[z, z^{−1}] which maps z to z^{−1}. Given a function f : GL(2, 𝔸) → ℂ((z)), denote by f_{0} the function GL(2, 𝔸) → ℂ such that f_{0}(x) is the constant term of f(x). Define an operator

M′′ : I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] → I_{0} ⊗_{ℂ} ℂ((z))  by  (M′′u)(z) = M(z)u(z^{−1}).

We claim that M_{0} = M′′. Consider M′′ as a map I_{c} → I_{+}. We have to show that for every f ∈ I_{c}, we have F M′′f = MF f, for the isomorphism F : I_{+} →^{∼} C_{+}^{∞}(Y_{α}). As I_{c} is the sum over μ_{1}, μ_{2} of I_{c}(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}), it suffices to consider f in one of these summands.

For x ∈ GL(2, 𝔸), we have (M′′f)(x) = ∫_{N(𝔸)} ιf (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dn. Then

(F M′′f)(x) = (M′′f)_{0}(x) = ∫_{N(𝔸)} f_{0} (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dn

(MF f)(x) = ∫_{N(𝔸)} F f (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dn = ∫_{N(𝔸)} f_{0} (( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) nx) dn

are equal, as required.

4.8 A second proof of the rationality of M(μ_{1}, μ_{2}, t) and of the functional equation M(μ_{1}, μ_{2}, t)M(μ_{2}, μ_{1}, t^{−1}) = 1

Let U, W^{U}, A be as in the proof of Proposition 4.35. Then W^{U} = ⊕_{μ_{1},μ_{2}}W_{μ_{1},μ_{2}}^{U}, where W_{μ_{1},μ_{2}}^{U} is the space of functions f ∈ W^{U} with

f (( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) x) = μ_{1}(a)^{−1}μ_{2}(b)^{−1}f(x)

whenever deg(a) = deg(b) = 0. The natural maps I_{0}(μ_{2}, μ_{1})^{U} →^{∼} W_{μ_{1},μ_{2}}^{U} permit one to identify W^{U} and the space I_{0}^{U}. The map

M : W^{U} ⊗_{ℂ} ℂ[z, z^{−1}] → W^{U} ⊗_{ℂ} ℂ((z))

is induced by the operator

M_{0} : I_{0} ⊗_{ℂ} ℂ[z, z^{−1}] → I_{0} ⊗_{ℂ} ℂ((z)).

The proof of Proposition 4.35 implies that the operator M satisfies the conditions of Proposition 4.31. Then M is given by a formula of the form (Mu)(z) = P (z)u(z^{−1}), where P (z) is an automorphism of V which depends on z rationally, and P (z^{−1}) = P (z)^{−1}. From Proposition 4.37 it follows that P (z) is just the restriction of M(z) to I_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}]. The group U may be arbitrarily small. Hence M(z) is a rational function of z, and M(z)M(z^{−1}) = 1. Hence for any characters μ_{1}, μ_{2}, of α^{ℤ} · F^{×}\𝔸^{×}, the operator M(μ_{1}, μ_{2}, z) depends rationally on z, and

M(μ_{1}, μ_{2}, z)M(μ_{1}, μ_{2}, z^{−1}) = 1.

The same is true for any characters μ_{1}, μ_{2} of 𝔸^{×}/F^{×}, which are not necessarily trivial at α. To see this, it suffices to use the identities M(μ_{1}ν_{t}, μ_{2}ν_{t}, z) = M(μ_{1}, μ_{2}, z) and M(μ_{1}ν_{t}, μ_{2}ν_{t^{−1}}, z) = M(μ_{1}, μ_{2}, tz). □

5 Proof of the trace formula

5.1 The geometric part

Our aim is to compute the trace tr r_{0}(f), where f ∈ C_{c}^{∞}(GL(2, 𝔸)) and r_{0} is the representation of GL(2, 𝔸) by right translation on the space A_{0,α} of cusp forms invariant under α. Recall that the space A_{c,α} of α-invariant automorphic forms is equal to the direct sum of A_{0,α} and A_{E,α} = Im(E : C_{c}^{∞}(Y_{α}) → A_{c,α}). The corresponding representations of GL(2, 𝔸) are denoted by r and r_{E}. Had r been admissible, we would have had tr r_{0}(f) = tr r(f) − tr r_{E}(f), and the computation of tr r_{0}(f) would have reduced to that of tr r(f) and tr r_{E}(f). But r and r_{E} are not admissible, so tr r(f) and tr r_{E}(f) make no sense. Suppose f is right invariant under the open subgroup U of GL(2, O). Denote by A_{0}^{U}, A_{c}^{U}, A_{E}^{U} the spaces of U-invariant vectors in A_{0,α}, A_{c,α}, A_{E,α}. Since Im r_{0}(f) ⊂ A_{0}^{U}, we have tr r_{0}(f) = tr r_{0}^{U}(f), where r_{0}^{U}(f) is the restriction of r_{0}(f) to A_{0}^{U}. Denote by χ_{m} the characteristic function of the set

{x ∈ α^{ℤ} · GL(2, F)\ GL(2, 𝔸); ht(x) < m}, m > 0.

Denote by θ_{m} the operator of multiplication by χ_{m} on A_{c,α}.

Proposition 5.1. (1) For any m > 0, dim θ_{m}(A_{c}^{U}) < ∞.

(2) If m >> 1 then (a) θ_{m} acts as the identity on A_{0}^{U}, and (b) θ_{m}(A_{E}^{U}) ⊂ A_{E}^{U}.

Proof. (1) The support of χ_{m} is compact mod Z(𝔸), the quotient by the open U is then finite. (2a) A_{0}^{U} is finite dimensional, consisting of compactly supported forms. (2b) By (2a), (1 − θ_{m})A_{E}^{U} = (1 − θ_{m})A_{c}^{U}. This lies in A_{E}^{U} as U-invariant cusp forms are uniformly compactly supported. Hence θ_{m}(A_{E}^{U}) ⊂ A_{E}^{U}.

Denote by r^{U}(f) and r_{E}^{U}(f) the restrictions of r(f) to A_{c}^{U} and A_{E}^{U}. For m such that θ_{m}(A_{E}^{U}) ⊂ A_{E}^{U}, denote the restriction of θ_{m} to A_{E}^{U} again by θ_{m}. Then for m >> 1,

tr r_{0}(f) = tr r_{0}^{U}(f) = tr(θ_{m}r^{U}(f)) − tr(θ_{m}r_{E}^{U}(f)) = tr(θ_{m}r(f)) − tr(θ_{m}r_{E}^{U}(f)).

We then proceed to compute tr(θ_{m}r(f)) and tr(θ_{m}r_{E}^{U}(f)).

Proposition 5.2. There exist c_{f} ∈ E and α_{m} ∈ E with lim_{n→∞} α_{m} = 0, and

tr(θ_{m}r(f)) = ∑_{1≤i≤4} S_{i}(f) + c_{f}(m − \frac{1}{2}) + α_{m}.

Proof. The map θ_{m}r(f) : A_{c,α} → A_{c,α} is an integral operator with kernel χ_{m}(y)K_{f}(x, y), where K_{f}(x, y) = ∑_{γ∈α^{ℤ}·GL(2,F)} f(x^{−1}γy). Then

tr(θ_{m}r(f)) = ∫_{α^{ℤ}·GL(2,F)\ GL(2,𝔸)} χ_{m}(x)K_{f}(x, x)dx.

Lemma 5.3. There exists m_{f} > 0 such that if x ∈ GL(2, 𝔸), γ ∈ α^{ℤ} GL(2, F), ht^{+}(x) > m_{f}, f(x^{−1}γx) ≠ 0, then γ ∈ α^{ℤ}A(F)N(F).

Proof. We have γx = xy, y in supp(f). Since ht^{+}(x) + ht^{+}(δx) ≤ 0 for δ ∈ GL(2, F) − B(F), we have that ht^{+}(x) > 0. If in addition we had ht^{+}(xy) > 0, we would conclude that γ ∈ α^{ℤ}B(F). The number m_{f} = − min{ht^{+}(z); z ∈ GL(2, O) · supp(f)} then has the property that ht^{+}(x) > m_{f}, y ∈ supp(f), implies ht^{+}(xy) = ht^{+}(x) + ht^{+}(ky) > 0, where x = bk and ky = b′k′ so that xy = bb′k (b, b′ ∈ B(𝔸); k, k′ ∈ GL(2, 𝔸)).

Denote by ξ_{m} the characteristic function of the set {x ∈ GL(2, 𝔸); ht^{+}(x) ≥ m}, by A′(F) the set of nonscalar diagonal matrices, and by Ell the set of elliptic matrices in GL(2, F), namely those whose eigenvalues are not in F.

Put w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ).

Lemma 5.4. If m is big enough, then χ_{m}(y)K_{f}(x, x) is the sum of

T_{1,m}(x) = χ_{m}(x) ∑_{γ∈α^{ℤ}·F^{×}} f(γ), T_{2,m}(x) = ∑_{γ∈α^{ℤ}·Ell} f(x^{−1}γx),

T_{3,m}(x) = \frac{1}{2} ∑_{γ∈α^{ℤ}·A′(F)} ∑_{δ∈A(F)\ GL(2,F)} f(x^{−1}δ^{−1}γδx) · (1 − ξ_{m}(δx) − ξ_{m}(wδx)),

T_{4,m}(x) = ∑_{a∈α^{ℤ}·F^{×}} ∑_{δ∈F^{×}N(F)\ GL(2,F)} f(x^{−1}δ^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) δx) · (1 − ξ_{m}(δx)).

Proof. T_{1,m}(x) is the contribution of the elements γ ∈ α^{ℤ}·F^{×} in χ_{m}(x)K_{f}(x, x). We claim that the contribution of the elements γ ∈ α^{ℤ} · Ell in χ_{m}(x)K_{f}(x, x) is T_{2,m}(x). To show this, we need to see that if x ∈ GL(2, 𝔸), γ ∈ α^{ℤ} · Ell and Φ(x^{−1}γx) ≠ 0, then ht^{+}(x) < m. Indeed, if ht(x) ≥ m then there is some δ ∈ GL(2, F) with ht^{+}(δx) ≥ m. Lemma 5.3 then implies that δγδ^{−1} ∈ α^{ℤ}A(F)N(F), contradicting γ ∈ α^{ℤ} · Ell. Denote by T′_{3,m}(x) the contribution into χ_{m}(x)K_{f}(x, x) of the elements γ of the form α^{j}γ, j ∈ ℤ, γ ∈ GL(2, F) with distinct eigenvalues in F. By T′_{4,m}(x) we denote the contribution of the elements α^{j}γ, j ∈ ℤ, γ ∈ GL(2, F), γ ∉ F^{×} but the eigenvalues of γ are equal. We have

T′_{3,m}(x) = \frac{1}{2}χ_{m}(x) ∑_{γ∈α^{ℤ}·A′(F)} ∑_{δ∈A(F)\ GL(2,F)} f(x^{−1}δ^{−1}γδx).

The factor \frac{1}{2} appears since diag(b, a) is conjugate to diag(a, b). To show that T′_{3,m}(x) = T_{3,m}(x) it suffices to show that when f(x^{−1}δ^{−1}γδx) ≠ 0,

χ_{m}(x) = 1 − ξ_{m}(δx) − ξ_{m}(wδx),

namely if ht(x) ≥ m then either ht^{+}(δx) ≥ m or ht^{+}(wδx) ≥ m. So if ht(x) ≥ m, then there is some η ∈ GL(2, F) with ht^{+}(ηx) ≥ m. By Lemma 5.3, ηδ^{−1}γδη^{−1} ∈ α^{ℤ}A(F)N(F), but this implies that ηδ^{−1} ∈ A(F)N(F) or ηδ^{−1}w ∈ A(F)N(F). Correspondingly, ht^{+}(δx) = ht^{+}(ηx) ≥ m or ht^{+}(wδx) = ht^{+}(ηx) ≥ m, but both inequalities cannot hold simultaneously if m > 0. Now

T′_{4,m}(x) = χ_{m}(x) ∑_{a∈α^{ℤ}·F^{×}} ∑_{δ∈F^{×}N(F)\ GL(2,F)} f(x^{−1}δ^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) δx).

To show that this equals T_{4,m}(x) we need to check that when

f(x^{−1}δ^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) δx) ≠ 0

and ht(x) ≥ m, then ht^{+}(δx) ≥ m. Suppose then that ht^{+}(ηx) ≥ m for η ∈ GL(2, F). Then by Lemma 5.3 we have

ηδ^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) δη^{−1} ∈ α^{ℤ}A(F)N(F).

Hence ηδ^{−1} ∈ A(F)N(F), so that ht^{+}(δx) = ht^{+}(ηx) ≥ m.

We conclude that tr θ_{m}r(f) = ∑_{1≤i≤4} t_{i,m} with

t_{i,m} = ∫_{α^{ℤ}·GL(2,F)\ GL(2,𝔸)} T_{i,m}(x)dx.

To prove the proposition it suffices to show that t_{i,m} = S_{i}(f)+c_{i}(2m−1)+β_{m} for all i (1 ≤ i ≤ 4), where c_{i} does not depend on m and lim β_{m} = 0. It is clear that t_{1,m} → S_{1}(f) as m → ∞. As T_{2,m}(x) is independent of m, t_{2,m} = S_{2}(f). Now

t_{3,m} = \frac{1}{2} ∑_{γ∈α^{ℤ}·A′(F)} ∫_{α^{ℤ}(A(F)\ GL(2,𝔸)} f(x^{−1}γx)(1 − ξ_{m}(x) − ξ_{m}(wx))dx

= \frac{1}{2} ∑_{γ∈α^{ℤ}·A′(F)} ∫_{A(𝔸)\ GL(2,𝔸)} f(x^{−1}γx)s(x)dx

where

s(x) = ∫_{α^{ℤ}A(F)\A(𝔸)}[1 − ξ_{m}(yx) − ξ_{m}(wyx)]dy
= vol{y ∈ α^{ℤ}A(F)\A(𝔸); ht^{+}(yx) < n, ht^{+}(wyx) < n}.

Note that for y ∈ A(𝔸), ht^{+}(yx) = ht^{+}(y)+ht^{+}(x) and ht^{+}(wyx) = ht^{+}(wx)− ht^{+}(y). Hence

s(x) = |{y ∈ A(𝔸)/α^{ℤ} · A(F); ht^{+}(wx) − m < ht^{+}(y) < m − ht^{+}(x)}|.

This is the number of integers between ht^{+}(wx) − m and m − ht^{+}(x). So s(x) = 2m − 1 − ht^{+}(x) − ht^{+}(wx).

Lemma 5.5. We have ht^{+}(x) + ht^{+}(wx) = −2r(x), where if x = a ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) k, a ∈ A(𝔸), k ∈ GL(2, O) and y ∈ 𝔸, we put r(x) = ∑_{v} max(0, log_{q} |y_{v}|_{v}).

Proof. Note that y is determined up to a change y ↦ by + c, b ∈ O^{×}, c ∈ O, so r(x) is well defined. The asserted relation does not change if x is replaced by axk, a ∈ A(𝔸), k ∈ GL(2, O), so we may assume x = ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) ∈ N(𝔸).

Then ht^{+}(x) = 0, and ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}− \frac{1}{y} & 1 \\ 0 & y \end{matrix} ) ( \begin{matrix}1 & 0 \\ \frac{1}{y} & 1 \end{matrix} ) implies that ht^{+}(wx) = −2r(x).

Lemma 5.5 implies that

t_{3,m} = S_{3}(f) + (m − \frac{1}{2}) ∑_{γ∈α^{ℤ}·A′(F)} ∫_{A(𝔸)\ GL(2,𝔸)} f(x^{−1}γx)dx.

Next

t_{4,m} = ∑_{a∈α^{ℤ}·F^{×}} ∫_{α^{ℤ}F^{×}N(F)\ GL(2,𝔸)} f (x^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) x) (1 − ξ_{m}(x))dx

= ∑_{a∈α^{ℤ}·F^{×}} ∫_{\{x∈α^{ℤ}F^{×}N(F)\ GL(2,𝔸); ht^{+}(x)<m\}} f (x^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) x) dx.

Recall that θ_{a,f}(t) = ∫_{α^{ℤ}F^{×}N(F)\ GL(2,𝔸)} f (x^{−1} ( \begin{matrix}a & a \\ 0 & a \end{matrix}) x) t^{ht^{+}(x)}dx is a Laurent

series at t = 0 of a rational function of t with ζ_{F} (q^{−1}t)^{−1}θ_{a,f}(t) ∈ ℂ[t, t^{−1}]. Suppose θ_{a,f}(t) = ∑_{k} u_{k}(a)t^{k}. Then t_{4,m} = ∑_{a∈α^{ℤ}·F^{×}} ∑_{k<m} u_{k}(a). Since ζ_{F} (q^{−1}t) has a simple pole at t = 1, we have that θ_{a,f}(t) = \frac{ρ(a)}{1−t} + \overline{θ}_{a,f}(t), with \overline{θ}_{a,f}(t) without poles on 0 < |t| ≤ 1. Then

θ̃_{a,f}(t) = \frac{1}{2}(θ_{a,f}(t) + θ_{a,f}(t^{−1})) = \frac{1}{2}(\overline{θ}_{a,f}(t) + \overline{θ}_{a,f}(t^{−1})) + \frac{1}{2}ρ(a),

θ̃_{a,f}(1) = \overline{θ}_{a,f}(1) + \frac{1}{2}ρ(a) = \frac{1}{2}ρ(a) + ∑_{k}(u_{k}(a) − ρ(a))

= lim_{m→∞}[ ∑_{k<m} u_{k}(a) − (m − \frac{1}{2})ρ(a)].

Then

t_{4,m} = ∑_{a∈α^{ℤ}·F^{×}} θ̃_{a,f}(1) + (m − \frac{1}{2})ρ(a) + β_{m}, β_{m} → 0 as m → ∞,

and S_{4}(f) = ∑_{a∈α^{ℤ}·F^{×}} θ̃_{a,f}(1). Proposition 5.2 follows. □

Note that β_{m} is 0 for sufficiently large m, as will be seen below.

5.2 The Eisenstein contribution

Next we turn to computing tr(θ_{m}r_{E}^{U}(f)) for large m. Put W_{c}^{U} = C_{c}^{∞}(Y_{α})^{U}, W_{M}^{U} = (1 + M)W_{c}^{U}.

Proposition 5.6. The operator E^{∗} maps A_{E}^{U} isomorphically onto W_{M}^{U}.

Proof. As A_{E}^{U} = E(W_{c}^{U}) and E^{∗}E = 1 + M, it suffices to show that ker E^{∗}E = ker E. For φ ∈ ker E^{∗}E we have (Eφ, Eφ) = (E^{∗}Eφ, φ) = 0, hence Eφ = 0.

Definition 1. Denote by W_{m}^{U} the space of f in W_{c}^{U} with f(x) = 0 if ht^{+}(x) < m. Denote by ξ_{m} also the operator W_{m}^{U} → W_{m}^{U} of multiplication by the characteristic function of the set {x ∈ Y_{α}; ht^{+}(x) ≥ m}. [If m > 0 then ξ_{m} is a left inverse to the operator 1 + M : W_{m}^{U} → W_{M}^{U}. Indeed, if f is in W_{m}^{U}, then (Mf)(x) = 0 already when ht^{+}(x) > −m since ht^{+}(wnx)+ht^{+}(nx) < 0 implies ht^{+}(wnx) < m and so f(wnx) = 0.] Hence π^{m} = (1 + M)ξ_{m} : W_{M}^{U} → W_{M}^{U} satisfies π^{m}π^{m} = π^{m}, for m > 0. Put π_{m} = 1 − π^{m}.

Proposition 5.7. For sufficiently large m, E^{∗} intertwines θ_{m} with π_{m}, thus π_{m}E^{∗} = E^{∗}θ_{m}, namely the diagram

A_{E}^{U} →^{E∗} W_{M}^{U}
θ_{m}↓  ↓π_{m}
A_{E}^{U} →^{E∗} W_{M}^{U}

is commutative.

Proof. Suppose f ∈ A_{E}^{U} and (1 − θ_{m})f = 0. Then f(x) = 0 for x with ht(x) ≥ m. As ξ_{m}(x) ≠ 0 only on x with ht^{+}(x) ≥ m, we have 0 = (1 + M)ξ_{m}E^{∗}f = (1 − π_{m})E^{∗}f, the last equality as 1 − π_{m} = π^{m} = (1 + M)ξ_{m}. For such f we have E^{∗}θ_{m}f = E^{∗}f and π_{m}E^{∗}f = E^{∗}f. If f ∈ A_{E}^{U} and θ_{m}f = 0, then by Proposition 4.34 there is φ ∈ W_{m}^{U} with f = Eφ. Then

π_{m}E^{∗}f = π_{m}E^{∗}Eφ = π_{m}(1 + M)φ = π_{m}(1 + M)ξ_{m}φ = π_{m}π^{m}φ = 0,

hence E^{∗}θ_{m}f = π_{m}E^{∗}f for such f. Any f ∈ A_{E}^{U} can be written as f = f_{1} + f_{2}, f_{1} = (1 − θ_{m})f, f_{2} = θ_{m}f, thus θ_{m}f_{1} = 0 and (1 − θ_{m})f_{2} = 0.

Definition 2. Recall that Y_{α} = α^{ℤ}A(F)N(𝔸)\ GL(2, 𝔸). Denote by σ_{c}, σ_{+}, σ_{M} the representations of GL(2, 𝔸) in the spaces

W_{c} = C_{c}^{∞}(Y_{α}),  W_{+} = C_{+}^{∞}(Y_{α}),  W_{M} = (1 + M)C_{c}^{∞}(Y_{α}).

Consider σ_{c}(f), σ_{+}(f), σ_{M}(f) as operators in the spaces W_{c}^{U}, W_{+}^{U}, W_{M}^{U}.

Corollary 5.8. We have tr(θ_{m} · r_{E}^{U}(f)) = tr(π_{m} · σ_{M}(f)).

Proof. The operator E^{∗} yields an isomorphism of A_{E}^{U} = E(W_{c}^{U}) with W_{M}^{U} intertwining θ_{m} with π_{m}.

In the proof of Proposition 4.35 we introduced a structure of ℂ[z, z^{−1}]-module on W_{c}^{U} and W_{+}^{U}, as well as isomorphisms W_{c}^{U} ≃ W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}] and W_{+}^{U} ≃ W_{0}^{U} ⊗_{ℂ} ℂ((z)), where W_{0}^{U} = {f ∈ W_{c}^{U}; f(x) = 0 if ht^{+}(x) ≠ 0}. Under these isomorphisms, the operator M : W_{c}^{U} → W_{+}^{U} corresponds to the operator

M : W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}] → W_{0}^{U} ⊗_{ℂ} ℂ((z)),

which satisfies the conditions of Proposition 4.31, hence has the form

(Mu)(z) = P (z)u(z^{−1})  for  u ∈ W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}]

which is viewed as a function of z with values in W_{0}^{U}. Here P (z) is a rational function in z with values in Aut W_{0}^{U}, and P (z^{−1}) = P (z)^{−1}. Now σ_{c}(f) is an endomorphism of W_{c}^{U} as a ℂ[z, z^{−1}]-module. The corresponding endomorphism of the module W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}] is determined by a function B(z) in End(W_{0}^{U}) ⊗_{ℂ} ℂ[z, z^{−1}]. The endomorphism of W_{0}^{U} ⊗_{ℂ} ℂ((z)) corresponding to the operator σ_{+}(f) is determined by the same function B(z). The relation

Mσ_{c}(f) = σ_{+}(f)M  becomes  P (z)B(z^{−1})u(z^{−1}) = B(z)P (z)u(z^{−1})

for any u ∈ W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}], thus B(z^{−1}) = P (z)^{−1}B(z)P (z).

Definition 3. Under the isomorphism W_{+}^{U} ≃ W_{0}^{U} ⊗_{ℂ} ℂ((z)), the subspace W_{M}^{U} = (1 + M)W_{c}^{U} is mapped onto the subspace L consisting of all rational functions of the form u(z) + P (z)u(z^{−1}), with u ∈ W_{0}^{U} ⊗_{ℂ} ℂ[z, z^{−1}]. Put

L_{m} = L ∩ (W_{0}^{U} ⊗_{ℂ} z^{−m+1}ℂ[[z]]).

Denote by L^{m} the set of rational functions of the form

u(z) + P (z)u(z^{−1})  with  u ∈ W_{0}^{U} ⊗_{ℂ} z^{−m}ℂ[z^{−1}].

For sufficiently large m we have L = L_{m} ⊕L^{m}. Under the isomorphism W_{M}^{U} →^{∼} L, the operator π_{m} : W_{M}^{U} → W_{M}^{U} corresponds to the idempotent operator L → L with kernel L^{m} and image L_{m}. This projection will also be denoted by π_{m}. Thus tr(π_{m}σ_{M}(f)) = tr(π_{m}B), where B : L → L is the operator of multiplication by B(z). On the left, π_{m} is an operator on W_{M}^{U}, on the right, on L. Fix Q_{1}, Q_{2} ∈ M(k, ℂ[z, z^{−1}]), k ≥ 1, such that det Q_{i} ≠ 0. Suppose the function Q_{2}(z)^{−1}Q_{1}(z) is regular at z = ∞, thus Q_{1}(z) ∈ Q_{2}(z)M(k, ℂ[[z^{−1}]]), and the function Q_{1}(z)^{−1}Q_{2}(z) is regular at z = 0, thus Q_{2}(z) ∈ Q_{1}(z)M(k, ℂ[[z]]). Put R = ℂ[z, z^{−1}]^{k}. For m ≥ 1, put

R_{m} = R ∩ z^{1−m}Q_{1}(z)ℂ[[z]]^{k} ∩ z^{m−1}Q_{2}(z)ℂ[[z^{−1}]]^{k}.

Also put

R_{−}^{m} = z^{−m}Q_{1}(z)ℂ[z^{−1}]^{k} R_{+}^{m} = z^{m}Q_{2}(z)ℂ[z]^{k}.

Then dim R_{m} is finite.

Proposition 5.9. We have R = R_{−}^{m} ⊕ R_{m} ⊕ R_{+}^{m},

R_{m} ⊕ R_{+}^{m} = R ∩ z^{1−m}Q_{1}(z)ℂ[[z]]^{k}

and

R_{m} ⊕ R_{−}^{m} = R ∩ z^{m−1}Q_{2}(z)ℂ[[z^{−1}]]^{k}.

Proof. The natural map φ : R_{−}^{m} → X_{−} = ℂ((z))^{k}/z^{1−m}Q_{1}(z)ℂ[[z]]^{k} is an isomorphism (note that ℂ((z))/z^{1−m}ℂ[[z]] ≃ z^{−m}ℂ[z^{−1}] and Q_{1}(z) is invertible in GL(k, ℂ((z))). The natural map

ψ : R_{+}^{m} → X_{+} = ℂ((z^{−1}))^{k}/z^{m−1}Q_{2}(z)ℂ[[z^{−1}]]^{k}

is then too. The natural map f : R/R_{m} → X_{−} ⊕ X_{+} is injective (by definition of R_{m} as the intersection of R and the denominators of X_{−}, X_{+}) and the composition of the natural map R_{+}^{m} ⊕ R_{−}^{m} → R/R_{m} with f is φ ⊕ ψ.

Definition 4. (1) Denote by pr_{m} : R → R the projection with kernel R_{+}^{m}⊕R_{−}^{m} and image R_{m}. (2) If A(z) is a matrix in M(k, ℂ[z, z^{−1}]), denote by A[z] also the corresponding automorphism of R = ℂ[z, z^{−1}]^{k}. Denote by A_{0} the constant term of A(z).

Proposition 5.10. The trace tr(pr_{m} ·A[z]) is equal to

(2m−1) tr A_{0}−res_{z=0} tr A(z)Q′_{1}(z)Q_{1}(z)^{−1}dz−res_{z=∞} tr A(z)Q′_{2}(z)Q_{2}(z)^{−1}dz.

Proof. Define a projection pr_{+}^{m} : R → R with image R_{+}^{m} and kernel R_{−}^{m} + R_{m}, and a projection pr_{−}^{m} : R → R with image R_{−}^{m} and kernel R_{+}^{m} + R_{m}. Analogously to the decomposition R = R_{−}^{m} ⊕ R_{m} ⊕ R_{+}^{m}, consider the decomposition

R = z^{−m}ℂ[z^{−1}]^{k} ⊕ (z^{1−m}ℂ[z]^{k} ∩ z^{m−1}ℂ[z^{−1}]^{k}) ⊕ z^{m}ℂ[z]^{k},

namely the case where Q_{1} = 1 = Q_{2}. Denote the associated projections by p_{−}^{m}, p_{m}, p_{+}^{m}. Since the space z^{−m}ℂ[z^{−1}]^{k}/R_{−}^{m} ∩ z^{−m}ℂ[z^{−1}]^{k} is finite dimensional, the operator pr_{+}^{m} −p_{+}^{m} has finite rank, and the operator pr_{−}^{m} −p_{−}^{m} has finite rank since z^{m}ℂ[z]^{k}/R_{+}^{m} ∩ z^{m}ℂ[z]^{k} is finite dimensional.

Lemma 5.11. We have

tr(pr_{−}^{m} ·A[z] − p_{−}^{m} · A[z]) =res_{z=0} tr A(z)Q′_{1}(z)Q_{1}(z)^{−1}dz,

as well as

tr(pr_{+}^{m} ·A[z] − p_{+}^{m} · A[z]) = res_{z=∞} tr A(z)Q′_{2}(z)Q_{2}(z)^{−1}dz.

Proof. Denote by Pr_{−}^{m} : ℂ((z))^{k} → ℂ((z))^{k} the projection with image z^{−m}Q_{1}(z)ℂ[z^{−1}]^{k} and kernel z^{1−m}Q_{1}(z)ℂ[[z]]^{k}. Denote by P_{−}^{m} : ℂ((z))^{k} → ℂ((z))^{k} the projection with image z^{−m}ℂ[z^{−1}]^{k} and kernel z^{1−m}ℂ[[z]]^{k} (thus the case of Q_{1} = 1). Denote by A((z)) the endomorphism of ℂ((z))^{k} which is defined by multiplication by A(z). Then Pr_{−}^{m} = Q_{1}((z)) · P_{−}^{m} · Q_{1}((z))^{−1}. Now

Im(Pr_{−}^{m} · A((z)) − P_{−}^{m} · A((z))) ⊂ ℂ[z, z^{−1}]^{k},

and the restriction of the operator

Pr_{−}^{m} · A((z)) − P_{−}^{m} · A((z))  to  ℂ[z, z^{−1}]^{k}  (⊂ ℂ((z))^{k})

is equal to pr_{−}^{m} ·A[z] − p_{−}^{m} · A[z]. Hence

tr(pr_{−}^{m} ·A[z] − p_{−}^{m} · A[z]) = tr(Pr_{−}^{m} · A((z)) − P_{−}^{m} · A((z)))
= tr(Q_{1}((z)) · P_{−}^{m} · Q_{1}((z))^{−1} · A((z)) − P_{−}^{m} · A((z)))

= tr(Q_{1}((z)) · P_{−}^{m} · C((z)) − P_{−}^{m} · Q_{1}((z)) · C((z))), C(z) = Q_{1}(z)^{−1}A(z).

As tr A(z)Q′_{1}(z)Q_{1}(z)^{−1} = tr C(z)Q′_{1}(z), to prove the first claim of the lemma it suffices to show that

tr(Q_{1}((z)) · P_{−}^{m} · C((z)) − P_{−}^{m} · Q_{1}((z))C((z))) = res_{z=0} tr C(z)Q′_{1}(z)dz

for any Q_{1}(z) ∈ M(k, ℂ[z, z^{−1}]), C(z) ∈ M(k, ℂ((z))). By linearity, it suffices to show this when the matrices Q_{1}(z) and C(z) have a single nonzero entry. Thus we may assume k = 1, and that Q_{1}(z) = z^{b}. Thus we need to verify that for any formal power series c(z) = ∑_{d} c_{d}z^{d} in ℂ((z)), we have

tr[(((z^{b})) · P_{−}^{m} − P_{−}^{m} · ((z^{b})))c((z))] = bc_{−b},

where the operations here are in ℂ((z)). The left side is equal to

tr[(((z^{b})) · P_{−}^{m} · ((z^{−b})) − P_{−}^{m}) · ((z^{b}))c((z))] = tr[(P_{−}^{m−b} − P_{−}^{m}) · ((z^{b}))c((z))]

= tr ( \begin{matrix}c_{−b} & c_{−b+1} & ... & c_{−1} \\ c_{−b−1} & c_{−b} & ... & c_{−2} \\ ⋮ & ⋮ & ... & ⋮ \\ c_{1−2b} & c_{2−2b} & ... & c_{−b} \end{matrix} ) = bc_{−b}.

The second claim of the lemma is similarly proven.

As pr_{m} −p_{m} = (1 − pr_{−}^{m} − pr_{+}^{m}) − (1 − p_{−}^{m} − p_{+}^{m}) = (p_{−}^{m} − pr_{−}^{m}) + (p_{+}^{m} − pr_{+}^{m}), Lemma 5.11 implies that tr(pr_{m} ·A[z] − p_{m} · A[z])

= − res_{z=0} tr[A(z)Q′_{1}(z)Q_{1}(z)^{−1}dz] − res_{z=∞} tr[A(z)Q′_{2}(z)A_{2}(z)^{−1}dz].

Since tr(p_{m} · A[z]) = (2m − 1) tr A_{0}, the proposition follows. □

Proposition 5.12. Let ι : ℂ[z, z^{−1}]^{k} → ℂ[z, z^{−1}]^{k} be the involution (ιu)(z) = u(z^{−1}). For sufficiently large m we have 2 tr(ι·pr_{m} ·A[z]) = tr A(1)+tr A(−1).

Proof. Write A(z) = ∑_{k} A_{k}z^{k}, A_{k} ∈ M(k, ℂ). Then tr(ι · p_{m} · A[z]) = ∑_{|i|<m} tr A_{2i}. If m is big enough the right side here is equal to \frac{1}{2}(tr A(1) + tr A(−1)). It remains to show that tr(ι · pr_{m} ·A[z]) = tr(ι · p_{m} · A[z]) for large enough m. As pr_{m} −p_{m} = p_{+}^{m} − pr_{+}^{m} +(p_{−}^{m} − pr_{−}^{m}), it suffices to show that for large enough m

tr(ι · (p_{+}^{m} − pr_{+}^{m}) · A[z]) = 0 = tr(ι · (p_{−}^{m} − pr_{−}^{m}) · A[z]).

Note that pr_{+}^{m} = [z^{m}] pr_{+}^{0}[z^{−m}] and p_{+}^{m} = [z^{m}]p_{+}^{0}[z^{−m}], where as usual [z^{m}] here means the operator of multiplication by z^{m}. The operators pr_{+}^{m} and p_{+}^{m} were defined only for m > 0, but the definition extends to m = 0 so that the two relations above hold. Now

tr(ι · (p_{+}^{m} − pr_{+}^{m}) · A[z]) = tr(ι · [z^{m}](p_{+}^{0} − pr_{+}^{0})[z^{−m}] · A[z])

= tr([z^{−m}]ι · (p_{+}^{0} − pr_{+}^{0})[z^{−m}] · A[z]) = tr(ι · (p_{+}^{0} − pr_{+}^{0})[z^{−m}] · A[z][z^{−m}])

= tr(ι · (p_{+}^{0} − pr_{+}^{0})[z^{−2m}] · A[z]).

Recall that dim V is finite, where V = im[ι(p_{+}^{0} − pr_{+}^{0})]. If m is big enough then

[z^{−2m}] · A[z]V ⊂ z^{−1}ℂ[z^{−1}]^{k} ∩ z^{−1}Q_{2}(z)ℂ[[z^{−1}]]^{k} ⊂ ker p_{+}^{0} ∩ ker pr_{+}^{0} .

Hence tr(ι · (p_{+}^{0} − pr_{+}^{0})[z^{−2m}] · A[z]) is zero. Hence tr(ι(p_{+}^{m} − pr_{+}^{m})A[z]) is zero. The proof of tr(ι(p_{−}^{m} − pr_{−}^{m})A[z]) = 0 for large m is analogous.

Definition 5. Fix P ∈ GL(k, ℂ(z)) such that P (z) is regular at z = 0 and P (z)^{−1} is regular at z = ∞. Put S = ℂ[z, z^{−1}]^{k} + P · ℂ[z, z^{−1}]^{k},

S_{m} = S ∩ z^{1−m}ℂ[[z]]^{k} ∩ z^{m−1}P · ℂ[[z]]^{k}, S^{m} = z^{−m}ℂ[z^{−1}]^{k} + z^{m}P · ℂ[z]^{k}.

Fix B in M(k, ℂ[z, z^{−1}]) such that P^{−1}BP lies in M(k, ℂ[z, z^{−1}]). Then BS ⊂ S. We denote by [B] or B[z] the operator S → S of multiplication by B(z).

Proposition 5.13. We have S = S_{m} ⊕ S^{m}. Denote by ps_{m} : S → S the projection with image S_{m} and kernel S^{m}. Then

tr(ps_{m} ·[B]) = (2m − 1) tr B_{0}
− res_{z=∞} tr[B(z)P′(z)P (z)^{−1}]dz + tr([B]; S/ℂ[z, z^{−1}]^{k}).

Here B_{0} is the constant term of B = B(z), and tr([B]; S/ℂ[z, z^{−1}]^{k}) denotes the trace of the endomorphism of S/ℂ[z, z^{−1}]^{k} induced by multiplication by B(z).

Proof. The space S is a k-dimensional free ℂ[z, z^{−1}]-submodule of ℂ(z)^{k}. Hence there exists a matrix D in GL(k, ℂ(z)) such that S = D·ℂ[z, z^{−1}]^{k}. Since S contains ℂ[z, z^{−1}]^{k}, D^{−1} lies in M(k, ℂ[z, z^{−1}]). Since S contains P · ℂ[z, z^{−1}]^{k} we deduce that D^{−1}P ∈ M(k, ℂ[z, z^{−1}]). Put Q_{1} = D^{−1}, Q_{2} = D^{−1}P. The function Q_{1}(z)^{−1}Q_{2}(z) = P (z) is regular at z = 0. The function Q_{2}(z)^{−1}Q_{1}(z) is regular at z = ∞. Under the isomorphism S ˜→ℂ[z, z^{−1}]^{k}, u ↦ D^{−1}u, the subspaces S_{m} and S^{m} correspond to the subspaces R_{m} and R^{m} of Proposition 5.9. The multiplication [B] : S → S corresponds to [A] : ℂ[z, z^{−1}]^{k} → ℂ[z, z^{−1}]^{k}, A = D^{−1}BD. Then Proposition 5.10 implies the first part of the proposition, as well as the equality

tr(ps_{m} ·B[z]) = (2m − 1) tr A_{0} − res_{z=0} tr A(z)Q′_{1}(z)Q_{1}(z)^{−1}dz
− res_{z=∞} tr A(z)Q′_{2}(z)Q_{2}(z)^{−1}dz.

Here A_{0} is the constant term of A(z). We have

tr(AQ′_{1}Q_{1}^{−1}) = − tr(D^{−1}BD′) = − tr(BD′D^{−1}),

tr(AQ′_{2}Q_{2}^{−1}) = − tr(D^{−1}BP′P^{−1}D − D^{−1}BD′) = tr(BP′P^{−1}) − tr(BD′D^{−1}).

As A = D^{−1}BD, we have tr A = tr B, and tr A_{0} = tr B_{0}. Hence

tr(ps_{m} ·B[z]) = (2m − 1) tr B_{0} − res_{z=∞} tr B(z)P′(z)P (z)^{−1}dz

+ res_{z=0} tr B(z)D′(z)D(z)^{−1}dz + res_{z=∞} tr B(z)D′(z)D(z)^{−1}dz +(2m − 1) tr B_{0} − res_{z=∞} tr B(z)P′(z)P (z)^{−1}dz

− ∑_{ζ∈ℂ^{×}} res_{z=ζ} tr B(z)D′(z)D(z)^{−1}dz.

Lemma 5.14. Suppose T ∈ GL(k, ℂ((z))), C ∈ M(k, ℂ[[z]]) and T^{−1}CT ∈ M(k, ℂ[[z]]). Then res_{z=0} tr C(z)T′(z)T (z)^{−1} = a−b, where a denotes the trace of the operator multiplication by C in the space (ℂ[[z]]^{k} + T ℂ[[z]]^{k})/T ℂ[[z]]^{k}, while b denotes the trace of multiplication by C in the space

(ℂ[[z]]^{k} + T ℂ[[z]]^{k})/ℂ[[z]]^{k}.

Proof. Both sides of the asserted equality do not change if (T, C) is replaced by (UT V, UCU^{−1}) where U, V ∈ GL(k, ℂ[[z]]). We may then assume that T is a diagonal matrix, hence that k = 1. When k = 1 both sides of the asserted relation are simply mC(0), where m is the multiplicity of zero of T (z) at z = 0.

It follows from the lemma that − res_{z=ζ} tr(B(z)D′(z)D(z)^{−1})dz is just the trace of the operator of multiplication by B(z) on the ζ component of the module S/ℂ[z, z^{−1}]^{k}. This, and the equality just before the lemma, implies the proposition. □

Suppose we have P (z^{−1}) = P (z)^{−1}. Replace the assumption P (z)^{−1}B(z)P (z) ∈ M(k, ℂ[z, z^{−1}]) in Proposition 5.13 with the stronger assumption

P (z)^{−1}B(z)P (z) = B(z^{−1}).

Recall that L is the space of all rational functions of the form u(z)+P (z)u(z^{−1}) with u ∈ ℂ[z, z^{−1}]^{m}. In view of the stronger assumption, L is invariant under multiplication by B.

Definition 6. Denote by B_{L} the operator of multiplication by B on L. Put L_{m} = L ∩ z^{1−m}ℂ[[z]]^{k}. Denote by L^{m} the set of rational functions of the form u(z) + P (z)u(z^{−1}) with u(z) ∈ z^{−m}ℂ[z^{−1}]^{k}.

Proposition 5.15. The space L_{m} is finite dimensional, and L = L_{m} ⊕ L^{m}. Denote by π_{m} : L → L the projection with image L_{m} and kernel L^{m}. Suppose the function P (z) is regular at z = ±1. Then for large enough m we have that tr(π_{m}B_{L}) equals

(m − \frac{1}{2}) tr B_{0} − \frac{1}{2} res_{z=∞} tr(B(z)P′(z)P (z)^{−1})dz

+\frac{c}{2} + \frac{1}{4}[tr(B(1)P (1)) + tr(B(−1)P (−1))].

Here B_{0} is the constant term of B(z), and c is the trace of the operator of multiplication by B(z) in the space (ℂ[z, z^{−1}]^{k} + P (z)ℂ[z, z^{−1}]^{k})/ℂ[z, z^{−1}]^{k}.

Proof. Let S, S_{m}, S^{m}, ps_{m}, B be as in Proposition 5.13. From P (z^{−1}) = P (z)^{−1} it follows that if u ∈ S then ũ, given by ũ(z) = P (z)u(z^{−1}), is also in S. Define τ : S → S by τ(u) = ũ. Then τ^{2} = 1, L = {u ∈ S; τ(u) = u}, L_{m} = S_{m} ∩ L, L^{m} = S^{m} ∩ L, and

tr(π_{m}B_{L}) = \frac{1}{2} tr(ps_{m} ·B[z]) + \frac{1}{2} tr(τ · ps_{m} ·B[z]).

The finite dimensionality of S_{m} and Proposition 5.13 then imply that the space L_{m} is finite dimensional, and L = L_{m} ⊕ L^{m}. To deduce the last claim of the proposition from Proposition 5.13, it remains to show that

tr(τ · ps_{m} ·[B]) = \frac{1}{2}(tr(B(1)P (1)) + tr(B(−1)P (−1)))

for large enough m. Let D, Q_{1}, Q_{2} be as in Proposition 5.13. Then under the isomorphism S ˜→ℂ[z, z^{−1}]^{k}, u ↦ D^{−1}u, the operator ps_{m} : S → S translates into the operator pr_{m} (of Proposition 5.9), and multiplication by B : S → S translates into multiplication by

A = D^{−1}BD, ℂ[z, z^{−1}]^{k} → ℂ[z, z^{−1}]^{k}.

The map τ : S → S translates into

[C]ι : ℂ[z, z^{−1}]^{k} → ℂ[z, z^{−1}]^{k}, (ιu)(z) = u(z^{−1}), C(z) = D(z)^{−1}P (z)D(z^{−1}).

Hence

tr(τ · ps_{m} ·B[z]) = tr(C[z]ι · pr_{m} ·A[z]) = tr(ι pr_{m} A[z]C[z]),

which – by Proposition 5.9 – is

\frac{1}{2}(tr A(1)C(1) + tr A(−1)C(−1)) = \frac{1}{2} tr(B(1)P (1) + tr B(−1)P (−1));

note that D(z) is regular at z = ±1, since so is P (z).

If F ∈ M(k, ℂ) and Y ⊂ ℂ^{k} is an F -invariant subspace, write tr(F, Y) for the trace of F on Y.

Proposition 5.16. Fix P (z) ∈ GL(k, ℂ(z)) with P (z^{−1}) = P (z)^{−1}. Suppose that the function P (z) is regular on |z| = 1 and at z = 0, and that it has order 1 at all its poles ζ_{1}, …, ζ_{s} inside {z ∈ ℂ; 0 < |z| < 1}. Denote by Y_{i} the image of the operator lim_{z→ζ_{i}}(z − ζ_{i})P (z) acting on ℂ^{k}. Fix B(z) ∈ M(k, ℂ[z, z^{−1}]) and suppose B_{1}(z) = P (z)^{−1}B(z)P (z) ∈ M(k, ℂ[z, z^{−1}]). Then

tr(ps_{m} ·[B]) = (2m − 1) tr B_{0} + \frac{1}{2πi} ∫_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz

+ ∑_{1≤i≤s} tr(B(ζ_{i}) + B_{1}(ζ_{i}^{−1}), Y_{i}),

with B_{0} being the constant term of B(z). If in addition B_{1}(z) = B(z^{−1}) then

tr(π_{m}B_{L}) = (m − \frac{1}{2}) tr B_{0} + \frac{1}{4πi} ∫_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz

+ ∑_{1≤i≤s} tr(B(ζ_{i}), Y_{i}) + \frac{1}{4}[tr(B(1)P (1)) + tr(B(−1)P (−1))].

Note that the subspace Y_{i} of ℂ^{k} is invariant under B(ζ_{i}) and B_{1}(ζ_{i}^{−1}).

Proof. In view of Propositions 5.13 and 5.15 it suffices to verify that

\frac{1}{2πi} ∮_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz + ∑_{1≤i≤s} tr(B(ζ_{i}) + B_{1}(ζ_{i}^{−1}), Y_{i})

= tr([B], S/ℂ[z, z^{−1}]^{k}) − res_{z=∞} tr B(z)P′(z)P (z)^{−1}dz,

where S = ℂ[z, z^{−1}]^{k} + P (z)ℂ[z, z^{−1}]^{k}.

For any ζ ≠ 0 in ℂ denote by M_{ζ} and N_{ζ} the ζ-components of the ℂ[z, z^{−1}]- modules S/ℂ[z, z^{−1}]^{k} and S/P (z)ℂ[z, z^{−1}]^{k}, respectively. From Cauchy’s formula and Lemma 5.14, it follows that

\frac{1}{2πi} ∮_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz = ∑_{1<|ζ|<∞} tr([B], M_{ζ})

− ∑_{1<|ζ|<∞} tr([B], N_{ζ}) − res_{z=∞} tr(B(z)P′(z)P (z)^{−1})dz.

On the other hand, tr([B], S/ℂ[z, z^{−1}]^{k}) = ∑_{ζ∈ℂ^{×}} tr([B], M_{ζ}). Hence the required identity follows from

∑_{0<|ζ|<1} tr([B], M_{ζ}) = ∑_{1≤i≤s} tr(B(ζ_{i}), Y_{i}),

∑_{1<|ζ|<∞} tr([B], N_{ζ}) = ∑_{1≤i≤s} tr(B_{1}(ζ_{i}^{−1}), Y_{i}).

If P (z) is regular at ζ then M_{ζ} = 0. At each ζ_{i}, P (z) has a pole of order one. Hence there exists isomorphisms M_{ζ_{i}} ˜→Y_{i} which translate the operator [B] : M_{ζ_{i}} → M_{ζ_{i}} to the operator of multiplication by B(ζ_{i}) on Y_{i}. This implies the first identity. For the second identity, for any ζ ∈ ℂ^{×}, denote by \overline{N}_{ζ} the ζ-component of the module (ℂ[z, z^{−1}]^{k} +P (z)^{−1}ℂ[z, z^{−1}]^{k})/ℂ[z, z^{−1}]^{k}. Multiplication by P (z)^{−1} induces an isomorphism N_{ζ} ˜→\overline{N}_{ζ}. Under this isomorphism, multiplication by B : N_{ζ} → N_{ζ} translates into multiplication by B_{1} : \overline{N}_{ζ} → \overline{N}_{ζ}, hence tr([B], N_{ζ}) = tr([B_{1}], \overline{N}_{ζ}). From P (z)^{−1} = P (z^{−1}) we deduce that \overline{N}_{ζ} = 0 if P (z) is regular at z = ζ^{−1}, and that tr([B_{1}], \overline{N}_{ζ_{i}^{−1}}) = tr(B_{1}(ζ_{i}^{−1}), Y_{i}). This implies the second identity, hence the proposition.

5.3 Spectral terms

To deduce the trace formula from Proposition 5.16, we use properties of the function M(μ_{1}, μ_{2}, t). Recall that we have the projection π_{m} : L → L with kernel L^{m} and image L_{m}, and B_{L} denotes the operator of multiplication by B(z) on L. The operator P (z) is the restriction to the subspace of U-invariant vectors of the operator M on the space I_{0} = ⊕I_{0}(μ_{1}, μ_{2}) (μ_{1}, μ_{2} range over the characters of 𝔸^{×}/F^{×} · α^{ℤ}), which maps I_{0}(μ_{1}, μ_{2}) to I_{0}(μ_{2}, μ_{1}) via M(μ_{1}, μ_{2}, z).

Proposition 5.17. There exists a_{f} ∈ ℂ such that for sufficiently large m,

tr(π_{m}B_{L}) = (m − \frac{1}{2})a_{f} − ∑_{5≤i≤8} S_{i}(f).

Proof. By Proposition 4.29 the function P (z) has two poles in the domain |z| ≤ 1, namely at z = ±q^{−1/2}, each of order 1. We have P (z^{−1}) = P (z)^{−1} and P (z)^{−1}B(z)P (z) = B(z^{−1}). Hence the final claim of Proposition 5.16 applies and implies that for large enough m,

tr(π_{m}[B]) = (m− \frac{1}{2}) tr B_{0}+ \frac{1}{4πi} ∮_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz+tr(B(q^{−1/2}), Y_{+})

+ tr(B(−q^{−1/2}), Y_{−}) + \frac{1}{4}[tr(B(1)P (1)) + tr(B(−1)P (−1))].

Here B_{0} is the constant term of B(z) and the image of the operator lim_{z→±q^{−1/2}}(z ∓q^{−1/2})P (z) is denoted by Y_{±}. The proposition follows once we show that

∮_{|z|=1} tr B(z)P′(z)P (z)^{−1}dz = −4πi(S_{5}(f) + S_{6}(f)), (1)

tr(B(q^{−1/2}), Y_{+}) + tr(B(−q^{−1/2}), Y_{−}) = −S_{8}(f), (2)
tr(B(1)P (1)) + tr(B(−1)P (−1)) = −4S_{7}(f). (3)

Denote by r(z) the representation of GL(2, 𝔸) by right translation in I(z) = ⊗_{μ_{1},μ_{2}}I(μ_{1}ν_{z^{−1}}, μ_{2}ν_{z}). Here μ_{1}, μ_{2} are characters of 𝔸^{×}/F^{×}·α^{ℤ}. Let r(z, f) be the convolution operator defined by r(z) and the compactly supported function f in C_{c}^{∞}(GL(2, 𝔸)). Identify, as usual, I(z) to the space I_{0}, and consider r(z, f) as an operator in I_{0}. From Proposition 4.36, B(z) coincides with the restriction of r(z, f) to I_{0}^{U}. Also, P (z) coincides with the restriction of M(z) to I_{0}^{U}. Hence the integral on the left of (1) equals

∮_{|z|=1} tr r(z, f)M′(z)M(z)^{−1}dz

= ∑_{μ_{1},μ_{2}} ∮_{|z|=1} tr I(μ_{2}ν_{z^{−1}}, μ_{1}ν_{z}, f)M′(μ_{1}, μ_{2}, z)M(μ_{1}, μ_{2}, z)^{−1}dz

= ∑_{μ_{1},μ_{2}} ∮_{|z|=1} tr M(μ_{1}, μ_{2}, z)^{−1}I(μ_{2}ν_{z^{−1}}, μ_{1}ν_{z}, f)M′(μ_{1}, μ_{2}, z)dz

= ∑_{μ_{1},μ_{2}} ∮_{|z|=1} tr I(μ_{1}ν_{z}, μ_{2}ν_{z^{−1}}, f)M(μ_{1}, μ_{2}, z)^{−1}M′(μ_{1}, μ_{2}, z)dz.

Then (1) follows from Proposition 4.9. For (2), it follows from Proposition 4.29 that Y_{+} = L^{U}, with L = ⊕L_{μ}, L_{μ} ⊂ I(μ, μ) being generated by the function x ↦ μ(x). The operator r(q^{−1/2}, f) acts in L_{μ} as the operator of multiplication by ∫_{GL(2,𝔸)} f(x)μ(det x)dx. Hence

tr(B(q^{−1/2}), Y_{+}) = tr(r(q^{−1/2}, f), L) = ∑_{μ} ∫_{GL(2,𝔸)} f(x)μ(det x)dx,

where μ ranges over the set of characters of 𝔸^{×}/F^{×} · α^{ℤ}. Similarly

tr(B(q^{−1/2}), Y_{−}) = tr(r(−q^{−1/2}, f), L)

= ∑_{μ} ∫_{GL(2,𝔸)} f(x)μ(det x)ν_{−1}(det x)dx.

Every character of 𝔸^{×} which is trivial on F^{×} · α^{2ℤ} is either trivial on F^{×} · α^{ℤ} or its product with ν_{−1} is, so (2) follows. For (3) note that

tr B(1)P (1) = tr r(1, f)M(1) = ∑_{μ} tr I(μ, μ, f)M(μ, μ, 1) = − ∑_{μ} tr I(μ, μ, f)

by Proposition 4.30. Similarly tr B(−1)P (−1) = − ∑_{μ} tr I(μν_{−1}, μν_{−1}, f).

This completes the proof of the trace formula.

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Yuval Z. Flicker

The Ohio State University Columbus Ohio 43210 USA and University of Ariel Ariel 40700 Israel yzflicker@gmail.com