Hecke Algebra Isomorphisms and

Adelic Points on Algebraic Groups

Gunther Cornelissen and Valentijn Karemaker

Received: May 20, 2016 Revised: January 20, 2017

Communicated by Dan Ciubotaru

Abstract. Let G denote a linear algebraic group over Q and K and L two number fields. We establish conditions on the group G, related to the structure of its Borel groups, under which the existence of a group isomorphism G(A_{K,f}) ≅ G(A_{L,f}) over the finite adeles implies that K and L have isomorphic adele rings. Furthermore, if G satisfies these conditions, K or L is a Galois extension of Q, and G(A_{K,f}) ≅ G(A_{L,f}), then K and L are isomorphic as fields.

We use this result to show that if for two number fields K and L that are Galois over Q, the finite Hecke algebras for GL(n) (for fixed n ≥ 2) are isomorphic by an isometry for the L^{1}-norm, then the fields K and L are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field, if it is Galois over Q.

2010 Mathematics Subject Classification: 11F70, 11R56, 14L10, 20C08, 20G35, 22D20 Keywords and Phrases: algebraic groups, adeles, Hecke algebras, arithmetic equivalence

1. Introduction

Suppose that G is a linear algebraic group over Q, and K and L are two number fields such that the (finite adelic) Hecke algebras for G over K and L are isomorphic. As we will see, a closely related hypothesis is: suppose that the groups of finite adelic points on G are isomorphic for K and L. What does this imply about the relationship between the fields K and L? Before stating the general result, let us discuss an example.

Part of this work was done when the authors visited the University of Warwick; we are grateful to Richard Sharp for hosting us. The second author would like to thank Jan Nekovář for interesting discussions. We thank Manfred Lochter for sending us a copy of [21], Dipendra Prasad for pointing us to existing results on isomorphisms of point groups over rings, and Julia Hartmann, Wilberd van der Kallen, Bart de Smit, Maarten Solleveld, and the referee for helpful comments on an earlier version of this paper.

Example A. If G = G_{a}^{r} × G_{m}^{s} for any integers r, s, then for any two distinct imaginary quadratic fields K and L of discriminant < −8 we have an isomorphism of topological groups G(A_{K,f}) ≅ G(A_{L,f}) while A_{K} ≅ A̸_{L} and A_{K,f} ≅ A̸_{L,f} (cf. Section 2). To prove this, one determines separately the abstract structures of the additive and multiplicative groups of the adele ring A_{K} and sees that they depend on only a few arithmetic invariants, allowing for a lot of freedom in “exchanging local factors”. This example illustrates that at least some condition on the rank, unipotent rank, and action of the torus on the unipotent part will be required to deduce that we have a ring isomorphism A_{K} ≅ A_{L}.

Let us now state the main technical condition, which we will elaborate on in Section 3.

Definition B. Let G denote a linear algebraic group over Q. We call G fertile for a field K/Q if G contains a Borel group B which is split over K as B = T ⋉ U with T ≠ {1} and U ≠ {0}, and such that over K, the split maximal torus T acts non-trivially by conjugation on the abelianisation of the maximal unipotent group U. In Appendix A, it is shown that G being fertile for K is equivalent to G containing a K-split maximal torus and having a connected component which is not a direct product of a torus and a unipotent group.

Split tori and unipotent groups are not fertile for any K. On the other hand, for n ≥ 2, GL(n) is fertile for all K. In general, fertility is slightly stronger than noncommutativity. Roughly speaking, it says that the group has semisimple elements that do not commute with certain unipotent elements.

Our first main result is:

Theorem C. Let K and L be two number fields, and let G denote a linear algebraic group over Q which is fertile for K and L. There is a topological group isomorphism of finite adelic point groups G(A_{K,f}) ≅ G(A_{L,f}) if and only if there is a topological ring isomorphism A_{K} ≅ A_{L}.

An isomorphism of adele rings A_{K} ≅ A_{L} implies (but is generally stronger than) arithmetic equivalence of K and L (Komatsu [20], cf. [18, VI.2]). Recall that K and L are said to be arithmetically equivalent if they have the same Dedekind zeta function: ζ_{K} = ζ_{L} . If K or L is a Galois extension of Q, then this is known to imply that K and L are isomorphic as fields (Gaßmann [9], cf. [18, III.1]). The question whether G(R) ≅ G(S) for algebraic groups G and rings R, S implies a ring isomorphism R ≅ S has been considered before (following seminal work of van der Waerden and Schreier from 1928 [36]), most notably when G = GL_{n} for n ≥ 3 or when G is a Chevalley group and R and S are integral domains (see, e.g., [6], [31] and the references therein). The methods employed there make extensive use of root data and Lie algebras. By contrast, our proof of Theorem C uses number theory in adele rings and, by not passing to Lie algebras, applies to a more general class of (not necessarily reductive) algebraic groups. First, we prove in general that maximal divisible subgroups D of G(A_{K,f}) and maximal unipotent point groups are the same up to conjugacy (Proposition 4.8; note that this does not apply at the archimedean places). The torus T (as a quotient of the normaliser N of the unipotent point group D by itself) acts on the abelian group V = [N, D]/[D, D], that decomposes as a sum of one-dimensional T-modules, on which T acts by multiplication with powers. Now we use a formula of Siegel, which allows us to express any adele as a linear combination of fixed powers, to show how this implies that the centre of the endomorphism ring of the T-module V is a a cartesian power of the finite adele ring. We then use the structure of the maximal principal ideals in the finite adele ring to find from these data the adele ring itself.

Example D. Consider G = GL(2). Then D = ( \begin{matrix}1 & A_{K,f} \\ 0 & 1 \end{matrix} ) ≅ (A_{K,f}, +) is (con-

jugate to) a group of strictly upper triangular matrices, N = ( \begin{matrix}A_{K,f}^{∗} & A_{K,f} \\ 0 & A_{K,f}^{∗} \end{matrix} ) and

T ≅ (A_{K,f}^{∗}, ·)^{2} (represented as diagonal matrices) acts on V ≅ D, represented as upper triangular matrices, by multiplication. Now End_{T}V ≅ A_{K,f} as (topological) rings.

By the (finite adelic) Hecke algebra for G over K, we mean the convolution algebra ℋ_{G}(K) := C_{c}^{∞}(G(A_{K,f}), R) of locally constant compactly supported real-valued functions on G(A_{K,f}). By an L^{1}-isomorphism of Hecke algebras we mean one that respects the L^{1}-norm. The second main result is the following:

Theorem E. Let K and L be two number fields, and let G denote a linear algebraic group over Q that is fertile for K and L. There is an L^{1}-isomorphism of Hecke algebras ℋ_{G}(K) ≅ ℋ_{G}(L) if and only if there is a ring isomorphism A_{K} ≅ A_{L}.

This follows from the previous theorem by using some density results in functional analysis and a theorem on the reconstruction of an isomorphism of groups from an isometry of L^{1}-group algebras due to Wendel [41]. It seems that the analytic condition of being an isometry for the L^{1}-norm is necessary (cf. [16], which uses the Bernstein decomposition to show that purely algebraic isomorphisms of Hecke algebras of local fields cannot distinguish local fields).

Let 𝔊_{K} denote the absolute Galois group of a number field K that is Galois over Q. Neukirch [26] has proven that 𝔊_{K} determines K (Uchida [39] later removed the condition that K is Galois over Q). The set of one-dimensional representations of 𝔊_{K}, i.e., the abelianisation 𝔊_{K}^{ab}, far from determines K (compare [29] or [2]). Several years ago, in connection with the results in [8], Jonathan Rosenberg asked the first author whether, in a suitable sense, two-dimensional irreducible—the “lowestdimensional non-abelian”— representations of 𝔊_{K} determine K. By the philosophy of the global Langlands programme, such representations of 𝔊_{K} in GL(n, C) should give rise to automorphic representations, i.e., to certain modules over the Hecke algebra ℋ_{GL(n)}(K). If we consider the analogue of this question in the setting of ℋ_{GL(n)}(K)-modules instead of n-dimensional Galois representations, our main theorem implies a kind of “automorphic anabelian theorem”:

Corollary F. Suppose that K and L are number fields that are Galois over Q. There is an L^{1}-algebra isomorphism of Hecke algebras ℋ_{GL(n)}(K) ≅ ℋ_{GL(n)}(L) for some n ≥ 2 if and only if there is a field isomorphism K ≅ L.

We now present some open problems:

(1) Is it possible to characterise precisely the linear algebraic groups for which G(A_{K}) ≅ G(A_{L}) implies A_{K} ≅ A_{L}? Is Theorem C true without imposing that a maximal torus T of G splits over K and L?

(2) What happens if G is not a linear algebraic group, but any algebraic group? (It follows from Chevalley’s structure theorem that such G have a unique maximal linear subgroup H; can we deduce H(A_{K}) ≅ H(A_{L}) from G(A_{K}) ≅ G(A_{L}))? What happens if there is no linear part, i.e., G is an abelian variety, e.g., an elliptic curve? For every number field, is there a sufficiently interesting elliptic curve E/Q such that E(A_{K}) determines all localisations of K?

(3) What happens over global fields of positive characteristic?

(4) The category of ℋ_{G}(K)-modules does not seem to determine the field K, cf. [16]. Can the category be enriched in some way so as to determine K? Our theorem suggests to try and keep track of some analytic information about ℋ_{G}(K) related to the L^{1}-norm.

The paper has the following structure. In Section 2, we discuss what happens if G is the additive or multiplicative group or a direct product thereof. In Section 3, we introduce and discuss the notion of fertility. In Section 4, we prove that maximal divisibility is equivalent to unipotency in finite-adelic point groups. In Section 5 we use this to prove Theorem C. In Section 6, we discuss Hecke algebras and prove Theorem E.

2. Additive and multiplicative groups of adeles

In this section, we elaborate on Example A from the introduction. We discuss the group structure of the additive and multiplicative groups of adeles of a number field, and we recall the notions of local isomorphism of number fields and its relation to isomorphism of adele rings and to arithmetic equivalence. We introduce local additive and multiplicative isomorphisms and prove that their existence implies arithmetic equivalence.

Arithmetic equivalence and local isomorphism.

2.1. Notations/Definitions. If K is a number field with ring of integers 𝒪_{K}, let M_{K} denote the set of all places of K, M_{K,f} the set of non-archimedean places of K, and M_{K,∞} the set of archimedean places. If 𝔭 ∈ M_{K,f} is a prime ideal, then K_{𝔭} denotes the completion of K at 𝔭, and 𝒪_{K,𝔭} its ring of integers. Let e(𝔭) and f(𝔭) denote the ramification and residue degrees of 𝔭 over the rational prime p below 𝔭, respectively. The decomposition type of a rational prime p in a field K is the sequence (f(𝔭))_{𝔭|p} of residue degrees of the prime ideals of K above p, in increasing order, with multiplicities. For an archimedean place 𝔭 of K, we have K_{𝔭} = R or C and we let 𝒪_{K,𝔭} = K_{𝔭}.

2.2. Definition. We use the notation ∏′(G_{i}, H_{i}) for the restricted product of the groups G_{i} with respect to the subgroups H_{i}. We denote by

A_{K} = ∏_{𝔭∈M_{K}}′ (K_{𝔭}, 𝒪_{K,𝔭})

the adele ring of K, and by

A_{K,f} = ∏_{𝔭∈M_{K,f}}′ (K_{𝔭}, 𝒪_{K,𝔭})

its ring of finite adeles.

Two number fields K and L are arithmetically equivalent if for all but finitely many prime numbers p, the decomposition types of p in K and L coincide. Two number fields K and L are called locally isomorphic if there is a bijection φ: M_{K,f} → M_{L,f} between their sets of prime ideals such that the corresponding local fields are topologically isomorphic, i.e. K_{𝔭} ≅ L_{φ(𝔭)} for all 𝔭 ∈ M_{K,f}. The Dedekind zeta function of a number field K is defined as

ζ_{K}(s) = ∑_{I⊆𝒪_{K}} (# 𝒪_{K}/I)^{−s},

where the sum ranges over the nonzero ideals of 𝒪_{K}.

The main properties are summarised in the following proposition (see e.g. [18, III.1 and VI.2]):

2.3. Proposition. Let K and L be number fields. Then:

(i) K and L are locally isomorphic if and only if the adele rings A_{K} and A_{L} are isomorphic as topological rings, if and only if the rings of finite adeles A_{K,f} and A_{L,f} are isomorphic as topological rings.

(ii) K and L are arithmetically equivalent if and only if ζ_{K} = ζ_{L}, if and only if there is a bijection φ: M_{K,f} → M_{L,f} such that the local fields K_{𝔭} ≅ L_{φ(𝔭)} are isomorphic for all but finitely many 𝔭 ∈ M_{K,f}.

(iii) We have K ≅ L ⇒ A_{K} ≅ A_{L} (as topological rings) ⇒ ζ_{K} = ζ_{L} and none of the implications can be reversed in general, but if K or L is Galois over Q, then all implications can be reversed. □

The additive group of adeles.

2.4. Proposition. If H is a number field, then there are topological isomorphisms of additive groups

(A_{H,f}, +) ≅ (A_{Q,f}^{[H:Q]}, +)

and

(A_{H}, +) ≅ (A_{Q}^{[H:Q]}, +).

Proof. If 𝔭 is a prime of H above the rational prime p, then 𝒪_{H,𝔭} is a free Z_{p}-module of rank e(𝔭)f(𝔭) (cf. [5, 5.3-5.4]). By tensoring with Q, summing over all 𝔭 | p for fixed p, and using that n := [H : Q] = ∑_{𝔭|p} e(𝔭)f(𝔭), we find that

(A_{H,f}, +) = ∏_{p∈M_{Q,f}}′(⨁_{𝔭|p}(H_{𝔭}, +), ⨁_{𝔭|p}(𝒪_{H,𝔭}, +))

≅ ∏_{p∈M_{Q,f}}′((Q_{p}^{n}, +), (Z_{p}^{n}, +)) ≅ (A_{Q,f}^{n}, +),

and then the second statement follows from (A_{H}, +) = (A_{H,f}, +) × (R^{n}, +). □

2.5. Corollary. The additive groups (A_{K}, +) and (A_{L}, +) are isomorphic (as topological groups) for two number fields K and L if and only if K and L have have the same degree over Q, if and only if the additive groups (A_{K,f}, +) and (A_{L,f}, +) are isomorphic (as topological groups).

Proof. By Proposition 2.4, we know that [K : Q] = [L : Q] implies that (A_{K,f}, +) ≅ (A_{L,f}, +) and (A_{K}, +) ≅ (A_{L}, +). Conversely, a topological isomorphism (A_{K}, +) ≅ (A_{L}, +) of additive groups induces a homeomorphism between their respective connected components of the identity, i.e., R^{[K:Q]} ≅ R^{[L:Q]}. It follows that [K : Q] = [L : Q] ([12, Theorem 2.26]). Finally, let W be a compact open subgroup of (A_{K,f}, +). Locally at each place 𝔭 | p, W_{𝔭} is a free Z_{p}-module of rank [K_{𝔭} : Q_{p}], so W is topologically isomorphic to Ẑ^{[K:Q]}. Hence, for a prime ℓ, we find that W/ℓW is isomorphic to (Z/ℓZ)^{[K:Q]}. The image of W under an isomorphism of topological groups (A_{K,f}, +) ≅ (A_{L,f}, +) has the same property, and we find [K : Q] = [L : Q]. □

If, additionally, the isomorphism is “local”, i.e. induced by local additive isomorphisms, then we have the following result:

2.6. Proposition. Let K and L be number fields such that there is a bijection φ: M_{K,f} → M_{L,f} with, for almost all places 𝔭, isomorphisms of topological groups Φ_{𝔭}: (K_{𝔭}, +) ≅ (L_{φ(𝔭)}, +). Then K and L are arithmetically equivalent.

Proof. A compact open subgroup W of K_{𝔭} is a free Z_{p}-module of rank e(𝔭)f(𝔭). It follows that for any prime number ℓ ≠ p, W/ℓW ≅ {0}, while W/pW ≅ (F_{p}^{e(𝔭)f(𝔭)}, +). Since the same holds for the image of W under Φ_{𝔭}, we conclude that for all non-archimedean places 𝔭 ∈ M_{K,f}, we must have e(𝔭)f(𝔭) = e(φ(𝔭))f(φ(𝔭)). At all but finitely many primes p, both K and L are unramified, so f(𝔭) = f(φ(𝔭)) for all but finitely many residue field degrees f(𝔭). This implies that K and L are arithmetically equivalent. □

The multiplicative group of adeles.

We start by quoting for future reference the following result from [11, Kapitel 15]: let H denote a number field and 𝔭 ∈ M_{H,f}. Let π_{𝔭} be a local uniformiser at 𝔭 and let \overline{H}_{𝔭} = 𝒪_{H,𝔭}/𝔭 denote the residue field; then the unit group of H_{𝔭} is

(1)  𝒪_{H,𝔭}^{∗} ≅ \overline{H}_{𝔭}^{∗} × (1 + π_{𝔭}𝒪_{H,𝔭})

and the one-unit group

(2)  1 + π_{𝔭}𝒪_{H,𝔭} ≅ Z_{p}^{[H_{𝔭}:Q_{p}]} × μ_{p^{∞}}(H_{𝔭})

where μ_{p^{∞}}(H_{𝔭}) is the group of p-th power roots of unity in H_{𝔭}.

2.7. Proposition. If H is a number field with r_{1} real and r_{2} complex places, then there is a topological group isomorphism

(A_{H}^{∗}, ·) ≅ (R^{∗})^{r_{1}} × (C^{∗})^{r_{2}} × (⨁_{Z} Z) × Ẑ^{[H:Q]} × ∏_{𝔭∈M_{H,f}}(\overline{H}_{𝔭}^{∗} × μ_{p^{∞}}(H_{𝔭})).

Proof. We have

A_{H}^{∗} ≅ (R^{∗})^{r_{1}} × (C^{∗})^{r_{2}} × A_{H,f}^{∗} and A_{H,f}^{∗} ≅ J_{H} × ∏_{𝔭∈M_{H,f}} 𝒪_{H,𝔭}^{∗},

where J_{H} is the topologically discrete group of fractional ideals of H, so J_{H} ≅ ⨁_{Z} Z with the index running over the set of prime ideals, and the entry of 𝔫 ∈ J_{H} corresponding to a prime ideal 𝔭 is given by ord_{𝔭}(𝔫). The result follows from (1) and (2). □

It remains to determine the exact structure of the p-th power roots of unity, e.g.:

2.8. Example. [2, Lemma 3.1 and Lemma 3.2] If H ≠ Q(i) and H ≠ Q(\sqrt{−2}), then there is an isomorphism of topological groups

∏_{𝔭∈M_{H,f}}(\overline{H}_{𝔭}^{∗} × μ_{p^{∞}}(H_{𝔭})) ≅ ∏_{n≥1} Z/nZ.

Hence we conclude: If K and L are two imaginary quadratic number fields different from Q(i) and Q(\sqrt{−2}), then we have a topological group isomorphism A_{K}^{∗} ≅ A_{L}^{∗}.

Combining Proposition 2.4 and Example 2.8, we obtain the claim made in Example A in the introduction:

2.9. Corollary. For any two imaginary quadratic number fields K and L different from Q(i) and Q(\sqrt{−2}) and for any integers r and s, there are topological group isomorphisms

(A_{K,f})^{r} × (A_{K,f}^{∗})^{s} ≅ (A_{L,f})^{r} × (A_{L,f}^{∗})^{s}.

and

(A_{K})^{r} × (A_{K}^{∗})^{s} ≅ (A_{L})^{r} × (A_{L}^{∗})^{s}. □

On the other hand, we again have a “local” result:

2.10. Proposition. Let K and L be number fields such that there is a bijection φ: M_{K,f} → M_{L,f} with, for almost all places 𝔭, group isomorphisms Φ_{𝔭}: (K_{𝔭}^{∗}, ·) →^{∼} (L_{φ(𝔭)}^{∗}, ·). Then K and L are arithmetically equivalent.

Proof. From K_{𝔭}^{∗} ≅ Z × 𝒪_{K,𝔭}^{∗} and the isomorphisms (1) and (2), it follows that if we set W = K_{𝔭}^{∗}/K_{𝔭,tors}^{∗}, then for a prime ℓ we obtain

(3)  W/ℓW = \begin{cases}(Z/pZ)^{[K_{𝔭}:Q_{p}]+1} & if ℓ = p \\ Z/ℓZ & if ℓ ≠ p \end{cases} .

It follows from K_{𝔭}^{∗} ≅ L_{φ(𝔭)}^{∗} that 𝔭 and 𝔮 lie above the same rational prime p, and [K_{𝔭} : Q_{p}] = [L_{φ(𝔭)} : Q_{p}]. For all but finitely many primes, the extensions K_{𝔭}/Q_{p} and L_{φ(𝔭)}/Q_{p} are unramified. Hence, we find that the bijection φ matches the decomposition types of all but finitely many primes. This implies that K and L are arithmetically equivalent. □

3. Set-up from algebraic groups and the notion of fertility

In this section, we set up notations and terminology from the theory of algebraic groups, and we elaborate on the notion of a group being fertile for a pair of number fields.

Algebraic groups.

3.1. Background (General references: [3] and [38]). Let G denote a linear (viz., affine) algebraic group over Q. We denote the multiplicative group over Q by G_{m} and the additive group over Q by G_{a}. A torus T is a linear algebraic group which is isomorphic, over \overline{Q}, to G_{m}^{r}, for some integer r, called the rank of T. When T is an algebraic subgroup of G and moreover a maximal torus inside G, then the rank of G is r as well. Suppose that a maximal torus splits over a field F/Q, meaning that there exists an isomorphism T ≅ G_{m}^{r} defined over F. All maximal F -split tori of G are G(F)-conjugate and have the same dimension, called the (F -)rank of G. A subgroup U of G is unipotent if U(\overline{Q}) consists of unipotent elements. Every unipotent subgroup of G/Q splits over Q, meaning that it has a composition series over Q in which every successive quotient is isomorphic to G_{a}. Alternatively, it is isomorphic over Q to a subgroup of a group of strictly upper triangular matrices. Any connected group G that is not unipotent contains a non-trivial torus. A Borel subgroup B of G is a maximal connected solvable subgroup of G. If all successive quotients in the composition series of B over F are isomorphic to G_{a} or G_{m}, then B is conjugate, over F, to a subgroup of an upper triangular matrix group, by the Lie-Kolchin theorem. Moreover, over F, for some split maximal torus T and maximal unipotent group U, we can write B ≅ T ⋉ U as a semi-direct product induced by the adjoint representation ρ: T → Aut(U) (i.e., by the conjugation action of T on U). Furthermore, given U, B is the normaliser of U in G, and T ≅ B/U.

3.2. Definition. We call a linear algebraic group G over Q fertile for a number field K if there exists a Borel K-subgroup B of G which is split over K, meaning that B ≅_{K} T ⋉ U for T a K-split maximal torus and U a maximal unipotent group, such that T ≠ {1} acts non-trivially (by conjugation) on the abelianisation U^{ab} of U ≠ {0}.

Appendix A contains an equivalent definition of fertility.

3.3. Examples.

(i) Tori and unipotent groups are not fertile for any K, and neither are direct product of such groups.

(ii) The general linear group GL(n) for n ≥ 2 is fertile for any K. Here, T is the group of diagonal matrices, split over Q, which acts non-trivially on the group of strictly upper triangular matrices. Similarly, the Borel group of (non-strictly) upper triangular matrices is fertile.

(iii) Let G = Res_{Q}^{F}(G_{m} ⋉ G_{a}) denote the “ax + b”-group of a number field F, as an algebraic group over Q. This group is fertile for any number field K that contains F.

Adelic point groups.

3.4. Definition. Let G denote a linear algebraic group over Q and let K a number field with adele ring A_{K}. As described in Section 3 of [28] (compare [22]) we may use any of the following equivalent definitions for the group of adelic points of G over K (also called the adelic point group), denoted G(A_{K}):

(1) Since A_{K} is a Q-algebra, G(A_{K}) is its scheme theoretic set of points.

(2) Let S be a suitable finite set of places of Q, including the archimedean place, and let 𝒢 be a smooth separated group scheme of finite type over the Sintegers Z_{S}, whose generic fibre is G. Then

G(A_{K}) = lim_{S′⊃S^{⟶}} ∏_{𝔭∈S′} G(K_{𝔭}) × ∏_{𝔭∉S′} 𝒢 (𝒪_{𝔭})

where S′ runs over subsets of M_{K,f} that contain divisors of primes in S.

(3) Choose a Q-isomorphism φ: G ↪ 𝔸^{N} of G onto a closed subvariety of a suitable affine space 𝔸^{N}. For every 𝔭 ∈ M_{K,f}, we define G(𝒪_{𝔭}) to be the set of points x ∈ G(K_{𝔭}) for which φ(x) ∈ 𝒪_{𝔭}^{N}. Then G(A_{K}) is the restricted product

G(A_{K}) = ∏_{𝔭∈M_{K,f}}′ (G(K_{𝔭}), G(𝒪_{𝔭}))) × ∏_{𝔭∈M_{K,∞}} G(K_{𝔭}).

The second and third definitions immediately provide G(A_{K}) with a topology induced from the 𝔭-adic topologies. Also, the group law on G induces a topological group structure on G(A_{K}). The definitions are, up to isomorphism, independent of the choices of S, 𝒢 and φ. We define the finite-adelic point group G(A_{K,f}) completely analogously.

In Section 2, we considered the group of adelic points on G_{a} and G_{m}, in the sense of the above definition, cf. Example A from the introduction.

4. Divisibility and unipotency

In this section, we show (Proposition 4.8) how to characterise maximal unipotent point groups inside finite-adelic point groups in a purely group theoretic fashion, using divisibility. This is used later to deduce an isomorphism of unipotent point groups from an isomorphism of ambient point groups. We first prove a series of results, relating divisibility and unipotency over local and global fields. Throughout, G will be a linear algebraic group over Q, K a number field, and 𝔭 ∈ M_{K,f} a finite place of K.

4.1. Definition. If H is a subgroup of a group G, an element h ∈ H is called divisible in G if for every integer n ∈ Z_{>0}, there exists an element g ∈ G such that h = g^{n}. The subgroup H ≤ G is called divisible (in G) if all of its elements are divisible in G.

4.2. Lemma. All divisible elements of G(K_{𝔭}) are unipotent.

Proof. We fix an embedding G ↪ GL_{N} throughout, and consider elements of G as matrices. Let v denote a divisible element of G(K_{𝔭}), and, for each n ∈ Z_{>0}, let w_{n} ∈ G(K_{𝔭}) satisfy w_{n}^{n} = v for n ∈ Z_{>0}. The splitting field L_{n} of the characteristic polynomial of w_{n} (seen as N × N-matrix) has bounded degree [L_{n} : K] ≤ N!. Since by Krasner’s Lemma (e.g., [27, 8.1.6]), there are only finitely many extensions of K_{𝔭} of bounded degree, the compositum L of all L_{n} is a discretely valued field, in which all the eigenvalues λ_{i} of v are n-th powers (namely, of eigenvalues of w_{n}) for all integers n. Since L is non-archimedean,

⋂_{n≥1} L^{n} = {1},

by discreteness of the absolute value and the structure of 𝒪_{H}^{∗} as described in (1) and (2). We conclude that all eigenvalues of v are 1 and v is unipotent. □

4.3. Remark. The lemma is not true for archimedean places. To give an example at a real place, the rotation group SO(2, R) ⊆ SL(2, R) is divisible but contains non-unipotent elements.

4.4. Lemma. Let U be a maximal unipotent algebraic subgroup of an algebraic group G, both defined over a field F of characteristic zero. Then U(F) is divisible in G(F).

Proof. Consider the exponential map

exp: 𝔲 → U(F)

from the nilpotent Lie algebra 𝔲 of U(F) to U(F), which is bijective since F has characteristic zero (cf. [25, Theorem 6.5]). For an integer n ∈ Z_{>0} and an element 𝔵 ∈ 𝔲 we find (by the Baker-Campbell-Hausdorff formula) that

exp(n𝔵) = exp(𝔵)^{n},

since multiples of the same 𝔵 commute. Hence (multiplicative) divisibility in the unipotent algebraic group corresponds to (additive) divisibility in the nilpotent Lie algebra 𝔲. Since the latter is an F -vector space and any integer n is invertible in F, we find the result. □

We will also need the following global version:

4.5. Lemma. The group U(A_{K,f}) is divisible in G(A_{K,f}).

Proof. Let v = (v_{𝔭})_{𝔭} ∈ U(A_{K,f}), let n ∈ Z_{≥0}, and for every 𝔭 ∈ M_{K,f}, let w_{𝔭} ∈ U(K_{𝔭}) be such that w_{𝔭}^{n} = v_{𝔭} (which exists by the previous lemma applied to F = K_{𝔭}). We claim that w_{𝔭} ∈ U(𝒪_{𝔭}) for all but finitely many 𝔭, which shows that w = (w_{𝔭})_{𝔭} ∈ U(A_{K,f}) and proves the lemma. Indeed, it suffices to prove that w_{𝔭} ∈ GL_{N}(𝒪_{𝔭}) for all but finitely many 𝔭. This follows from the Taylor series

w_{𝔭} = \sqrt[n]{1 + (v_{𝔭} − 1)} = ∑_{k=0}^{∞} \binom{1/n}{k}(v_{𝔭} − 1)^{k},

which is a finite sum since v_{𝔭} − 1 is nilpotent, by noting that for fixed n, the binomial coefficients introduce denominators at only finitely many places. □

4.6. Lemma. Any maximal divisible subgroup of G(K_{𝔭}) is conjugate to U(K_{𝔭}) in G(K_{𝔭}).

Proof. Let D denote a maximal divisible subgroup of G(K_{𝔭}). By Lemma 4.2, it consists of unipotent elements. Since unipotency is defined by polynomial equations in the affine space of N × N matrices, the Zariski closure of D in G_{K_{𝔭}} is a unipotent algebraic subgroup U′ of G_{K_{𝔭}}. Moreover, Lemma 4.4 implies that U′(K_{𝔭}) consists of divisible elements, so by maximality of D, we find that U′ is a maximal unipotent algebraic subgroup of G_{K_{𝔭}}. Theorem 8.2 of Borel-Tits [4] implies that there exists an element γ_{𝔭} ∈ G(K_{𝔭}) such that γ_{𝔭}U′γ_{𝔭}^{−1} = U, for U any chosen maximal unipotent algebraic subgroup of G. Hence, γ_{𝔭}Dγ_{𝔭}^{−1} ⊆ U(K_{𝔭}). Since γ_{𝔭}^{−1}U(K_{𝔭})γ_{𝔭} is maximal divisible, the result follows. □

The proof to the following lemma was suggested to us by Maarten Solleveld.

4.7. Lemma. Let B ⊂ G denote a K-split Borel subgroup of G. Then for all but finitely many 𝔭 ∈ M_{K,f}, we have

G(K_{𝔭}) = B(K_{𝔭})G(𝒪_{𝔭}).

Proof. Since G/B is projective, the space (G/B)(A_{K,f}) is compact and equals (G/B)(A_{K,f}) = G(A_{K,f})/B(A_{K,f}) (both follow from [32, p. 258]). The open compact group ∏_{𝔭} G(𝒪_{𝔭}) (where the product runs over all finite places of K) of G(A_{K,f}) acts on (G/B)(A_{K,f}) by left multiplication. By compactness, the action has only finitely many orbits, i.e.,

(∏_{𝔭} G(𝒪_{𝔭})) \G(A_{K,f})/B(A_{K,f}) = ∏_{𝔭} (G(𝒪_{𝔭})\G(K_{𝔭})/B(K_{𝔭}))

is finite. In particular, for all but finitely many finite places 𝔭 of K, G(𝒪_{𝔭})\G(K_{𝔭})/B(K_{𝔭}) is trivial, so that G(K_{𝔭}) = B(K_{𝔭})G(𝒪_{𝔭}). □

In Appendix B, we provide an alternative cohomological proof that gives some more information about the set of excluded places. Finally, we prove the main result of this section:

4.8. Proposition. Suppose that G contains a K-split Borel subgroup. Let U be a maximal unipotent algebraic subgroup of G. Then any maximal divisible subgroup of G(A_{K,f}) is conjugate to U(A_{K,f}) in G(A_{K,f}).

Proof. Let D denote a maximal divisible subgroup of G(A_{K,f}) and D_{𝔭} = D∩G(K_{𝔭}) its local component for 𝔭 ∈ M_{K,f}. Let γ_{𝔭} ∈ G(K_{𝔭}) be as in Lemma 4.6, i.e., such that γ_{𝔭}D_{𝔭}γ_{p}^{−1} = U(K_{𝔭}). Let B = N_{G}(U) be a Borel subgroup containing U; since G is assumed to be fertile, we may choose these such that B is a split Borel K-subgroup of G. Lemma 4.7 implies that for all but finitely many 𝔭, we may replace γ_{𝔭} by an element in 𝒢 (𝒪_{𝔭}), which we again denote by γ_{𝔭} for ease of notation. This way, we

find γ = ∏_{𝔭∈M_{K,f}} γ_{𝔭} ∈ G(A_{K,f}) with

D ⊆ γ^{−1} ∏ U(K_{𝔭})γ ∩ G(A_{K,f}) = γ^{−1}U(A_{K,f})γ ⊆ D,

where the last inclusion holds by Lemma 4.5. □

Note that when G is fertile for K, it satisfies the hypotheses of Proposition 4.8.

5. Proof of Theorem C

We now turn to the proof of Theorem C. Let G := G(A_{K,f}) as a topological group. We will apply a purely group theoretic construction to G, to end up with the adele ring A_{K}; this shows that the isomorphism type of the adele ring is determined by the topological group G. We first reconstruct some cartesian power of A_{K,f}. Let D denote a maximal divisible subgroup of G. Consider the normaliser N := N_{G}D of D in G. Let V := [N, D]/[D, D] ≤ D^{ab}, and let T := N/D. Note that T acts naturally on V by conjugation. Since V is locally compact Hausdorff, we can give EndV, the endomorphisms of the abelian group V, the compact-open topology.

5.1. Proposition. There exists an integer ℓ ≥ 1 such that there is a topological ring isomorphism

Z(End_{T}V) ≅ A_{K,f}^{ℓ},

where the left hand side is the centre of the ring of continuous endomorphism of the T-module V.

Proof. First, we relate the subgroups of G to points groups of algebraic subgroups of G. From Proposition 4.8, we may assume that D = U(A_{K,f}) for a fixed maximal unipotent algebraic subgroup of G. The normaliser of U in G as an algebraic group is a Borel group B inside the fertile group G (Theorem of Chevalley, e.g. [3, 11.16]); again, we choose U such that B is split over K. Since taking points and taking normalisers commute ([24, Proposition 6.3]), we obtain that

N = N_{G}D = N_{G(A_{K,f})}U(A_{K,f}) = (N_{G}U)(A_{K,f}) = B(A_{K,f}).

and T ≅ T (A_{K,f}) for T any maximal torus in B, which is K-split by assumption. Next, we analyse the action of T on V, knowing the action of T on U. The hypothesis that T splits over K implies that T ≅ G_{m}^{r} over K for some r. The adjoint action of T by conjugation on U maps commutators to commutators, so it factors through the abelianisation U^{ab}, and we can consider the linear adjoint action ρ: T → Aut(U^{ab}) over K. Note that U^{ab} ≅ G_{a}^{k} for some integer k, so we have an action over K

(4)  ρ: T (≅ G_{m}^{r}) → Aut(G_{a}^{k}) = GL(k),

which is diagonalisable over K as a direct sum ρ = ⊕χ_{i} of k characters χ_{i} ∈ Hom_{K}(T, G_{m}) of algebraic groups. In coordinates t = (t_{1}, …, t_{r}) ∈ G_{m}^{r} = T, any such character is of the form

(5)  χ(t) = χ(t_{1}, …, t_{r}) = t_{1}^{n_{1}} · . . . · t_{r}^{n_{r}}

for some n_{1}, …, n_{r} ∈ Z. Since the action of T on V is given by specialisation from the action of T on a subspace of U^{ab}, we find an isomorphism of T-modules

V ≅ ⨁_{i=1}^{ℓ} A_{K,f,χ_{i}}^{μ_{i}},

where χ_{i} (i = 1, … ℓ) are the distinct non-trivial characters that occur in V, μ_{i} is the multiplicity of χ_{i} in V, and A_{K,f,χ_{i}} is the T-module A_{K,f} where T acts via χ_{i}. Hence,

(6)  End_{T}V = ∏_{i=1}^{ℓ} ∏_{j=1}^{ℓ} Mat_{μ_{j}×μ_{i}} (Hom_{T}(A_{K,f,χ_{i}}, A_{K,f,χ_{j}})) .

The assumption of fertility means precisely that ℓ ≥ 1. To conclude that

Z (End_{T}V) = Z (∏_{i=1}^{ℓ} Mat_{μ_{i}×μ_{i}} (A_{K,f})) = A_{K,f}^{ℓ},

as required, we now prove that if χ_{i} and χ_{j} are non-trivial characters occurring in this decomposition, then there is a topological ring isomorphism

Hom_{T}(A_{K,f,χ_{i}}, A_{K,f,χ_{j}}) ≅ \begin{cases}A_{K,f} & if χ_{i} = χ_{j} \\ \{0\} & otherwise . \end{cases}

Indeed, a homomorphism of additive groups f: (A_{K,f,χ_{i}}, +) → (A_{K,f,χ_{j}}, +) is Tequivariant precisely if f(χ_{i}(t)(u)) = χ_{j}(t)f(u) for all t ∈ T and u ∈ A_{K,f}. The χ are specialisations of algebraic characters as in (5), and some powers are non-zero by the assumption of fertility. If χ_{i} ≠ χ_{j}, this means that

(7)  f(t^{n}u) = t^{m}f(u), ∀t ∈ A_{K,f}^{∗}, ∀u ∈ A_{K,f}

for some n, m > 0, n ≠ m, which is impossible unless f = 0: indeed, choose t ∈ Z_{>0}; then the equation says that t^{n}f(u) = t^{m}f(u) for any u, so m = n. So we must have χ_{i} = χ_{j}, and we find that

(8)  f(t^{n}u) = t^{n}f(u), ∀t ∈ A_{K,f}^{∗}, ∀u ∈ A_{K,f}

for some n > 0. We now reinterpret a formula of Siegel [37, p. 134] as saying the following: Let R denote a ring and n a positive integer such that n! is invertible in R. Then any element of R belongs to the Z-linear span of the n-th powers in R. In particular, we have the following explicit formula for any z ∈ R:

z = ∑_{k=0}^{n−1}(−1)^{n−k−1}\binom{n − 1}{k} \{(\frac{z}{n!} + k)^{n} − k^{n}\} .

Applied to R = A_{K,f}, in which n! is invertible, Siegel’s formula expresses any element of A_{K,f} as Z-linear combination of n-th powers in A_{K,f}. We conclude from (8) and additivity of f that

(9)  f(tu) = tf(u), ∀t ∈ A_{K,f}^{∗}, ∀u ∈ A_{K,f}.

Hence, f(t) = tf(1) is completely determined by specifying a value for f(1) ∈ A_{K,f}, and

End_{T}(A_{K,f,χ}) → A_{K,f}: f ↦ f(1)

is the required ring isomorphism. It is continuous, since evaluation maps (such as this one) are continuous in the compact-open topology on End_{T}(A_{K,f,χ}). The inverse map is α ↦ (x ↦ αx), which is also continuous in the finite-adelic topology on A_{K,f}. Hence, we find an isomorphism of topological groups, as required. □

5.2. Theorem (Theorem C). Let K and L be two number fields, and let G denote a linear algebraic group over Q, fertile for K and L. There is a topological group isomorphism of adelic point groups G(A_{K,f}) ≅ G(A_{L,f}) if and only if there is a topological ring isomorphism A_{K} ≅ A_{L}.

Proof. Using Proposition 5.1, from G = G(A_{K,f}) we group theoretically construct Z (End_{T}V) = A_{K,f}^{ℓ}. Now consider the maximal ideals in A_{K,f}^{ℓ}. For any ring R, let ℳ (R) denote its set of principal maximal ideals. Observe that ℳ (R^{ℓ}) = ℳ (R) × Z/ℓZ, since a maximal ideal in R^{ℓ} is of the form R^{ℓ_{1}} × 𝔪 × R^{ℓ_{2}} for some maximal ideal 𝔪 of R and a decomposition ℓ = ℓ_{1} + ℓ_{2} + 1. We recall the description of the principal maximal ideals in an adele ring A_{K,f} as given by Iwasawa and Lochter ([21, Satz 8.6] and [14, p. 340–342], cf. [18, VI.2.4]):

ℳ (A_{K,f}) = {𝔪_{𝔭} = ker (A_{K,f} → K_{𝔭})}.

Note that A_{K,f}/𝔪_{𝔭} ≅ K_{𝔭}. Hence the multiset

{A_{K,f}^{ℓ}/𝔪: 𝔪 ∈ ℳ (A_{K,f}^{ℓ})}

contains a copy of the local field K_{𝔭} exactly ℓr_{𝔭} times, where r_{𝔭} is the number of local fields of K isomorphic to K_{𝔭}. Thus, we have constructed the multiset of local fields

{K_{𝔭}: 𝔭 ∈ M_{K,f}}

of K, up to isomorphism of local fields. If K and L are two number fields with G(A_{K,f}) ≅ G(A_{L,f}) as topological groups, then these multisets are in bijection, i.e., there exists a bijection of places φ: M_{K,f} → M_{L,f} such that K_{𝔭} ≅ L_{φ(𝔭)} for all 𝔭 ∈ M_{K,f}. Hence K and L are locally isomorphic (in the sense of Section 2.1), and we find ring isomorphisms A_{K,f} ≅ A_{L,f} and A_{K} ≅ A_{L}, by Proposition 2.3. For the reverse implication A_{K} ≅ A_{L} ⇒ G(A_{K,f}) ≅ G(A_{L,f}), we use that a topological ring isomorphism A_{K} ≅ A_{L} implies the existence of topological isomorphisms Φ_{𝔭}: K_{𝔭} ≅ L_{φ(𝔭)} of local fields for some bijection of places φ: M_{K,f} → M_{L,f} (again by Proposition 2.3). The fact that all Φ_{𝔭} are homeomorphisms implies in particular that Φ_{𝔭}(𝒪_{K,𝔭}) = 𝒪_{L,φ(𝔭)} for all 𝔭. Now G(A_{K,f}) ≅ G(A_{L,f}) is immediate from the definition of finite-adelic point groups (with topology) in 3.4(2) or, equivalently, 3.4(3). This finishes the proof of Theorem C. □

6. Hecke algebras and proof of Theorem E

6.1. Definition. Let G denote a linear algebraic group over Q. Because A_{K,f} is locally compact, G{K} := G(A_{K,f}) is a locally compact topological group for the topology described in Definition 3.4, equipped with a (left) invariant Haar measure μ_{G{K}}. The finite (or non-archimedean) real Hecke algebra ℋ_{G}(K) = C_{c}^{∞}(G{K}, R) of G over K is the algebra of all real-valued locally constant compactly supported continuous functions Φ : G{K} → R with the convolution product

Φ_{1} ∗ Φ_{2} : g ↦ ∫_{G{K}} Φ_{1}(gh^{−1})Φ_{2}(h)dμ_{G{K}}(h).

(Replacing R by C yields the finite-adelic complex Hecke algebra; the results in this section also hold in the complex setting.)

Every element of ℋ_{G}(K) is a finite linear combination of characteristic functions on double cosets KhK, for h ∈ G{K} and K a compact open subgroup of G{K}. Alternatively, we may write

ℋ_{G}(K) = lim_{\overrightarrow{K}} ℋ (G\{K\}//K),

where ℋ (G{K}//K) is the Hecke algebra of K-biinvariant smooth functions on G{K} (for example, if K is maximally compact, this is the spherical Hecke algebra).

6.2. Definition. We define the L^{1}-norm on functions on G{K} through

||f||_{1} = ∫_{G{K}} |f|dμ_{G{K}}.

Then let L^{1}(G{K}) denote the group algebra, i.e., the algebra of real-valued L^{1}- functions on G{K} with respect to the Haar measure μ_{G{K}}, under convolution.

6.3. Definition. An isomorphism of Hecke algebras Ψ: ℋ_{G}(K) →^{∼} ℋ_{G}(L) which is an isometry for the L^{1}-norms arising from the Haar measures (i.e., which satisfies ||Ψ(f)||_{1} = ||f||_{1} for all f ∈ ℋ_{G}(K)) is called an L^{1}-isomorphism. We will denote this by

ℋ_{G}(K) ≅_{L^{1}} ℋ_{G}(L).

Before we give its proof, let us recall the statement of Theorem E:

6.4. Theorem (Theorem E). Let K and L be two number fields, and let G denote a linear algebraic group over Q, fertile for K and L. There is an L^{1}-isomorphism of Hecke algebras ℋ_{G}(K) ≅_{L^{1}} ℋ_{G}(L) if and only if there is a ring isomorphism A_{K} ≅ A_{L}.

Proof. The proof consists of two steps: first we show, using the Stone-Weierstrass theorem, that the Hecke algebras are dense in the group algebras, and then we use results on reconstructing a locally compact group from its group algebra due to Wendel.

Step 1: ℋ_{G}(K) ≅_{L^{1}} ℋ_{G}(L) implies L^{1}(G{K}) ≅_{L^{1}} L^{1}(G{L}). By the locally compact real version of the Stone-Weierstrass theorem [13, 7.37(b)], ℋ_{G}(K) is dense in C_{0}(G{K}) for the sup-norm, where C_{0}(G{K}) denotes the functions that vanish at infinity, i.e., such that |f(x)| < ε outside a compact subset of G{K}. Indeed, one needs to check the nowhere vanishing and point separation properties of the algebra. Since ℋ_{G}(K) contains the characteristic function of any compact subset K ⊆ G{K}, the algebra vanishes nowhere, and the point separating property follows since G{K} is Hausdorff. A fortiori, ℋ_{G}(K) is dense in the compactly supported functions C_{c}(G{K}) for the sup-norm, and hence also in the L^{1}-norm. Now C_{c}(G{K}) is dense in L^{1}(G{K}), and the claim follows.

Step 2: L^{1}(G{K}) ≅_{L^{1}} L^{1}(G{L}) implies G{K} ≅ G{L}. Indeed, an L^{1}-isometry ℋ_{G}(K) ≅_{L^{1}} ℋ_{G}(L) implies an L^{1}-isometry of group algebras L^{1}(G{K}) ≅_{L^{1}} L^{1}(G{L}). Hence the result follows from a theorem due to Wendel [41, Theorem 1], which says that an L^{1}-isometry of group algebras of locally compact topological groups is always induced by an isomorphism of the topological groups.

Step 3: If G is fertile, G{K} ≅ G{L} implies A_{K} ≅ A_{L}. This is Theorem C. □

6.5. Corollary. If G is a connected linear algebraic group over Q which is fertile for K and L, where K and L are two numbers fields which are Galois over Q, then an L^{1}-isomorphism of Hecke algebras ℋ_{G}(K) ≅ ℋ_{G}(L) implies that the fields K and L are isomorphic.

Proof. Since the hypotheses imply that the fields are arithmetically equivalent, the result follows from Proposition 2.3.(iii). □

Since GL(n) is fertile if n ≥ 2, we obtain Corollary F.

Variations on Theorem E.

(1) The theorem is also true if the real-valued Hecke algebra is replaced by the complex-valued Hecke algebra (using the complex versions of Stone-Weierstrass and Kawada/Wendel).

(2) It seems that the theorem also holds for the full Hecke algebra ℋ_{G} ⊗ ℋ_{G}^{∞}, where ℋ_{G}^{∞} is the archimedean Hecke algebra for G, viz., the convolution algebra of distributions on G(R ⊗_{Q} K) supported on a maximal compact subgroup of G(R ⊗_{Q} K), but we have not checked the analytic details.

Appendix A. An equivalent definition of fertility

The following was observed by Wilberd van der Kallen:

A.1. Proposition. G is fertile over K if and only if it contains a K-split maximal torus, and the connected component of the identity G^{0} is not a direct product of a torus and a unipotent group.

Proof. Suppose for the entire proof that G contains a K-split maximal torus. Since Borel groups are connected, the identity component G^{0} contains a Borel group, and since Borel groups are conjugate over \overline{K}, the condition of fertility in Definition 3.2 is equivalent to G^{0} containing a Borel subgroup with non-trivial action of its torus T on the abelianization U^{ab} of its maximal unipotent subgroup U. We claim that T acts non-trivially on U^{ab} if and only if T acts non-trivially on U. Hence, G is not fertile if and only if G^{0} contains a Borel subgroup in which the torus acts trivially on the unipotent part. If this happens, we conclude from the short exact sequence of algebraic groups

1 → R_{u}(G) → G^{0} → S → 1,

where R_{u}(G) is the unipotent radical of G and S is a reductive group, that S has only trivial roots. The classification of reductive groups in characteristic zero implies that S is a torus, so R_{u}(G) = U, S = T, and G^{0} is itself a Borel group T × U, which is what we wanted to prove. Let us now prove the claim. Since we are in characteristic zero and T is reductive, it is linearly reductive. Hence, by [7, Corollary A.8.11], T acts trivially on a connected algebraic group if and only if it acts trivially on its Lie algebra. Thus, it suffices to show that T acts non-trivially on Lie(U) if and only if it acts non-trivially on Lie(U^{ab}). By linear reductivity, Lie(U^{ab}) is a direct factor of 𝔲 := Lie(U) as a T -module, so necessity is clear. For the converse, observe that since T acts via the adjoint representation, it preserves the lower central series 𝔲_{0} := 𝔲; 𝔲_{i} := [𝔲, 𝔲_{i−1}] (i ≥ 1) of 𝔲. We will show that if T acts trivially on 𝔲/𝔲_{j−1} for some j ≥ 2, then it acts trivially on 𝔲/𝔲_{j}. The result will then follow, since Lie(U^{ab}) = 𝔲/𝔲_{1} and 𝔲/𝔲_{k} = 𝔲 for sufficiently large k, as 𝔲 is nilpotent. To prove our claim, we use that, by assumption, T acts trivially on the subalgebra 𝔲_{j−2}/𝔲_{j−1} of 𝔲/𝔲_{j−1}. The bracket [·, ·]: 𝔲 × 𝔲_{j−2} → 𝔲_{j−1} factors through to give a surjective T -equivariant map

[·, ·]: 𝔲/𝔲_{j−1} ⊗ 𝔲_{j−2}/𝔲_{j−1} → 𝔲_{j−1}/𝔲_{j}.

Hence, T acts trivially on 𝔲_{j−1}/𝔲_{j}. By linear reductivity of T, the short exact sequence of T -modules

0 → 𝔲_{j−1}/𝔲_{j} → 𝔲/𝔲_{j} → 𝔲/𝔲_{j−1} → 0,

splits, and we conclude that T acts trivially on the middle term, too. □

Appendix B. Alternative proof of Lemma 4.7

The following is an alternative proof of Lemma 4.7 using cohomological methods and providing some insight into the set of exceptional places. It was inspired by an answer

by Bhargav Bhatt on mathoverflow.net/a/2231.

Let ℬ ⊂ 𝒢 denote any inclusion of smooth finite-type separated group schemes over the ring of S-integers 𝒪_{K,S} for a suitable finite set of primes S, so that the generic fibre of ℬ is B and that of 𝒢 is G. By [1, Theorem 4.C], the fppf sheaf quotient 𝒢 /ℬ is a scheme over 𝒪_{K,S}; choose S such that it contains the (finitely many, cf. [10, Troisième partie, 9.6.1]) places where the special fibre of 𝒢 /ℬ is not proper. Since fppf quotients commute with base change, the generic fibre of 𝒢 /ℬ is G/B.

We will show that for 𝔭 ∈ M_{K,f} not contained in S, we have G(K_{𝔭}) = B(K_{𝔭})𝒢 (𝒪_{𝔭}). It suffices to show that

(10)  G(K_{𝔭})/B(K_{𝔭}) = 𝒢 (𝒪_{𝔭})/ℬ(𝒪_{𝔭})

for all 𝔭 ∉ S. We will prove this by arguing that both sides of (10) equal (𝒢 /ℬ)(𝒪_{𝔭}). First consider the long exact sequence in fppf-cohomology associated to the exact sequence

1 → ℬ → 𝒢 → 𝒢 /ℬ → 1

of smooth group schemes (cf. [33, p. 151-152 and Theorem 6.5.10]), over M = K_{𝔭} or M = 𝒪_{𝔭}:

1 → ℬ(M) → 𝒢 (M) → (𝒢 /ℬ)(M) → H^{1}(M, ℬ) → …

To rewrite the left hand side of (10), we take M = K_{𝔭}, so we are dealing with Gal(\overline{K}_{𝔭}/K_{𝔭})-cohomology. The short exact sequence

1 → U → B → T → 1,

for U a unipotent group and T a maximal K-split torus, induces a long exact sequence

\begin{aligned}1 → U(K_{𝔭}) → B(K_{𝔭}) → T (K_{𝔭}) → H^{1}(K_{𝔭}, U) → H^{1}(K_{𝔭}, B) & → H^{1}(K_{𝔭}, T) \\ & → …. \end{aligned}

Since T is split over K, hence split (and a fortiori quasisplit) over K_{𝔭}, and K_{𝔭} is a perfect field, applying [32, Lemma 2.4] yields H^{1}(K_{𝔭}, T) = 1. Moreover, since K_{𝔭} has characteristic zero, H^{1}(K_{𝔭}, U) = 1 by [32, Lemma 2.7]. Thus, we find that

H^{1}(K_{𝔭}, B) = 1.

Hence,

G(K_{𝔭})/B(K_{𝔭}) = (G/B)(K_{𝔭}).

Since G/B is projective and 𝒢 /ℬ is proper, the valuative criterion of properness implies that

(G/B)(K_{𝔭}) = (𝒢 /ℬ)(𝒪_{𝔭}).

For the right hand side of (10), we set M = 𝒪_{𝔭} and argue as in Step 3 of [33, Theorem 6.5.12]: H^{1}(𝒪_{𝔭}, ℬ) classifies ℬ-torsors over 𝒪_{𝔭}; let 𝒯 → Spec 𝒪_{𝔭} denote such a torsor. By Lang’s theorem, its special fibre 𝒯_{𝔭} → Spec F_{𝔭} over the finite residue field F_{𝔭} has a rational point. Since ℬ smooth, so is 𝒯 , so we can lift the rational point by Hensel’s Lemma. Hence, 𝒯 is also trivial, so H^{1}(𝒪_{𝔭}, ℬ) = 1. We conclude that

(𝒢 /ℬ)(𝒪_{𝔭}) = 𝒢 (𝒪_{𝔭})/ℬ(𝒪_{𝔭}).

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Gunther Cornelissen Mathematisch Instituut Universiteit Utrecht Postbus 80.010 3508 TA Utrecht, Nederland g.cornelissen@uu.nl Valentijn Karemaker Department of Mathematics University of Pennsylvania David Rittenhouse Labs 209 South 33rd Street Philadelphia, PA 19104-6395 U.S.A. vkarem@math.upenn.edu 872

Documenta Mathematica 22 (2017)