Equivariant Counts of Points of the

Moduli Spaces of Pointed Hyperelliptic Curves

Jonas Bergström

Received: June 18, 2008 Revised: March 27, 2009

Communicated by Thomas Peternell

Abstract. We consider the moduli space H_{g,n} of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make 𝕊_{n}-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make 𝕊_{n}-equivariant counts of H_{g,n} for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for H_{g,n} for all g and for n ≤ 7. These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the 𝕊_{n}-equivariant Galois (resp. Hodge) structure of their ℓ-adic (resp. Betti) cohomology.

2000 Mathematics Subject Classification: 14H10, 11G20 Keywords and Phrases: Cohomology of moduli spaces of curves, curves over finite fields.

1. Introduction

By virtue of the Lefschetz trace formula, counting points defined over finite fields of a space gives a way of finding information on its cohomology. In this article we wish to count points of the moduli space H_{g,n} of n-pointed smooth hyperelliptic curves of genus g. On this space we have an action of the symmetric group 𝕊_{n} by permuting the marked points of the curves. To take this action into account we will make 𝕊_{n}-equivariant counts of the numbers of points of H_{g,n} defined over finite fields. For every n we will find simple recurrence relations in the genus, for the equivariant number of points of H_{g,n} defined over a finite field. Thus, if we can count these numbers for low genus, we will know the answer for every genus. The hyperelliptic curves will need to be separated according to whether the characteristic is odd or even and the respective recurrence relations will in some cases be different. When the number of marked points is at most 7 we use the fact that the base cases of the recurrence relations only involve the genus 0 case, which is easily computed, and previously known 𝕊_{n}-equivariant counts of points of M_{1,n}, to get equivariant counts for every genus. If we consider the odd and even cases separately, then all these counts are polynomials when considered as functions of the number of elements of the finite field. For up to five points these polynomials do not depend upon the characteristic. But for six-pointed hyperelliptic curves there is a dependence, which appears for the first time for genus 3. By the Lefschetz trace formula, the 𝕊_{n}-equivariant count of points of H_{g,n} is equivalent to the trace of Frobenius on the ℓ-adic 𝕊_{n}-equivariant Euler characteristic of H_{g,n}. But this information can also be formulated as traces of Frobenius on the Euler characteristic of some natural local systems 𝕍_{λ} on H_{g}. By Theorem 3.2 in [1] we can use this connection to determine the Euler characteristic, evaluated in the Grothendieck group of absolute Galois modules, of all 𝕍_{λ} on H_{g} ⊗ \overline{ℚ} of weight at most 7. These result are in agreement with the results on the ordinary Euler characteristic and the conjectures on the motivic Euler characteristic of 𝕍_{λ} on H_{3} by Bini-van der Geer in [5], the ordinary Euler characteristic of 𝕍_{λ} on H_{2} by Getzler in [16], and the 𝕊_{2}-equivariant cohomology of H_{g,2} for all g ≥ 2 by Tommasi in [20]. The moduli stack \overline{M}_{g,n} of stable n-pointed curves of genus g is smooth and proper, which implies purity of the cohomology. If the 𝕊_{n}-equivariant count of points of this space, when considered as a function of the number of elements of the finite field, gives a polynomial, then using the purity we can determine the 𝕊_{n}-equivariant Galois (resp. Hodge) structure of its individual ℓ-adic (resp. Betti) cohomology groups (see Theorem 3.4 in [2] which is based on a result of van den Bogaart-Edixhoven in [6]). All curves of genus 2 are hyperelliptic and hence we can apply this theorem to \overline{M}_{2,n} for all n ≤ 7. These results on genus 2 curves are all in agreement with the ones of Faber-van der Geer in [9] and [10]. Moreover, for n ≤ 3 they were previously known by the work of Getzler in [14, Section 8].

Acknowledgements

The method I shall use to count points of the moduli space of pointed hyperelliptic curves follows a suggestion by Nicholas M. Katz. I thank Bradley Brock for letting me read an early version of the article [7], and Institut Mittag-Leffler for support during the preparation of this article. I would also like to thank my thesis advisor Carel Faber.

Outline

Let us give an outline of the paper, where ⋆_{·} denotes the section.

⋆_{2} In this section we define 𝕊_{n}-equivariant counts of points of H_{g,n} over a finite field k, and we formulate the counts in terms of numbers a_{λ}|_{g}, which are connected to the H^{1}’s of the hyperelliptic curves.

⋆_{3} The hyperelliptic curves of genus g, in odd characteristic, are realized as degree 2 covers of P^{1} given by square-free polynomials of degree 2g +2 or 2g +1. The numbers a_{λ}|_{g} are then expressed in terms of these polynomials in equation (3.2). The expression for a_{λ}|_{g} is decomposed into parts denoted u_{g}, which are indexed by pairs of tuples of numbers (n; r). The special cases of genus 0 and 1 are discussed in Section 3.1.

⋆_{4} A recurrence relation is found for the numbers u_{g} (Theorem 4.12). The first step is to use the fact that any polynomial can be written uniquely as a monic square times a square-free one. This results in an equation which gives U_{g} in terms of u_{h} for h less than or equal to g, where U_{g} denotes the expression corresponding to u_{g}, but in terms of all polynomials instead of only the square-free ones. The second step is to use that, if g is large enough, U_{g} can be computed using a simple interpolation argument.

⋆_{5} The recurrence relations for the u_{g}’s are put together to form a linear recurrence relation for a_{λ}|_{g}, whose characteristic polynomial is given in Theorem 5.2.

⋆_{6} It is shown how to compute u_{0} for any pair (n; r). ⋆_{7} Information on the cases of genus 0 and 1 is used to compute, for all g,

u_{g} for tuples (n; r) of degree at most 5, and a_{λ}|_{g} of weight at most 7. ⋆_{8} The hyperelliptic curves are realized, in even characteristic, as pairs

(h, f) of polynomials fulfilling three conditions. The numbers u_{g} and U_{g} are then defined to correspond to the case of odd characteristic.

⋆_{9} In even characteristic, a recurrence relation is found for the numbers u_{g} (Theorem 9.11). Lemmas 9.6 and 9.7 show that one can do something in even characteristic corresponding to uniquely writing a polynomial as a monic square times a square-free one in odd characteristic. This results in a relation between U_{g} and u_{h} for h less than or equal to g. Then, as in odd characteristic, a simple interpolation argument is used to compute U_{g} for g large enough.

⋆_{10} The same amount of information as in Section 7 is obtained in the case of even characteristic. It is noted that a_{λ}|_{g} is independent of the characteristic for weight at most 5 (Theorem 10.3). This does not continue to hold for weight 6 where there is dependency for genus at least 3 (see Example 10.6).

⋆_{11} The counts of points of the previous sections are used to get cohomo-

logical information. This is, in particular, applied to \overline{M}_{2,n} for n ≤ 7. ⋆_{12} In the first appendix, a more geometric interpretation is given of the

information contained in all the numbers u_{g} of at most a certain degree (see Lemma 12.8).

⋆_{13} In the second appendix, we find that for sufficiently large g we can compute the Euler characteristic, with Gal(\overline{ℚ}/ℚ)-structure, of the part of the cohomology of sufficiently high weight, of some local systems 𝕍_{λ} on H_{g}. We will also see that these results are, in a sense, stable in g.

2. Equivariant counts

Let k be a finite field with q elements and denote by k_{m} a degree m extension. Define H_{g,n} to be the coarse moduli space of H_{g,n}⊗k̄ and let F be the geometric Frobenius morphism. The purpose of this article is to make 𝕊_{n}-equivariant counts of the number of points defined over k of H_{g,n}. With this we mean a count, for each element σ ∈ 𝕊_{n}, of the number of fixed points of F σ acting on H_{g,n}. Note that these numbers only depend upon the cycle type c(σ) of the permutation σ. Define R_{σ} to be the category of hyperelliptic curves of genus g that are defined over k together with marked points (p_{1}, …, p_{n}) defined over k̄ such that (F σ)(p_{i}) = p_{i} for all i. Points of H_{g,n} are isomorphism classes of n-pointed hyperelliptic curves of genus g defined over k̄. For any pointed curve X that is a representative of a point in H_{g,n}^{F σ}, the set of fixed points of F σ acting on H_{g,n}, there is an isomorphism from X to the pointed curve (F σ)X. Using this isomorphism we can descend to an element of R_{σ} (see [17, Lem. 10.7.5]). Therefore, the number of k̄-isomorphism classes of the category R_{σ} is equal to |H_{g,n}^{F σ}|. Fix an element Y = (C, p_{1}, …, p_{n}) in R_{σ}. We then have the following equality (see [12] or [17]):

∑_{\begin{matrix}[X]∈R_{σ}/≅k \\ X≅k̄Y \end{matrix}} \frac{1}{|Aut_{k}(X)|} = 1.

This enables us to go from k̄-isomorphism classes to k-isomorphism classes:

|H_{g,n}^{F σ}| = ∑_{[Y]∈R_{σ}/≅k̄} 1 = ∑_{[Y]∈R_{σ}/≅k̄} ∑_{\begin{matrix}[X]∈R_{σ}/≅k \\ X≅k̄Y \end{matrix}} \frac{1}{|Aut_{k}(X)|} = ∑_{[X]∈R_{σ}/≅_{k}} \frac{1}{|Aut_{k}(X)|}.

For any curve C over k, define C(σ) to be the set of n-tuples of distinct points (p_{1}, …, p_{n}) in C(k̄) that fulfill (F σ)(p_{i}) = p_{i}.

Notation 2.1. A partition λ of an integer m consists of a sequence of nonnegative integers λ_{1}, …, λ_{ν} such that |λ| := ∑_{i=1}^{ν} iλ_{i} = m. We will write λ = [1^{λ_{1}}, …, ν^{λ_{ν}}].

Say that τ ∈ 𝕊_{n} consists of one n-cycle. The elements of C(τ) are then given by the choice of p_{1} ∈ C(k_{n}) such that p_{1} ∉ C(k_{i}) for every i < n. By an inclusion-exclusion argument it is then straightforward to show that

|C(τ)| = ∑_{d|n} μ(n/d) |C(k_{d})|,

where μ is the Möbius function. Say that λ is any partition and that σ ∈ 𝕊_{|λ|} has the property c(σ) = λ. Since C(σ) consists of tuples of distinct points it directly follows that

(2.1)  |C(σ)| = ∏_{i=1}^{ν} ∏_{j=0}^{λ_{i}−1}(∑_{d|i}(μ(i/d) |C(k_{d})| − ji)).

Fix a curve C over k and let X_{1}, …, X_{m} be representatives of the distinct kisomorphism classes of the subcategory of R_{σ} of elements (D, q_{1}, …, q_{n}) where D ≅_{k} C. For each X_{i} we can act with Aut_{k}(C) which gives an orbit lying in R_{σ} and where the stabilizer of X_{i} is equal to Aut_{k}(X_{i}). Together the orbits of X_{1}, …, X_{m} will contain |C(σ)| elements and hence we obtain

(2.2)  |H_{g,n}^{F σ}| = ∑_{[X]∈R_{σ}/≅_{k}} \frac{1}{|Aut_{k}(X)|} = ∑_{[C]∈H_{g}(k)/≅_{k}} \frac{|C(σ)|}{|Aut_{k}(C)|}.

We will compute slightly different numbers than |H_{g,n}^{F σ}|, but which contain equivalent information. Let C be a curve defined over k. The Lefschetz trace formula tells us that for all m ≥ 1, (2.3)

|C(k_{m})| = |CF^{m} ; k̄| = 1 + q^{m} − a_{m}(C) where a_{m}(C) = Tr(F^{m}, H^{1}(Ck̄, ℚ_{ℓ})).

If we consider equations (2.1) and (2.2) in view of equation (2.3) we find that

|H_{g,n}^{F σ}| = ∑_{[C]∈H_{g}(k)/≅_{k}} \frac{1}{|Aut_{k}(C)|} f_{σ}(q, a_{1}(C), …, a_{n}(C)),

where f_{σ}(x_{0}, …, x_{n}) is a polynomial with coefficients in ℤ. Give the variable x_{i} degree i. Then there is a unique monomial in f_{σ} of highest degree, namely x_{1}^{λ_{1}} ⋯ x_{ν}^{λ_{ν}}. The numbers which we will pursue will be the following.

Definition 2.2. For g ≥ 2 and any partition λ define

(2.4)  a_{λ}|_{g} := ∑_{[C]∈H_{g}(k)/≅_{k}} \frac{1}{|Aut_{k}(C)|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}}.

This expression will be said to have weight |λ|. Let us also define

a_{0}|_{g} := ∑_{[C]∈H_{g}(k)/≅_{k}} \frac{1}{|Aut_{k}(C)|},

an expression of weight 0.

3. Representatives of hyperelliptic curves in odd characteristic

Assume that the finite field k has an odd number of elements. The hyperelliptic curves of genus g ≥ 2 are the ones endowed with a degree 2 morphism to P^{1}. This morphism induces a degree 2 extension of the function field of P^{1}. If we consider hyperelliptic curves defined over the finite field k and choose an affine coordinate x on P^{1}, then we can write this extension in the form y^{2} = f(x), where f is a square-free polynomial with coefficients in k of degree 2g + 1 or 2g + 2. At infinity, we can describe the curve given by the polynomial f in the coordinate t = 1/x by y^{2} = t^{2g+2} f(1/t). We will therefore let f(∞), which corresponds to t = 0, be the coefficient of f of degree 2g + 2.

Definition 3.1. Let P_{g} denote the set of square-free polynomials with coefficients in k and of degree 2g + 1 or 2g + 2, and let P′_{g} ⊂ P_{g} consist of the monic polynomials. Write C_{f} for the curve corresponding to the element f in P_{g}.

By construction, there exists for each k-isomorphism class of objects in H_{g}(k) an f in P_{g} such that C_{f} is a representative. Moreover, the k-isomorphisms between curves corresponding to elements of P_{g} are given by k-isomorphisms of their function fields. By the uniqueness of the linear system g_{2}^{1} on a hyperelliptic curve, these isomorphisms must respect the inclusion of the function field of P^{1}. The k-isomorphisms are therefore precisely (see [16, p. 126]) the ones induced by elements of the group G := GL_{2}^{op}(k) × k^{∗}/D where

D := \{(( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ), a^{g+1}) : a ∈ k^{∗}\} ⊂ GL_{2}^{op}(k) × k^{∗}

and where an element

γ = [(( \begin{matrix}a & b \\ c & d \end{matrix} ), e)] ∈ G

induces the isomorphism

(x, y) ↦ (\frac{ax + b}{cx + d}, \frac{ey}{(cx + d)^{g+1}} ) .

This defines a left group action of G on P_{g}, where γ ∈ G takes f ∈ P_{g} to f̃ ∈ P_{g}, with

(3.1)  f̃(x) = \frac{(cx + d)^{2g+2}}{e^{2}} f(\frac{ax + b}{cx + d}).

Notation 3.2. Let us put I := 1/|G| = (q^{3} − q)^{−1}(q − 1)^{−1}.

Definition 3.3. Let χ_{2,m} be the quadratic character on k_{m}. Recall that it is the function that takes α ∈ k_{m} to 1 if it is a square, to −1 if it is a nonsquare and to 0 if it is 0. With a square or a nonsquare we will always mean a nonzero element.

Lemma 3.4. If C_{f} is the hyperelliptic curve corresponding to f ∈ P_{g} then

a_{m}(C_{f}) = − ∑_{α∈P^{1}(k_{m})} χ_{2,m}(f(α)).

Proof: The fiber of C_{f} → P^{1} over α ∈ A^{1}(k_{m}) will consist of two points defined over k_{m} if f(α) is a square in k_{m}, no point if f(α) is a nonsquare in k_{m}, and one point if f(α) = 0. By the above description of f in terms of the coordinate t = 1/x, the same holds for α = ∞. The lemma now follows from equation (2.3). □

We will now rephrase equation (2.4) in terms of the elements of P_{g}. By what was said above, the stabilizer of an element f in P_{g} under the action of G is equal to Aut_{k}(C_{f}) and hence

(3.2) a_{λ}|_{g} = ∑_{[f]∈P_{g}/G} \frac{1}{|Stab_{G}(f)|} ∏_{i=1}^{ν} a_{i}(C_{f})^{λ_{i}} =

= \frac{1}{|G|} ∑_{f∈P_{g}} ∏_{i=1}^{ν} a_{i}(C_{f})^{λ_{i}} = I ∑_{f∈P_{g}} ∏_{i=1}^{ν}(− ∑_{α∈P^{1}(k_{i})} χ_{2,i}(f(α)))^{λ_{i}}.

This can up to sign be rewritten as

(3.3)  I ∑_{f∈P_{g}} ∑_{(α_{1,1},...,α_{ν,λν})∈S} ∏_{i=1}^{ν} ∏_{j=1}^{λ_{i}} χ_{2,i}(f(α_{i,j})),

where S := ∏_{i=1}^{ν} P^{1}(k_{i})^{λ_{i}}, in other words, α_{i,j} ∈ P^{1}(k_{i}) for each 1 ≤ i ≤ ν and 1 ≤ j ≤ λ_{i}. The sum (3.3) will be split into parts for which we, in Section 4, will find recurrence relations in g.

Definition 3.5. For any tuple n = (n_{1}, …, n_{m}) ∈ ℕ_{≥1}^{m}, let the set A(n) consist of the tuples α = (α_{1}, …, α_{m}) ∈ ∏_{i=1}^{m} P^{1}(k_{n_{i}}) such that for any 1 ≤ i, j ≤ m and any s ≥ 0,

F^{s}(α_{i}) = α_{j} ⟹ n_{i}|s and i = j.

Let us also define A′(n) := A(n) ∩ ∏_{i=1}^{m} A^{1}(k_{n_{i}}).

Definition 3.6. Let N_{m} denote the set of pairs (n; r) such that n = (n_{1}, …, n_{m}) ∈ ℕ_{≥1}^{m} and r = (r_{1}, …, r_{m}) ∈ {1, 2}^{m}.

Definition 3.7. For any g ≥ −1, (n; r) ∈ N_{m} and α = (α_{1}, …, α_{m}) ∈ A(n) define

u_{g,α}^{(n;r)} := I ∑_{f∈P_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}}

and

u_{g}^{(n;r)} := ∑_{α∈A(n)} u_{g,α}^{(n;r)}.

Construction-Lemma 3.8. For each λ, there are positive integers c_{1}, …, c_{s} and m_{1}, …, m_{s}, and moreover pairs (n^{(i)}; r^{(i)}) ∈ N_{m_{i}} for each 1 ≤ i ≤ s, such that for any finite field k,

a_{λ}|_{g} = ∑_{i=1}^{s} c_{i} u_{g}^{(n^{(i)};r^{(i)})}.

Proof: The lemma will be proved by writing the set S as a disjoint union of parts that only depend upon the partition λ, and which therefore are independent of the chosen finite field k. For each positive integer i, let i = d_{i,1} > … > d_{i,δ_{i}} = 1 be the divisors of i.

⋆ For each 1 ≤ i ≤ ν, let T_{i,1}, …, T_{i,δ_{i}} be an ordered partition of the set

{1, …, λ_{i}} into (possibly empty) subsets. ⋆ For each 1 ≤ i ≤ ν and each 1 ≤ j ≤ δ_{i}, let Q_{i,j,1}, …, Q_{i,j,κ_{i,j}} be an unordered partition (where κ_{i,j} is arbitrary) of the set T_{i,j} into non-empty subsets.

From such a choice of partitions we define a subset S′ = S({T_{i,j}}, {Q_{i,j,k}}) of S consisting of the tuples (α_{1,1}, …, α_{ν,λ_{ν}}) ∈ S fulfilling the following two properties.

⋆ If x ∈ T_{i,j} then: α_{i,x} ∈ k_{j} and ∀s < j, α_{i,x} ∉ k_{s}.

⋆ If x ∈ Q_{i,j,k} and x′ ∈ Q_{i′,j′,k′} then:
∃s : F^{s}(α_{i,x}) = α_{i′,x′} ⟺ (i, j, k) = (i′, j′, k′).

Define n to be equal to the tuple

(\overbrace{d_{1,1}, …, d_{1,1}}^{κ_{1,1}}, \overbrace{d_{1,2}, …, d_{1,2}}^{κ_{1,2}}, …, \overbrace{d_{1,δ_{1}}, …, d_{1,δ_{1}}}^{κ_{1,δ_{1}}}, \overbrace{d_{2,1}, …, d_{2,1}}^{κ_{2,1}}, …, \overbrace{d_{ν,δ_{ν}} , …, d_{ν,δ_{ν}}}^{κ_{ν,δν}}).

Let ρ_{i,j,k} be equal to 2 if either i/d_{i,j} or |Q_{i,j,k}| is even, and 1 otherwise. Define r to be equal to

(ρ_{1,1,1}, ρ_{1,1,2}, …, ρ_{1,1,κ_{1,1}}, ρ_{1,2,1}, …, ρ_{1,δ_{1},κ_{1,δ_{1}}} , ρ_{2,1,1}, …, ρ_{ν,δ_{ν},κ_{ν,δν}}).

The equality

u_{g}^{(n;r)} = I ∑_{f∈P_{g}} ∑_{(α_{1,1},...,α_{ν,λν})∈S′} ∏_{i=1}^{ν} ∏_{j=1}^{λ_{i}} χ_{2,i}(f(α_{i,j}))

is clear in view of the following three simple properties of the quadratic character.

⋆ Say that α ∈ P^{1}(k_{s}), then if s̃/s is even we have χ_{2,s̃}(f(α)) =

χ_{2,s}(f(α))^{2} and if s̃/s is odd we have χ_{2,s̃}(f(α)) = χ_{2,s}(f(α)).

⋆ If for any α, β ∈ P^{1} we have F^{s}(α) = β for some s, then χ_{2,i}(f(α)) =

χ_{2,i}(f(β)) for all i.

⋆ Finally, for any α ∈ P^{1} and any s, we have χ_{2,s}(f(α))^{r} = χ_{2,s}(f(α))^{2}

if r is even and χ_{2,s}(f(α))^{r} = χ_{2,s}(f(α)) if r is odd.

The lemma now follows directly from the fact that the sets S({T_{i,j}}, {Q_{i,j,k}}) ⊂ S (for different choices of partitions {T_{i,j}} and {Q_{i,j,k}}) are disjoint and cover S. □

The set of data {(c_{i}, (n^{(i)}; r^{(i)}))} resulting from the procedure given in the proof of Construction-Lemma 3.8 is, after assuming the pairs (n^{(i)}; r^{(i)}) to be distinct, unique up to simultaneous reordering of the elements of n^{(i)} and r^{(i)} for each i, and it will be called the decomposition of a_{λ}|_{g}.

Definition 3.9. For a partition λ, the pair

(n; r) = ((\overbrace{1, …, 1}^{λ_{1}}, \overbrace{2, …, 2}^{λ_{2}}, …, \overbrace{ν, …, ν}^{λ_{ν}}); (1, …, 1))

will appear in the decomposition of a_{λ}|_{g} (corresponding to the partitions T_{i,1} = {1, …, λ_{i}} for 1 ≤ i ≤ ν, and Q_{i,1,k} = {k} for 1 ≤ i, k ≤ ν) with coefficient equal to 1, and it will be called the general case. All other pairs (n; r) appearing in the decomposition of a_{λ}|_{g} will be refered to as degenerations of the general case.

Definition 3.10. For any (n; r) ∈ N_{m}, the number |n| := ∑_{i=1}^{m} n_{i} will be called the degree of (n; r).

Lemma 3.11. The general case is the only case in the decomposition of a_{λ}|_{g} which has degree equal to the weight of a_{λ}|_{g}.

Proof: If (n; r) appears in the decomposition of a_{λ}|_{g} and is associated to the partitions {T_{i,j}} and {Q_{i,j,k}}, then |n| = ∑_{i=1}^{ν} ∑_{j=1}^{δ_{i}} κ_{i,j}d_{i,j}. Since λ_{i} = ∑_{j=1}^{δ_{i}} κ_{i,j} and 1 ≤ d_{i,j} ≤ i, the equality |λ| = |n| implies that κ_{i,1} = λ_{i} and κ_{i,j} = 0 if j ≠ 1. □

Lemma 3.12. If (n; r) appears in the decomposition of a_{λ}|_{g} then ∑_{i=1}^{m} r_{i}n_{i} ≤ |λ| and these two numbers have the same parity.

Proof: If (n; r) appears in the decomposition of a_{λ}|_{g} and is associated to the partitions {T_{i,j}} and {Q_{i,j,k}}, then ∑_{i=1}^{m} r_{i}n_{i} = ∑_{i=1}^{ν} ∑_{j=1}^{δ_{i}} ∑_{k=1}^{κ_{i,j}} ρ_{i,j,k}d_{i,j}. Let us prove the lemma by induction on m, starting with the case that m = ∑_{i=1}^{ν} λ_{i}. In this case we must have |Q_{i,j,k}| = 1 for all 1 ≤ i ≤ ν, 1 ≤ j ≤ δ_{i} and 1 ≤ k ≤ κ_{i,j}, and hence ρ_{i,j,k} is only equal to two if i/d_{i,j} is even. This directly tells us that ρ_{i,j,k}d_{i,j} ≤ i, and that these two numbers have the same parity. Since λ_{i} = ∑_{j=1}^{δ_{i}} κ_{i,j}, it follows that ∑_{i=1}^{m} r_{i}n_{i} ≤ |λ| and that these two numbers have the same parity. Assume now that m = k and that the lemma has been proved for all pairs (ñ; r̃) with m̃ > k. Since m < ∑_{i=1}^{ν} λ_{i} we know that there exists numbers i_{0}, j_{0}, k_{0} such that |Q_{i_{0},j_{0},k_{0}}| ≥ 2. Let us fix an element x ∈ Q_{i_{0},j_{0},k_{0}} and define a new pair (n′; r′) associated to the partitions {T′_{i,j}} and {Q′_{i,j,k}} by putting:

⋆ T′_{i,j} = T_{i,j} for all 1 ≤ i ≤ ν and 1 ≤ j ≤ δ_{i},

⋆ Q′_{i_{0},j_{0},k_{0}} = Q_{i_{0},j_{0},k_{0}} \ {x},
⋆ κ′_{i_{0},j_{0}} = κ_{i_{0},j_{0}} + 1 and Q′_{i_{0},j_{0},κ′_{i_{0},j_{0}}} = {x},
⋆ Q′_{i,j,k} = Q_{i,j,k} in all other cases.

The pair (n′, r′) thus appears in the decomposition of λ, and m′ = k + 1. Moreover, we directly find that ∑_{i=1}^{m} r_{i}n_{i} ≤ ∑_{i=1}^{m′} r′_{i}n′_{i} and that these two numbers have the same parity. By the induction hypothesis the lemma is then also true for (n; r). □

Example 3.13. Let us decompose a_{[2^{2}]}|_{g} starting with the general case:

a_{[2^{2}]}|_{g} = I ∑_{f∈P_{g}}(− ∑_{α∈P^{1}(k_{2})} χ_{2,2}(f(α)))^{2} = I ∑_{f∈P_{g}} ∑_{α,β∈P^{1}(k_{2})} χ_{2,2}(f(α)f(β)) =

= u_{g}^{((2,2);(1,1))} + 2u_{g}^{((2,1);(1,2))} + 2u_{g}^{((2);(2))} + u_{g}^{((1,1);(2,2))} + u_{g}^{((1);(2))}.

Example 3.14. The decomposition of a_{[1^{4},2]}|_{g}, starting with the general case:

a_{[1^{4},2]}|_{g} = −u_{g}^{((2,1,1,1,1);(1,1,1,1,1))} − 6u_{g}^{((2,1,1,1);(1,2,1,1))} − 3u_{g}^{((2,1,1);(1,2,2))}

− 4u_{g}^{((2,1,1);(1,1,1))} − u_{g}^{((2,1);(1,2))} − u_{g}^{((1,1,1,1,1);(2,1,1,1,1))} − 6u_{g}^{((1,1,1,1);(2,2,1,1))}

− 4u_{g}^{((1,1,1,1),(1,1,1,1))} − 3u_{g}^{((1,1,1);(2,2,2))} − 22u_{g}^{((1,1,1);(2,1,1))}

− 7u_{g}^{((1,1);(2,2))} − 8u_{g}^{((1,1);(1,1))} − u_{g}^{((1);(2))}.

3.1. The cases of genus 0 and 1. We would like to have an equality of the same kind as in equation (3.2), but for curves of genus 0 and 1. Every curve of genus 0 or 1 has a morphism to P^{1} of degree 2 and in the same way as for larger genera, it then follows that every k-isomorphism class of curves of genus 0 or 1 has a representative among the curves coming from polynomials in P_{0} and P_{1} respectively. But there is a difference, compared to the larger genera, in that for curves of genus 0 or 1 the g_{2}^{1} is not unique. In fact, the group G induces (in the same way as for g ≥ 2) all k-isomorphisms between curves corresponding to elements of P_{0} and P_{1} that respect their given morphisms to P^{1} (i.e a fixed g_{2}^{1}), but not all k-isomorphisms between curves of genus 0 or 1 are of this form. Let us, for all r ≥ 0, define the category A_{r} consisting of tuples (C, Q_{0}, …, Q_{r}) where C is a curve of genus 1 defined over k and the Q_{i} are, not necessarily distinct, points on C defined over k. The morphisms of A_{r} are, as expected, isomorphisms of the underlying curves that fix the marked points. Note that A_{0} is isomorphic to the category M_{1,1}(k). We also define, for all r ≥ 0, the category B_{r} consisting of tuples (C, L, Q_{1}, …, Q_{r}) of the same kind as above, but where L is a g_{2}^{1}. A morphism of B_{r} is an isomorphism ϕ of the underlying curves that fixes the marked points, and such that there is an isomorphism τ making the following diagram commute:

C ⟶^{ϕ} C′
L↓  ↓_{L′}
P^{1} ⟶^{τ} P^{1}.

Consider P_{1} as a category where the morphisms are given by the elements of G. To every element of P_{1} there corresponds, precisely as for g ≥ 2, a curve C_{f} together with a g_{2}^{1} given by the morphism to P^{1}, thus an element of B_{0}. Since every morphism in B_{0} between objects corresponding to elements of P_{1} is induced by an element of G, and since for every k-isomorphism class of an element in B_{0} there is a representative in P_{1}, the two categories P_{1} and B_{0} are equivalent. For all r ≥ 1 there are equivalences of the categories A_{r} and B_{r} given by

(C, Q_{0}, …, Q_{r}) ↦ (C, |Q_{0} + Q_{1}|, Q_{1}, …, Q_{r}),

with inverse

(C, L, Q_{1}, …, Q_{r}) ↦ (C, |L − Q_{1}|, Q_{1}, …, Q_{r}).

We therefore have the equality

∑_{[X]∈A_{r}/≅_{k}} \frac{1}{|Aut_{k}(X)|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}} = ∑_{[Y]∈B_{r}/≅_{k}} \frac{1}{|Aut_{k}(Y)|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}}.

The Riemann hypothesis tells us that |a_{r}(C)| ≤ 2g\sqrt{q^{r}}, for any finite field k with q elements and for any curve C defined over k of genus g. For genus 1 this implies that |C(k)| ≥ q + 1 − 2\sqrt{q} > 0, and thus every genus 1 curve has a point defined over k. There is therefore a number s such that 1 ≤ |C(k)| ≤ s for all genus 1 curves C. As in the argument preceding equation (2.2) we can take a representative (C, Q_{0}, …, Q_{r}) for each element of A_{r}/ ≅_{k} and act with Aut_{k}(C, Q_{0}), respectively for each representative (C, L, Q_{1}, …, Q_{r}) of B_{0}/ ≅_{k} act with Aut_{k}(C, L), and by considering the orbits and stabilizers we get

∑_{j=1}^{s} j^{r} ∑_{\begin{matrix}[X]∈A_{0}/≅_{k} \\ |C(k)|=j \end{matrix}} \frac{1}{|Aut_{k}(X)|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}} = ∑_{j=1}^{s} j^{r} ∑_{\begin{matrix}[Y]∈B_{0}/≅_{k} \\ |C(k)|=j \end{matrix}} \frac{1}{|Aut_{k}(Y)|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}}.

Since this holds for all r ≥ 1 we can, by a Vandermonde argument, conclude that we have an equality as above for each fixed j. We can therefore extend Definition 2.2 to genus 1 in the following way:

(3.4) a_{λ}|_{1} := ∑_{\begin{matrix}[(C,Q_{0})]∈ \\ M_{1,1}(k)/≅_{k} \end{matrix}} \frac{1}{|Aut_{k}(C, Q_{0})|} ∏_{i=1}^{ν} a_{i}(C)^{λ_{i}} =

= ∑_{[f]∈P_{1}/G} \frac{1}{|Stab_{G}(f)|} ∏_{i=1}^{ν} a_{i}(C_{f})^{λ_{i}} = I ∑_{f∈P_{1}} ∏_{i=1}^{ν} a_{i}(C_{f})^{λ_{i}},

which gives an agreement with equation (3.2). All curves of genus 0 are isomorphic to P^{1} and a_{r}(P^{1}) = 0 for all r ≥ 1. In this trivial case we just let equation (3.2) be the definition of a_{λ}|_{0}.

4. Recurrence relations for u_{g} in odd characteristic

This section will be devoted to finding, for a fixed finite field k with an odd number of elements and for a fixed pair (n; r) ∈ N_{m}, a recurrence relation for u_{g}. Notice that we will often suppress the pair (n; r) in our notation and for instance write u_{g} instead of u_{g}^{(n;r)}. Fix a nonsquare t in k and an α = (α_{1}, …, α_{m}) ∈ A(n). Multiplying with the element t gives a fixed point free action on the set P_{g} and therefore

(4.1) u_{g,α} = I ∑_{f∈P_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}} = I ∑_{f∈P_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(t f(α_{i}))^{r_{i}} =

= I ∑_{f∈P_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(t)^{r_{i}} χ_{2,n_{i}}(f(α_{i}))^{r_{i}} = (−1)^{∑_{i=1}^{m} r_{i}n_{i}} u_{g,α}.

This computation and Lemmas 3.8 and 3.12 proves the following lemma.

Lemma 4.1. For any g ≥ −1, (n; r) ∈ N_{m} and α ∈ A(n), if ∑_{i=1}^{m} r_{i}n_{i} is odd then u_{g,α} = 0. Consequently, a_{λ}|_{g} is equal to 0 if it has odd weight.

Thus, the only interesting cases are those for which ∑_{i=1}^{m} r_{i}n_{i} is even.

Remark 4.2. The last statement of Lemma 4.1 can also be found as a consequence of the existence of the hyperelliptic involution.

We also see from equation (4.1) that

(4.2)  u_{g,α} = I (q − 1) ∑_{f∈P′_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}} if ∑_{i=1}^{m} r_{i}n_{i} is even.

Definition 4.3. Let Q_{g} denote the set of all polynomials (that is, not necessarily square-free) with coefficients in k and of degree 2g + 1 or 2g + 2, and let Q′_{g} ⊂ Q_{g} consist of the monic polynomials. For a polynomial h ∈ Q_{g} we let h(∞) be the coefficient of the term of degree 2g + 2 (which extends the earlier definition for elements in P_{g}). For any g ≥ −1, (n; r) ∈ N_{m} and α ∈ A(n), define

U_{g,α}^{(n;r)} := I ∑_{h∈Q_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(h(α_{i}))^{r_{i}},

U_{g}^{(n;r)} := ∑_{α∈A(n)} U_{g,α}^{(n;r)} and Û_{g}^{(n;r)} := ∑_{i=−1}^{g} U_{i}^{(n;r)}.

We will find an equation relating U_{g} to u_{i} for all −1 ≤ i ≤ g. Moreover, for g large enough we will be able to compute U_{g}. Together, this will give us our recurrence relation for u_{g}. With the same arguments as was used to prove equation (4.2) one shows that

(4.3)  U_{g,α} = I (q − 1) ∑_{h∈Q′_{g}} ∏_{i=1}^{m} χ_{2,n_{i}}(h(α_{i}))^{r_{i}} if ∑_{i=1}^{m} r_{i}n_{i} is even.

Definition 4.4. For any α = (α_{1}, …, α_{m}) ∈ A′(n), let b_{j} = b_{j}^{n} be the number of monic polynomials l of degree j such that l(α_{i}) is nonzero for all i. Let us also put b̂_{j} = b̂_{j}^{n} := ∑_{i=0}^{j} b_{i}^{n}.

Lemma 4.5. For each j ≥ 0 and n ∈ ℕ_{≥1}^{m}, we have the equality

(4.4)  b_{j} = q^{j} + ∑_{i=1}^{j}(−1)^{i} ∑_{\begin{matrix}1≤m_{1}<...<m_{i}≤m \\ ∑_{l=1}^{i} n_{m_{l}} ≤j \end{matrix}} q^{j−∑_{l=1}^{i} n_{m_{l}}}

from which it follows that b_{j} does not depend upon the choice of α ∈ A′(n).

Proof: The numbers b_{j} can be computed by inclusion-exclusion, where the choice of 1 ≤ m_{1} < … < m_{i} ≤ m corresponds to demanding the polynomial to be 0 in the points α_{m_{1}}, …, α_{m_{i}}. □

Notation 4.6. For any α ∈ A′(n), let p_{α_{i}} denote the minimal polynomial of α_{i} and put p_{α} := ∏_{i=1}^{m} p_{α_{i}}.

Lemma 4.7. For any α ∈ A′(n) there is a one-to-one correspondence between polynomials f defined over k with deg(f) ≤ |n| − 1, and tuples (f(α_{1}), …, f(α_{m})) ∈ ∏_{i=1}^{m} k_{n_{i}}.

Proof: For any α ∈ A′(n) we have deg(p_{α_{i}}) = n_{i} and gcd(p_{α_{i}}, p_{α_{j}}) = 1 if i ≠ j. The lemma now follows from the Chinese remainder theorem, which tells us that the morphism k[x]/p_{α} → ∏_{i=1}^{m} k[x]/p_{α_{i}} ≅ ∏_{i=1}^{m} k_{n_{i}} given by f(x) ↦ (f(α_{1}), …, f(α_{m})) is an isomorphism. □

Notation 4.8. Let R_{j} denote the set of polynomials of degree j and let R′_{j} be the subset containing the monic polynomials.

We will divide into two cases.

4.1. The case α ∈ A′(n). Fix an element α ∈ A′(n). Any nonzero polynomial h can be written uniquely in the form h = f l^{2} where f is a square-free polynomial and l is a monic polynomial. This statement translates directly into the equality

U_{s,α} = I ∑_{j+k=s} ∑_{l∈R′_{j}} ∑_{f∈P_{k}} ∏_{i=1}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}}χ_{2,n_{i}}(l(α_{i}))^{2r_{i}} = ∑_{j=0}^{s+1} b_{j}u_{s−j,α},

because for any β ∈ A^{1}(k_{s}), χ_{2,s}((fl^{2})(β)) = χ_{2,s}(f(β)) if l(β) ≠ 0. Summing this equality over all s between −1 and g gives

(4.5)  Û_{g,α} = ∑_{j=0}^{g+1} b̂_{j}u_{g−j,α}.

If r_{i} = 2 for all i, then it follows from equation (4.3) that

U_{s,α} = I (q − 1) ∑_{h∈Q′_{s}} ∏_{i=1}^{m} χ_{2,n_{i}}(h(α_{i}))^{2} = I (q − 1)(b_{2s+2} + b_{2s+1}).

Summing this equality over all s between −1 and g gives

(4.6)  Û_{g,α} = I (q − 1)b̂_{2g+2} for g ≥ −1 if ∀i : r_{i} = 2.

In Û_{g,α} we are summing over all polynomials h of degree less than or equal to 2g + 2, and every h can uniquely be written on the form h_{1} + p_{α}h_{2}, with deg h_{1} ≤ |n| − 1 and deg h_{2} ≤ 2g + 2 − |n|. Hence if 2g + 2 ≥ |n| − 1 we find that

Û_{g,α} = I q^{2g+3−|n|} ∑_{s=1}^{|n|−1} ∑_{h_{1}∈R_{s}} ∏_{i=1}^{m} χ_{2,n_{i}}(h_{1}(α_{i}))^{r_{i}}.

Using Lemma 4.7 we can reformulate this equality as

Û_{g,α} = I q^{2g+3−|n|} ∑ ; (β_{1},...,β_{m})∈∏_{i=1}^{m} k_{n_{i}} ∏_{i=1}^{m} χ_{2,n_{i}}(β_{i})^{r_{i}}.

For any j, half of the nonzero elements in k_{j} are squares and half are nonsquares, and thus if r_{i} = 1 for some i, we can conclude from this equality that

(4.7)  Û_{g,α} = 0 for g ≥ (|n| − 3)/2 if ∃i : r_{i} = 1.

4.2. The case α ∈ A(n) \ A′(n). Fix an element α ∈ A(n) \ A′(n). We can assume that α_{1} = ∞, and then α̃ := (α_{2}, …, α_{m}) ∈ A′(ñ) where ñ := (n_{2}, …, n_{m}). If h ∈ Q_{g} and f ∈ P_{j} such that h = f l^{2} for some monic polynomial l (which is then unique), then h(∞) = f(∞), because the coefficient of h of degree 2g + 2 must equal the coefficient of f of degree 2j + 2. As in Section 4.1 we get (4.8)

U_{g,α} = I ∑_{j+k=g} ∑_{l∈R′_{j}} ∑_{f∈P_{k}} f(∞) ∏_{i=2}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}}χ_{2,n_{i}}(l(α_{i}))^{2r_{i}} = ∑_{j=0}^{g+1} bñ ; ju_{g−j,α}.

If ∑_{i=1}^{m} r_{i}n_{i} is even, equation (4.3) and the definition of h(∞) shows that

(4.9)  U_{g,α} = I (q − 1) ∑_{h∈R′_{2g+2}} ∏_{i=2}^{m} χ_{2,n_{i}}(h(α_{i}))^{r_{i}}.

If r_{i} = 2 for all i, then equation (4.9) tells us that

(4.10)  U_{g,α} = I (q − 1)bñ ; 2g+2 for g ≥ −1, ∀i : r_{i} = 2.

If 2g + 2 ≥ |n| − 1, an element h ∈ R′_{2g+2} can be written uniquely as h = h_{1} + pα̃h_{2}, where deg(h_{1}) ≤ |n| − 2, deg(h_{2}) ≥ 0 and h_{2} monic. In the same way as in Section 4.1 we can (if ∑_{i=1}^{m} r_{i}n_{i} is even) use this together with equation (4.9) and Lemma 4.7 to conclude that

(4.11)  U_{g,α} = 0 for g ≥ (|n| − 3)/2, ∃i : r_{i} = 1,

which of course also holds if ∑_{i=1}^{m} r_{i}n_{i} is odd by Lemma 4.1 and equation (4.8).

Remark 4.9. Fix an α ∈ A(n). If there is an element β ∈ A^{1}(k) such that β ∉ {α_{1}, …, α_{n}}, then T (α) := (T (α_{1}), …, T (α_{n})) is in A′(n), where T is the projective transformation of P_{k}^{1} defined by x ↦ βx/(x − β). In the notation of equation (3.1), χ_{2,n_{i}}(f(T (α_{i}))) = χ_{2,n_{i}}(f̃(α_{i})) (with e = 1). Since this induces a permutation of P_{g}, we find that u_{g,α} = u_{g,T (α)} and similarily that U_{g,α} = U_{g,T (α)}. So, if q ≥ |n|, then equations (4.5), (4.6) and (4.7) will also hold for α ∈ A(n) \ A′(n). By Lemma 4.10 in the next section, we will see that this is true even if q < |n|.

4.3. The two cases joined. In this section we will put the results of the two previous sections together using the following lemma.

Lemma 4.10. For any ñ = (n_{2}, …, n_{m}), if n = (1, n_{2}, …, n_{m}) then b̂_{j}^{n} = bñ ; j.

Proof: Fix any tuple n = (n_{1}, …, n_{m}) and put n := |n|. If we let t_{i} = q^{n_{i}} in the formula

∏_{i=1}^{m}(t_{i} − 1) = t_{1} ⋯ t_{m} + ∑_{i=1}^{m}(−1)^{i} ∑_{1≤m_{1}<...<m_{i}≤m} t_{1} ⋯ t_{m} \frac{1}{t_{m_{1}}} ⋯ \frac{1}{t_{m_{i}}} ,

then the right hand side is equal to the right hand side of equation (4.4), and hence

(4.12)  ∏_{i=1}^{m}(q^{n_{i}} − 1) = b_{n}^{n}.

Say that b_{j}^{n} = ∑_{i=0}^{j} c_{j,i}^{n}q^{i} and b̂_{j}^{n} = ∑_{i=0}^{j} ĉ_{j,i}^{n}q^{i}. If i ≤ j then equation (4.4) implies that c_{j,i}^{n} = c_{n,n+i−j}^{n} and hence ĉ_{j,i}^{n} = ∑_{s=0}^{j} c_{n,n+i−s}^{n}. By equation (4.12) we know that q − 1 divides b_{n}^{n}, and if b_{n}^{n}/(q − 1) = ∑_{i=0}^{n−1} d_{i}q^{i} then ĉ_{j,i}^{n} = d_{n−1+i−j}. So, if n_{1} = 1 and ñ = (n_{2}, …, n_{m}) then b_{n}^{n}/(q − 1) = bñ ; n−1 and thus ĉ_{j,i}^{n} =

cñ ; n−1,n−1+i−j = cñ ; j,i. □

Notation 4.11. Let us write J := I (q − 1) |A(n)|.

Theorem 4.12. For any pair (n; r) ∈ N_{m},

∑_{j=0}^{g+1} b̂_{j}u_{g−j} = \begin{cases}J b̂_{2g+2} & if ∀i : r_{i} = 2, g ≥ −1; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−3}{2}. \end{cases}

Proof: The theorem follows from combining equations (4.5), (4.6), (4.7) and equations (4.8), (4.10), (4.11), using Lemma 4.10. □

Note that with this theorem we can, for any (n; r) ∈ N_{m} such that r_{i} = 2 for all i, compute u_{g} for any g. Moreover, for any pair (n; r) we can compute u_{g} for any g, if we already know u_{g} for all g < (|n| − 3)/2.

Lemma 4.13. For any n, q − 1 divides b_{|n|}^{n}, and if we write b_{|n|}^{n}/(q − 1) =

∑_{i=0}^{|n|−1} d_{i}q^{i} then b̂_{j} − qb̂_{j−1} = d_{|n|−1−j}.

Proof: The first claim is shown in the proof of Lemma 4.10. Using the notation of that proof we find that b̂_{j} − qb̂_{j−1} = ∑_{i=0}^{j} d_{n−1+i−j}q^{i} − ∑_{i=0}^{j−1} d_{n+i−j}q^{i+1} = d_{n−1−j}. Note that d_{n−1−j} only depends upon n and not on q. □

Theorem 4.14. For any pair (n; r) ∈ N_{m},

∑_{j=0}^{min(|n|−1,g+1)}(b̂_{j} − qb̂_{j−1})u_{g−j} = \begin{cases}J (b̂_{2g+2} − qb̂_{2g}) & if ∀i : r_{i} = 2, g ≥ 0; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−1}{2}. \end{cases}

Proof: Let us temporarily put F (s) := ∑_{j=0}^{s+1} b̂_{j}u_{s−j}. From Lemma 4.13 we find that b̂_{j} − qb̂_{j−1} = 0 if j > |n| − 1. The theorem then follows from applying Theorem 4.12 to the expression F (g) − qF (g − 1). □

For g ≥ (|n| − 1)/2, Theorem 4.14 presents us with a linear recurrence relation for u_{g} which has coefficients that are independent of the finite field k.

Example 4.15. If (n; r) = ((2, 1, 1, 1); (1, 2, 1, 1)) then b_{5}^{n}/(q−1) = (q^{2} −1)(q− 1)^{2} = q^{4} − 2q^{3} + 2q − 1. Applying Lemma 4.13 and then Theorem 4.14 we get

u_{g} − 2u_{g−1} + 2u_{g−3} − u_{g−4} = 0 for g ≥ 3.

Example 4.16. Let us compute u_{g}, for all g ≥ −1, when (n; r) = ((1, 1, 1), (2, 2, 2)). We have that u_{−1} = J = 1 and since r_{i} = 2 for all i, Theorem 4.14 gives the equality u_{0} = 2u_{−1} + J(q^{2} − 3q + 1) = q^{2} − 3q + 3. Applying Theorem 4.14 again we get

u_{g} − 2u_{g−1} + u_{g−2} = q^{2g−1}(q − 1)^{3} for g ≥ 1.

Solving this recurrence relation gives

u_{g}^{((1,1,1);(2,2,2))} = \frac{q^{2g+3}(q − 1) − (2g + 2)(q^{2} − 1) + 3q + 1}{(q + 1)^{2}} for g ≥ −1.

5. Linear recurrence relations for a_{λ}|_{g}

Remark 5.1. From a sequence v_{n} that fulfills a linear recurrence relation with characteristic polynomial C we can, for any polynomial D, in the obvious way construct a linear recurrence relation for v_{n} with characteristic polynomial CD. Thus, from two sequences v_{n} and w_{n} that each fulfill linear recurence relation with characteristic polynomial C and D respectively, we can construct a linear recurence relation for the sequence v_{n} + w_{n} with characteristic polynomial lcm(C, D).

Theorem 5.2. By applying Theorem 4.14 to each pair (n; r) appearing in the decomposition (given by Lemma 3.8) of a_{λ}|_{g}, we get a linear recurrence relation for a_{λ}|_{g}. The characteristic polynomial C(X) of this linear recurrence relation equals

(5.1)  \frac{1}{X − 1} ∏_{i=1}^{ν}(X^{i} − 1)^{λ_{i}}.

Proof: Fix any pair (n; r) in the decomposition of a_{λ}|_{g} and put n = |n|. Lemma 4.13 tells us that b̂_{j} − qb̂_{j−1} is equal to the coefficient of q^{n−1−j} in b_{n}/(q − 1). If g ≥ n − 1, then these numbers are also the coefficients in the recurrence relation given by Theorem 4.14. By equation (4.12), the characteristic polynomial C_{(n;r)} of this linear recurrence relation is equal to (∏_{i=1}^{m}(X^{n_{i}} − 1))/(X − 1). We find that the linear recurrence relation in the general case (see Definition 3.9) will have characteristic polynomial equal to C. Moreover, we find (by their construction in the proof of Lemma 3.8) that if (n; r) is a degenerate case then C_{(n;r)}|C. The theorem now follows from Remark 5.1. □

Theorem 5.2 tells us that if we can compute a_{λ}|_{g} for g < |λ| − 1 then we can compute it for every g. But note that by considering the individual cases in the decomposition of a_{λ}|_{g} we will do much better in Section 7, in the sense that we will be able to use information from curves of only genus 0 and 1 to compute a_{λ}|_{g} for any λ such that |λ| ≤ 6.

Example 5.3. For λ = [1^{4}, 2] the characteristic polynomial equals (X−1)^{4}(X+ 1), so if V_{g} is a particular solution to the linear recurrence relation for a_{[1^{4},2]}|_{g} then

a_{[1^{4},2]}|_{g} = V_{g} + A_{3}g^{3} + A_{2}g^{2} + A_{1}g + A_{0} + B_{0}(−1)^{g},

where A_{0}, A_{1}, A_{2}, A_{3} and B_{0} do not depend upon g.

6. Computing u_{0}

In this section we will see that we can compute u_{0} for any choice of a pair (n; r) ∈ N_{m}. This is due to the fact that if C is a curve of genus 0 then, for all r, |C(k_{r})| = 1 + q^{r} or equivalently a_{r}(C) = 0.

Construction-Lemma 6.1. For each (n; r) ∈ N_{m}, there are numbers c_{1}, …, c_{s} and pairs (n^{(1)}; r^{(1)}), …, (n^{(s)}; r^{(s)}), where r^{(i)} = (2, …, 2) for all i, such that for any finite field k,

u_{0}^{(n;r)} = ∑_{i=1}^{s} c_{i}u_{0}^{(n^{(i)};r^{(i)})}.

Proof: Fix a pair (n; r) ∈ N_{m}. We will use induction over the number n := |n|, where the base case n = 0 is trivial. Let us put (ñ; r̃) = ((n_{2}, …, n_{m}); (r_{2}, …, r_{m})). For an α̃ = (α_{2}, …, α_{m}) ∈ A(ñ) let P̂1 ; α̃(k_{i}) be the set of all points in P^{1}(k_{i}) \ {α_{2}, …, α_{m}} that are not defined over a proper subfield of k_{i}. The set of α_{1} ∈ P^{1}(k_{n_{1}}) such that (α_{1}, …, α_{m}) ∈ A(n) then equals

(6.1)  P^{1}(k_{n_{1}}) \ ( ⋃_{i|n_{1}} P̂1 ; α̃(k_{i}) ⋃_{n_{i}|n_{1}}{α_{i}, …, F^{n_{i}−1}α_{i}}).

Assume now that the lemma has been proved for all pairs of degree strictly less than n. By reordering the elements of the pair (n; r) we can assume that r_{1} = 1, because otherwise r = (2, …, 2) and we are done. By applying equation (6.1) we get

(6.2) I ∑_{α∈A(n)} ∏_{i=1}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}} = I ∑_{α̃∈A(ñ)} ∏_{i=2}^{m} χ_{2,n_{i}}(f(α_{i}))^{r_{i}}·

· (−a_{n_{1}}(C_{f}) − ∑_{i|n_{1}} ∑_{β∈P̂1 ; α̃(k_{i})} χ_{2,n_{1}}(f(β)) − ∑_{n_{i}|n_{1}} n_{i} χ_{2,n_{1}}(f(α_{i}))).

Let us put (n^{(i)}; r^{(i)}) = ((i, n_{2}, …, n_{m}); (n_{i}/i, r_{2}, …, r_{m})) for all i that divides n_{i} and r̃^{(i)} = (r_{2}, …, r_{i−1}, r_{i} n_{1}/n_{i}, r_{i+1}, …, r_{m}) for all n_{i} that divides n_{1}.

Summing both sides of equation (6.2) over polynomials f ∈ P_{0} and using that a_{n_{1}}(C_{f}) = 0 then gives

(6.3)  u_{0}^{(n;r)} = − ∑_{i|n_{1}} u_{0}^{(n^{(i)};r^{(i)})} − ∑_{n_{i}|n_{1}} n_{i}u_{0}^{(ñ;r̃^{(i)})}.

Since |ñ| < n and |n^{(i)}| < n, the lemma follows by induction from equation (6.3). □

Example 6.2. In the case (n; r) = ((6, 6, 3, 1, 1); (1, 1, 2, 2, 2)), the first step in the procedure in the proof of Lemma 6.1 equals

u_{0}^{(n;r)} = −u_{0}^{((6,3,3,1,1);(1,2,2,2,2))} − u_{0}^{((6,3,2,1,1);(1,2,1,2,2))} − u_{0}^{((6,3,1,1,1);(1,2,2,2,2))}

− 5u_{0}^{((6,3,1,1);(1,2,2,2))} − 6u_{0}^{((6,3,1,1);(2,2,2,2))}.

Example 6.3. In the case (n; r) = ((4, 1, 1, 1); (1, 2, 1, 1)), the procedure in the proof of Lemma 6.1 gives

u_{0}^{(n;r)} = u_{0}^{((2,1,1);(2,2,2))}+u_{0}^{((1,1,1);(2,2,2))}+u_{0}^{((1,1);(2,2))}−u_{0}^{((2,1);(2,2))}−u_{0}^{((1);(2))}

7. Results for weight up to 7 in odd characteristic

We will in this section show that we, for any number g and any finite field k of odd characteristic, can compute all a_{λ}|_{g} of weight at most 7. This is achieved by decomposing a_{λ}|_{g} using Lemma 3.8 and employing the recurrence relation of Theorem 4.12 on the different parts. This involves finding the necessary base cases for the recurrence relations and that will be possible with the help of results on genus 0 curves obtained in Section 6, and on genus 1 curves obtained in the article [1]. We will write a_{λ}|_{g,odd} and u_{g,odd} to stress that all results are in the case of odd characteristic. See Section 10 for results in the case of even characteristic.

Example 7.1. Theorem 4.12 is applicable even if the degree is 0 (if considered as a case when r_{i} = 2 for all i) and with b̂_{j} = ∑_{i=0}^{j} q^{i}. From Theorem 4.12 we find that a_{0}|_{0,odd} = Jq^{2} = q/(q^{2} − 1) and again from Theorem 4.12 that

a_{0}|_{g,odd} = J(q^{2g+2} − q^{2g}) = q^{2g−1} for g ≥ 1.

This result can also be found in [7, Proposition 7.1].

7.1. Degree at most 3. When the degree of the pair (n; r) is at most 3 we find using Theorem 4.12 that we do not need any base cases to compute u_{g} for every g.

Example 7.2. Let us consider (n; r) = ((2); (1)). We have u_{−1} = J = 1/(q+1) and applying Theorem 4.12 we get u_{0} = −(q +1)u_{−1} = −1. Theorem 4.14 tells us that u_{g} = −u_{g−1} for g ≥ 1 and thus

u_{g,odd}^{((2);(1))} = (−1)^{g+1} for g ≥ 0.

Example 7.3. The result for a_{[2]}|_{g,odd} is

a_{[2]}|_{g,odd} = −u_{g}^{((2);(1))} − u_{g}^{((1);(2))} = (−1)^{g} − q^{2g} for g ≥ 0.

Example 7.4. The result for a_{[1^{2}]}|_{g,odd} is

a_{[1^{2}]}|_{g,odd} = u_{g}^{((1,1);(1,1))} + u_{g}^{((1);(2))} = −1 + q^{2g} for g ≥ 0.

Remark 7.5. The result for (q^{2} + 1) a_{0}|_{g,odd} − a_{[2]}|_{g,odd} can be found in lecture notes by Bradley Brock and Andrew Granville from 28 July 2003.

Example 7.6. Consider the case (n; r) = ((1, 1, 1); (2, 1, 1)). We have u_{−1} = J = 1 and from Theorem 4.12 we get u_{0} = −(q−2)u_{−1} = −q+2. Theorem 4.14 gives the recurrence relation u_{g} = 2u_{g−1} − u_{g−2} for g ≥ 1 and hence

u_{g,odd}^{((1,1,1);(2,1,1))} = g(−q + 1) − q + 2.

7.2. Degree 4 or 5. From Theorem 4.12 we find that when the degree of the pair (n; r) is 4 or 5 we need the base case of genus 0. But the genus 0 case is always computable using Lemma 6.1 and then Theorem 4.12, and hence the same is true for u_{g} for all g.

Example 7.7. For (n; r) = ((2, 1, 1); (1, 1, 1)) we have u_{−1} = q and from Lemma 6.1 it follows that

u_{0}^{((2,1,1);(1,1,1))} = −u_{0}^{((2,1);(1,2))} = u_{0}^{((1,1);(2,2))} + u_{0}^{((1);(2))} = q.

Using Theorem 4.12 we get u_{1} = −(q − 1)u_{0} − (q^{2} − q − 1)u_{−1} = −q^{3} + 2q. Solving the recurrence relation u_{g} = u_{g−1} −u_{g−2} −u_{g−3} for g ≥ 2, coming from Theorem 4.14, gives

u_{g,odd}^{((2,1,1);(1,1,1))} = \frac{1}{4} (q^{3} − q)(−2g + (−1)^{g} − 1) + q.

Example 7.8. The result for a_{[1^{2},2]}|_{g,odd} is

a_{[1^{2},2]}|_{g,odd} = −u_{g}^{((2,1,1);(1,1,1))} − u_{g}^{((2,1);(1,2))} − u_{g}^{((1,1,1);(2,1,1))}

− u_{g}^{((1,1);(2,2))} − 2u_{g}^{((1,1);(1,1))} − u_{g}^{((1);(2))} =

= −\frac{q^{2g+2} − 1}{q + 1} − q^{2g} + \frac{1}{2} g(q^{3} + q − 2) + \frac{1}{2} \begin{cases}2q & if g ≡ 0 mod 2 \\ q^{3} − q − 2 & if g ≡ 1 mod 2 \end{cases}

7.3. Weight 6. We will not be able to compute u_{g} for all pairs (n; r) of degree 6. But we will be able to compute u_{g} for all pairs (n; r) that are general cases in the decomposition of a_{λ}|_{g} for λ’s of weight 6. This will be sufficient to compute all a_{λ}|_{g} of weight 6, because we saw in Lemma 3.11 that only the general case will have degree 6 and therefore all degenerate cases are covered in Sections 7.1 and 7.2. Let u_{g} be the general case in the decomposition of a_{λ}|_{g}. When the degree is equal to 6 we see from Theorem 4.12 that we need the base cases of genus 0 and 1 to compute u_{g} for all g. As we know, we can always compute u_{0} using Lemma 6.1. For genus 1, the numbers a_{λ}|_{1} have been computed for weight up to 6 by the author. This was done by embedding every genus 1 curve with a given point as a plane cubic curve, see [1, Section 15]. Since we know all the degenerate cases in the decomposition of a_{λ}|_{1} we can then compute the general case u_{1}.

Example 7.9. Let us deal with (n; r) = ((6); (1)) which is the generic case in the decomposition of a_{[6]}|_{g,odd} and for which we have u_{−1} = J = q^{3} + q − 1. Using Lemma 6.1 we get

u_{0}^{((6);(1))} = −u_{0}^{((3);(2))} − u_{0}^{((2);(1))} − u_{0}^{((1);(2))} = −u_{0}^{((3);(2))} = −q^{2}.

Using the results of [1, Section 15] we find that a_{[6]}|_{1} = q − 1. Decomposing a_{[6]}|_{g} gives a_{[6]}|_{1} = −u_{1}^{((6);(1))} − u_{1}^{((3);(2))} − u_{1}^{((2);(1))} − u_{1}^{((1);(2))}. Thus, using Example 7.2, we get u_{1} = −(q − 1) − (q^{4} − q^{2} − q − 1) − 1 − q^{2} = −q^{4} + 1. We can now apply Theorem 4.12 which gives u_{2} = −(q + 1)u_{1} − (q^{2} + q + 1)u_{0} − (q^{3} + q^{2} + q + 1)u_{−1} = −q^{6} + q^{2} − q, u_{3} = −u_{2} − u_{1} − u_{0} − u_{−1} = q^{6} + q^{4} − q^{3} and u_{4} = −u_{3} − u_{2} − u_{1} − u_{0} − u_{−1} = 0. If we then multiply the characteristic polynomial for the linear recurrence relation of u_{g} by X − 1 we get u_{g} = u_{g−6} for all g ≥ 5.

Example 7.10. The result for a_{[6]}|_{g,odd} is

a_{[6]}|_{g,odd} = −u_{g}^{((6);(1))} − u_{g}^{((3);(2))} − u_{g}^{((2);(1))} − u_{g}^{((1);(2))} = −q^{2g} − \frac{q^{2g+3}(q − 1)}{q^{2} − q + 1} +

+ \frac{1}{q^{2} − q + 1} \begin{cases}q^{2} & if g ≡ 0 mod 3 \\ −q^{2} − 1 & if g ≡ 1 mod 3 \\ 1 & if g ≡ 2 mod 3 \end{cases} + \begin{cases}q^{2} + 1 & if g ≡ 0 mod 6 \\ q^{4} − 2 & if g ≡ 1 mod 6 \\ q^{6} − q^{2} + q + 1 & if g ≡ 2 mod 6 \\ −q^{6} − q^{4} + q^{3} − 1 & if g ≡ 3 mod 6 \\ 1 & if g ≡ 4 mod 6 \\ −q^{3} − q & if g ≡ 5 mod 6 \end{cases}

Remark 7.11. For any choice of λ and g, consider a_{λ}|_{g,odd} as a function of the number q of elements of the finite field k of odd characteristic. If λ is of weight at most 7 it follows from our computations that this function is a polynomial in the variable q. This will not continue to hold when considering for instance a_{[1^{6}]}|_{3}, that is, also including finite fields of even characteristic, see Example 10.6. But it will also not hold for instance for a_{[1^{10}]}|_{1,odd}, which for prime fields will be a polynomial function minus the Ramanujan τ-function, compare [15, Corollary 5.4].

8. Representatives of hyperelliptic curves in even characteristic

Let k be a finite field with an even number of elements. We will again describe the hyperelliptic curves of genus g ≥ 2 defined over k by their degree 2 morphism to P^{1}. If we choose an affine coordinate x on P^{1} we can write the induced degree 2 extension of the function field of P^{1} in the form y^{2} +h(x)y +f(x) = 0, where h and f are polynomials defined over k that fulfill the following conditions:

(8.1) 2g + 1 ≤ max(2 deg(h), deg(f)) ≤ 2g + 2;

gcd(h, f^{′2} + fh^{′2}) = 1; (8.2)

t ∤ gcd(h_{∞}, f_{∞}^{′2} + f_{∞}h_{∞}^{′2}). (8.3)

The last condition comes from the nonsingularity of the point(s) in infinity, around which the curve can be described in the variable t = 1/x as y^{2} + h_{∞}(t)y + f_{∞}(t) = 0, where h_{∞} := t^{g+1}h(1/t) and f_{∞} := t^{2g+2}f(1/t). We therefore define h(∞) and f(∞) to be equal to the degree g + 1 and 2g + 2 coefficient respectively. For a reference see for instance [19, p. 294].

Definition 8.1. Let P_{g} denote the set of pairs (h, f) of polynomials defined over k, where h is nonzero, that fulfill all three conditions (8.1), (8.2) and (8.3). Write C_{(h,f)} for the curve corresponding to the element (h, f) in P_{g}.

To each k-isomorphism class of objects in H_{g}(k) there is a pair (h, f) in P_{g} such that C_{(h,f)} is a representative. All k-isomorphisms between the curves represented by elements of P_{g} are given by k-isomorphisms of their function fields, and since the g_{2}^{1} of a hyperelliptic curve is unique the k-isomorphisms must respect the inclusion of the function field of P^{1}. Identify the set of polynomials l(x) defined over k and of degree at most g + 1 with k^{g+2}, and define the group homomorphism

ϕ_{g} : GL_{2}^{op}(k) × k^{∗} → Aut(k^{g+2}), ϕ_{g}(( \begin{matrix}a & b \\ c & d \end{matrix} ), e)(l(x)) :=

e^{−1}(cx + d)^{g+1}l(\frac{ax + b}{cx + d}).

Now define the group G_{g} := (k^{g+2} ⋊_{ϕ_{g}} (GL_{2}^{op}(k) × k^{∗}))/D where

D := \{(0, ( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ), a^{g+1}) : a ∈ k^{∗}\} ⊂ k^{g+2} ⋊_{ϕ_{g}} (GL_{2}^{op}(k) × k^{∗}).

The k-isomorphisms between curves corresponding to elements of P_{g} are then precisely the ones induced by elements of G_{g} by letting

γ = [(l(x), ( \begin{matrix}a & b \\ c & d \end{matrix} ), e)] ∈ G_{g}

induce the isomorphism

(x, y) ↦ (\frac{ax + b}{cx + d}, \frac{e(y + l(x))}{(cx + d)^{g+1}} ) .

This defines a left group action of G_{g} on P_{g}, where γ = [(l, Λ, e)] ∈ G_{g} takes (h, f) ∈ P_{g} to (h̃, f̃) ∈ P_{g}, with

(h̃, f̃) = (ϕ_{g}(Λ, e)(h), e^{−1}ϕ_{2g}(Λ, e)(f) + l ϕ_{g}(Λ, e)(h) + l^{2}).

Definition 8.2. Let τ_{m} be the function that takes (a, b) ∈ k_{m}^{2} to 1 if the equation y^{2} + ay + b has two roots defined over k_{m}, 0 if it has one root and −1 if it has none.

Lemma 8.3. If C_{(h,f)} is the hyperelliptic curve corresponding to (h, f) ∈ P_{g} then

a_{m}(C_{(h,f)}) = − ∑_{α∈P^{1}(k_{m})} τ_{m}(h(α), f(α)).

Proof: Follows in the same way as Lemma 3.4. □

Notation 8.4. Let us put I_{g} := 1/|G_{g}| = q^{−(g+2)}(q^{3} − q)^{−1}(q − 1)^{−1}.

In the same way as in the case of odd characteristic we get the equality

a_{λ}|_{g} = I_{g} ∑_{(h,f)∈P_{g}} ∏_{i=1}^{ν}(− ∑_{α∈P^{1}(k_{i})} τ_{i}(h(α), f(α)))^{λ_{i}}.

All results of Section 3.1 are independent of the characteristic and hence we extend the definition of a_{λ}|_{g} to genus 0 and 1 in the same way as in that section.

Definition 8.5. For any g ≥ −1, (n; r) ∈ N_{m} and α ∈ A(n) define

u_{g,α}^{(n;r)} := I_{g} ∑_{(h,f)∈P_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), f(α_{i}))^{r_{i}}

and

u_{g}^{(n;r)} := ∑_{α∈A(n)} u_{g,α}^{(n;r)}.

Construction-Lemma 8.6. For each λ we have (in even characteristic) the same decomposition of a_{λ}|_{g} as given by Construction-Lemma 3.8.

Proof: The following properties of τ_{m} for (h, f) ∈ P_{g} correspond precisely to the ones for the quadratic character.

⋆ Say that α ∈ P^{1}(k_{s}), then τs̃(h(α), f(α)) = τ_{s}(h(α), f(α))^{2} if s̃/s is

even, and τs̃(h(α), f(α)) = τ_{s}(h(α), f(α)) if s̃/s is odd.

⋆ If for any α, β ∈ P^{1} we have F^{s}(α) = β for some s, then

τ_{i}(h(α), f(α)) = τ_{i}(h(α), f(β)) for all i.

⋆ Finally, for any α ∈ P^{1} and any s, τ_{,s}(h(α), f(α))^{r} = τ_{s}(h(α), f(α))^{2}

if r is even and τ_{s}(h(α), f(α))^{r} = τ_{s}(h(α), f(α)) if r is odd.

With this established we can use the same proof as for Construction-Lemma 3.8. □

Since the decompositions are the same, Lemmas 3.11 and 3.12 also hold in even characteristic.

9. Recurrence relations for u_{g} in even characteristic

Analogously to Section 4, this section will be devoted to finding for a fixed pair (n; r) ∈ N_{m}, a recurrence relation for u_{g}. Fix an s ∈ k which does not lie in the set {r^{2} +r : r ∈ k}, that is, such that τ_{1}(1, s) = −1. We define an involution on P_{g} sending (h, f) to (h, f + s h^{2}). This involution is fixed point free and hence

u_{g,α} = I_{g} ∑_{(h,f)∈P_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), f(α_{i}))^{r_{i}} =

= I_{g} ∑_{(h,f)∈P_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), f(α_{i}) + s h^{2}(α_{i}))^{r_{i}} = (−1)^{∑_{i=1}^{m} r_{i}n_{i}}u_{g,α}.

Thus, Lemma 4.1 also holds in the case of even characteristic.

Definition 9.1. Let Q_{g} denote the set of pairs (h, f) of polynomials over k, where h is nonzero and h, f are of degree at most g + 1, 2g + 2 respectively. Extending the definition for P_{g} above to a pair (h, f) ∈ Q_{g}, let h(∞) and f(∞) be equal to the degree g + 1 and 2g + 2 coefficient of h and f respectively. For any g ≥ −1, (n; r) ∈ N_{m} and α ∈ A(n) define

Û_{g,α}^{(n;r)} := I_{g} ∑_{(h,f)∈Q_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), f(α_{i}))^{r_{i}}

and

Û_{g}^{(n;r)} := ∑_{α∈A(n)} Û_{g,α}^{(n;r)}.

Remark 9.2. The connection between the sets Q_{g} and P_{g} which we will present below is due to Brock and Granville and can be found in an early version of [7]. There the connection is used to count the number of hyperelliptic curves in even characteristic, which is a_{0}|_{g,even} in our terminology.

Lemma 9.3. Let h and f be polynomials over k. For any irreducible polynomial m over k, the following two statements are equivalent:

⋆ m| gcd(h, f′^{2} + fh′^{2});

⋆ there is a polynomial l over k, such that m|h and m^{2}|f + hl + l^{2}.

Proof: Say that α ∈ k_{n} is a root of an irreducible polynomial m and of the polynomial gcd(h, f′^{2}+fh′^{2}). Let l be equal to f^{q^{n}/2}. Working modulo (x−α)^{2} we then get

f + hl + l^{2} = f + hf^{q^{n}/2} + f^{q^{n}}

≡ f(α) + f′(α)(x − α) + h′(α)f(α)^{q^{n}/2}(x − α) + f(α)^{q^{n}}

≡ (x − α)(f′(α) + h′(α)f(α)^{q^{n}/2}) ≡ (x − α)(f′(α)^{2} + h′(α)^{2}f(α))^{1/2} = 0,

which tells us that m^{2}|f +hl +l^{2}. For the other direction, assume that we have an irreducible polynomial m and a polynomial l such that m|h and m^{2}|f+hl+l^{2}.

Differentiating the polynomial f + hl + l^{2} gives m^{2}|f′ + h′l + hl′, and thus m|f′ + h′l. Taking squares we get m^{2}|f^{′2} + h^{′2}l^{2} and then it follows that m^{2}|f^{′2} + h^{′2}(f + hl) and hence m|f^{′2} + h^{′2}f. □

Let (h, f) be an element of Q_{g}. In the first part of the proof of Lemma 9.3, we may take for l any representative of f^{q^{n}/2} modulo h, because for these l we have f + hl + l^{2} ≡ f + hf^{q^{n}/2} + f^{q^{n}} modulo (x − α)^{2}. In the second part it does not matter which degree l has. We conclude from this that Lemma 9.3 also holds if we assume that l is of degree at most g + 1. Choose g ≥ −1 and let (h, f) ∈ Q_{g}. Lemma 9.3 gives the following alternative formulation of the conditions (8.1), (8.2) and (8.3). For all polynomials l of degree at most g + 1:

m|h, m^{2}|f + hl + l^{2} ⟹ deg(m) = 0; (9.1)

(9.2) deg(h) = g + 1 or deg(f + hl + l^{2}) ≥ 2g + 1.

Here we used that t| gcd(h_{∞}, f_{∞}^{′2} + f_{∞}h_{∞}^{′2}) if and only if t|h_{∞} and there exists a polynomial l_{∞} such that deg(l_{∞}) ≤ g + 1 and t^{2}|f_{∞} + h_{∞}l_{∞} + l_{∞}^{2}. In turn, this happens if and only if deg(h) ≤ g and there exists a polynomial l of degree at most g + 1 such that deg(f + hl + l^{2}) ≤ 2g, where we connect l and l_{∞} using the definitions l := x^{g+1}l_{∞}(1/x) and l_{∞} := t^{g+1}l(1/t). This reformulation leads us to making the following definition.

Definition 9.4. Let ∼_{g} be the relation on Q_{g} given by (h, f) ∼_{g} (h, f +hl+l^{2}) if l is a polynomial of degree at most g + 1. This is an equivalence relation and since (h, f) = (h, f + hl + l^{2}) if and only if l = 0 or l = h, the number of elements of each equivalence class [(h, f)]_{g} is q^{g+2}/2. If (h, f) ∈ P_{g} ⊂ Q_{g} then [(h, f)]_{g} ⊂ P_{g} and we get an induced equivalence relation on P_{g} which we also denote ∼_{g}.

We will now construct all ∼_{g} equivalence classes of elements of Q_{g} in terms of the ∼_{i} equivalence classes of the elements in P_{i}, where i is between −1 and g. This is the counterpart of factoring a polynomial into a square-free part and a squared part in the case of odd characteristic.

Definition 9.5. For z := [(h, f)]_{i} ∈ P_{i}/ ∼_{i} let V_{z} be the set of all equivalence classes [(mh, m^{2}f)]_{g} in Q_{g} for all monic polynomials m of degree at most g − i. This is well defined since if (h_{1}, f_{1}) ∼_{i} (h_{2}, f_{2}) then (mh_{1}, m^{2}f_{1}) ∼_{g} (mh_{2}, m^{2}f_{2}).

Lemma 9.6. The sets V_{z} for all z ∈ P_{i}/ ∼_{i} where −1 ≤ i ≤ g are disjoint.

Proof: Say that for some z_{1} and z_{2} the intersection V_{z_{1}} ∩ V_{z_{2}} is nonempty. That is, there exist (h_{1}, f_{1}) ∈ P_{i_{1}}, (h_{2}, f_{2}) ∈ P_{i_{2}} and monic polynomials m_{1}, m_{2} such that m_{1}h_{1} = m_{2}h_{2} and m_{1}^{2}f_{1} = m_{2}^{2}f_{2} + m_{2}h_{2}l + l^{2}. If for some irreducible polynomial r we have r|m_{1} but r ∤ m_{2}, it follows that r|h_{2} and r^{2}|m_{2}^{2}f_{2} + m_{2}h_{2}l + l^{2}. By the equivalence of conditions (8.2) and (9.1), this implies that r|(m_{2}^{2}f_{2})^{′2} +m_{2}^{2}f_{2}(m_{2}h_{2})^{′2} which in turn implies that r|f_{2}^{′2} +f_{2}h_{2}^{′2}. Since (h_{2}, f_{2}) ∈ P_{i_{2}} we see that r must be constant. Hence every irreducible factor of m_{1} is a factor of m_{2}. The situation is symmetric and therefore the converse also holds. So far we have not ruled out the possibility that a factor in m_{1} appears with higher multiplicity than in m_{2}, or vice versa. Let m be the product of all irreducible factors of m_{1} and put m̃_{1} := m_{1}/m, m̃_{2} := m_{2}/m and l̃:= l/m. We are then in the same situation as above, that is m̃_{1}h_{1} = m̃_{2}h_{2} and m̃_{1}^{2}f_{1} = m̃_{2}^{2}f_{2} + m̃_{2}h_{2}l̃+ l̃^{2}. Thus, if r is an irreducible polynomial such that r|m̃_{1} but r ∤ m̃_{2} we can argue as above to conclude that r is constant. By a repeated application of this line of reasoning we can conclude that m_{1} and m_{2} must be equal. It now follows that h_{1} = h_{2} and that m_{2}|l, thus (h_{1}, f_{1}) ∼_{i_{1}} (h_{2}, f_{2}). This tells us that V_{z_{1}} ∩ V_{z_{2}} is only nonempty when z_{1} = z_{2}. □

Lemma 9.7. The sets V_{z} for all z ∈ P_{i}/ ∼_{i} where −1 ≤ i ≤ g cover Q_{g}/ ∼_{g}.

Proof: Pick any element (h_{1}, f_{1}) ∈ Q_{g} and put g_{1} := g. We define a procedure, where at the ith step we ask if there are any polynomials m_{i} and l_{i} such that deg(m_{i}) > 0, deg(l_{i}) ≤ g_{i} + 1, m_{i}|h_{i} and m_{i}^{2}|f_{i} + h_{i}l_{i} + l_{i}^{2}. If so, take any such polynomials m_{i}, l_{i} and define h_{i+1} := h_{i}/m_{i}, f_{i+1} := (f_{i} + h_{i}l_{i} + l_{i}^{2})/m_{i}^{2} and g_{i+1} := g_{i} − deg(m_{i}). This procedure will certainly stop. Assume that the procedure has been carried out in some way and that it has stopped at the jth step, leaving us with some pair of polynomials (h_{j}, f_{j}). Next, we take (h_{j}, f_{j+1}) to be any element of the set [(h_{j}, f_{j})]_{g_{j}} for which deg(f_{j+1}) is minimal. Say that f_{j+1} = f_{j} + h_{j}l_{j} + l_{j}^{2} where deg(l_{j}) ≤ g_{j} + 1 and let us define g_{j+1} to be the number such that 2g_{j+1} + 1 ≤ max(2 deg(h_{j}), deg(f_{j+1})) ≤ 2g_{j+1} + 2. The claim is now that (h_{j}, f_{j+1}) ∈ P_{g_{j+1}}. By definition, condition (8.1) holds for (h_{j}, f_{j+1}). If there were polynomials m_{j+1} and l_{j+1} such that m_{j+1}|h_{j} and m_{j+1}^{2}|f_{j+1} +h_{j}l_{j+1} +l_{j+1}^{2} then the pair of polynomials m_{j+1} and l_{j} +l_{j+1} would contradict that the process above stopped at the jth step. Hence condition (9.1) is fulfilled for (h_{j}, f_{j+1}). Condition (9.2) is fulfilled if 2 deg(h_{j}) ≥ deg(f_{j+1}) because then deg(h_{j}) = g_{j+1} + 1. On the other hand, if 2 deg(h_{j}) < deg(f_{j+1}) and there were a polynomial l_{j+1} such that deg(l_{j+1}) ≤ g_{j+1} + 1 and deg(f_{j+1} + h_{j}l_{j+1} + l_{j+1}^{2}) ≤ 2g_{j+1} then this would contradict the minimality of deg(f_{j+1}). We conclude that (h_{j}, f_{j+1}) ∈ P_{g_{j+1}}. Finally we see that if we put m̂_{r} := ∏_{i=1}^{r−1} m_{i} and l := ∑_{i=1}^{j} m̂_{i}l_{i}, then deg(l) ≤ g + 1, h_{1} = m̂_{j}h_{j} and f_{1} = m̂_{j}^{2}f_{j+1} + h_{1}l + l^{2}. This shows that V_{z} contains [(h_{1}, f_{1})]_{g} where z := [(h_{j}, f_{j+1})]_{g_{j+1}} ∈ P_{g_{j+1}}/ ∼_{g_{j+1}}. □

Using the lemmas above we will be able to write Û_{g} in terms of u_{i} for i between −1 and g. After this we will determine Û_{g} for large enough values of g. We divide into two cases.

Notation 9.8. Let S_{j} denote all polynomials of degree at most j, and let S′_{j} ⊂ S_{j} consist of the monic polynomials.

9.1. The case α ∈ A′(n). Fix an element α ∈ A′(n). It follows from Lemma 9.6 and Lemma 9.7 that

(9.3) Û_{g,α} = \frac{I_{g}}{2} ∑_{l∈S_{g+1}} ∑_{j=−1}^{g} ∑_{z∈P_{j}/∼_{j}} ∑_{[(h,f)]_{j}∈V_{z}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), (f +hl+l^{2})(α_{i}))^{r_{i}}.

Lemma 9.9. Choose any s ≥ 1 and t_{1}, t_{2} in k_{s}. We then have

τ_{s}(vt_{1}, v^{2}t_{2}) = τ_{s}(t_{1}, t_{2}) for all v ≠ 0 ∈ k_{s};
τ_{s}(t_{1}, t_{2} + vt_{1} + v^{2}) = τ_{s}(t_{1}, t_{2}) for all v ∈ k_{s}.

Proof: Clear. □

Fix elements z = [(h_{0}, f_{0})]_{i} ∈ P_{i}/ ∼_{i} and β ∈ A^{1}(k_{s}) and define V′_{z} to be the subset of V_{z} of classes [(m̃h_{0}, m̃^{2}f_{0})]_{g}, where m̃ is a monic polynomial with m̃(β) ≠ 0. Lemma 9.9 shows that τ_{s}(h(β), f(β)) is constant for all s and (h, f) such that [(h, f)]_{g} ∈ V′_{z}. Applying this to equation (9.3) after recalling Definition 4.4 we find that

(9.4) Û_{g,α} = I_{g} \frac{q^{g+1}}{2} ∑_{j=−1}^{g} ∑_{z∈P_{j}/∼_{j}} ∑_{m̃∈S′_{g−j}} ∏_{i=1}^{m} τ_{n_{i}}((m̃h)(α_{i}), (m̃^{2}f)(α_{i}))^{r_{i}} =

= I_{g} \frac{q^{g+1}}{2} ∑_{j=−1}^{g} b̂_{g−j}u_{j,α} \frac{2}{q^{j+1}I_{j}} = ∑_{i=0}^{g+1} b̂_{i}u_{g−i,α},

where we have taken into account that the group of isomorphisms depends upon g and that the numbers of elements of the equivalence classes of the relations ∼_{g−j} and ∼_{g} differ by a factor q^{j}. From the definitions we see that q^{g−j} I_{g}/I_{j} = 1. For any g ≥ −1 and any h_{0} ∈ S_{g+1} it is clear that (9.5)

∑_{(h_{0},f)∈Q_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), f(α_{i}))^{r_{i}} = \begin{cases}0 & if ∀i : r_{i} = 2, ∃j : h_{0}(α_{j}) = 0; \\ q^{2g+3} & if ∀i : r_{i} = 2, ∀j : h_{0}(α_{j}) ≠ 0. \end{cases}

For any g such that 2g + 2 ≥ |n| − 1, and any nonzero polynomial h_{0} of degree at most g + 1, Lemma 4.7 tells us that

(9.6) ∑_{(h_{0},f_{1}+p_{α}f_{2})∈Q_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), (f_{1} + p_{α}f_{2})(α_{i}))^{r_{i}} =

= q^{2g+3−|n|} ∑_{f_{1}∈S_{|n|−1}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), f_{1}(α_{i}))^{r_{i}} =

= q^{2g+3−|n|} ∑ ; (β_{1},...,β_{m})∈∏_{i=1}^{m} k_{n_{i}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), β_{i})^{r_{i}} = 0 if ∃i : r_{i} = 1,

because for all a ∈ k_{s} there are as many b ∈ k_{s} for which τ_{s}(a, b) = 1 as there are b ∈ k_{s} for which τ_{s}(a, b) = −1.

Summing equations (9.5) and (9.6) over all h_{0} ∈ S_{g+1} and using that q^{2g+3}I_{g} = Iq^{g+1} we get

(9.7)  Û_{g,α} = \begin{cases}I (q − 1)q^{g+1}b̂_{g+1} & if ∀i : r_{i} = 2, g ≥ −1; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−3}{2}. \end{cases}

9.2. The case α ∈ A(n) \ A′(n). Fix an α ∈ A(n) \ A′(n). We can assume that α_{1} = ∞, and then α̃ := (α_{2}, …, α_{m}) ∈ A′(ñ) where ñ := (n_{2}, …, n_{m}).

Lemma 9.10. For any element (h, f) ∈ P′_{i} and any monic polynomial m of degree g − i,

τ_{s}((mh)(∞), (m^{2}f)(∞)) = τ_{s}(h(∞), f(∞));
τ_{s}((mh)(∞), (f + lh + l^{2})(∞)) = τ_{s}(h(∞), f(∞)).

Proof: Clear. □

For any (h, f) ∈ Q_{g} it holds that if deg(h) < g +1 then τ_{s}(h(∞), f(∞)) = 0 for all s. Define therefore P′_{g} and Q′_{g} to be the subsets of P_{g} and Q_{g} respectively, that consist of pairs (h, f) such that deg(h) = g+1. We get an induced relation ∼_{i} on P′_{i} and Q′_{i} and we let V′′_{z} be the set of all equivalence classes [(mh, m^{2}f)]_{g} in Q′_{g} for all monic polynomials m of degree g −i, where z := [(h, f)]_{i} ∈ P′_{i}/ ∼_{i}. In the same way as in Lemma 9.6 and 9.7 we see that the sets V′′_{z} for all z ∈ P′_{i}/ ∼_{i}, where −1 ≤ i ≤ g, are disjoint and cover Q′_{g}/ ∼_{g}. Using this together with Lemma 9.10 and the arguments showing equation (9.4) we find that

(9.8) Û_{g,α} = \frac{I_{g}}{2} ∑_{l∈S_{g+1}} ∑_{z∈Q′_{g}/∼_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h(α_{i}), (f + hl + l^{2})(α_{i}))^{r_{i}} =

= I_{g} \frac{q^{g+1}}{2} ∑_{j=−1}^{g} ∑_{z∈P′_{j}/∼_{j}} ∑_{m̃∈R′_{g−j}} ∏_{i=1}^{m} τ_{n_{i}}((m̃h)(α_{i}), (m̃^{2}f)(α_{i}))^{r_{i}} = ∑_{i=0}^{g+1} bñ ; iu_{g−i,α}.

If we choose g such that 2g + 2 ≥ |n| − 1, h_{0} ∈ R_{g+1} and we put p_{α}(x) := x pα̃, then we find in the same way as for equation (9.6) that

(9.9) ∑_{(h_{0},f_{1}+p_{α}f_{2})∈Q_{g}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), (f_{1} + p_{α}f_{2})(α_{i}))^{r_{i}} =

= q^{2g+3−|n|} ∑ ; (β_{1},...,β_{m})∈∏_{i=1}^{m} k_{n_{i}} ∏_{i=1}^{m} τ_{n_{i}}(h_{0}(α_{i}), β_{i})^{r_{i}} = 0 if ∃i : r_{i} = 1.

Since equation (9.5) also hold for α ∈ A(n) \ A′(n) we find, by summing over all polynomials h_{0} ∈ R_{g+1}, that

(9.10)  Û_{g,α} = \begin{cases}I (q − 1)q^{g+1}bñ ; g+1 & if ∀i : r_{i} = 2, g ≥ −1; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−3}{2}. \end{cases}

9.3. The two cases joined. Recall that J = (q − 1) I |A(n)|.

Theorem 9.11. For any pair (n; r) ∈ N_{m},

∑_{j=0}^{g+1} b̂_{j}u_{g−j} = \begin{cases}J q^{g+1}b̂_{g+1} & if ∀i : r_{i} = 2, g ≥ −1; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−3}{2}. \end{cases}

Proof: The theorem follows from combining equations (9.4), (9.7), (9.8) and (9.10), using Lemma 4.10. □

Theorem 9.12. For any pair (n; r) ∈ N_{m},

∑_{j=0}^{min(|n|−1,g+1)}(b̂_{j} − qb̂_{j−1})u_{g−j} = \begin{cases}J q^{g+1}(b̂_{g+1} − b̂_{g}) & if ∀i : r_{i} = 2, g ≥ 0; \\ 0 & if ∃i : r_{i} = 1, g ≥ \frac{|n|−1}{2}. \end{cases}

Proof: In the notation of the proof of Theorem 4.14, the theorem follows from applying Theorem 9.11 to the expression F (g) − qF (g − 1). □

Theorem 9.13. By applying Theorem 9.12 to each pair (n; r) appearing in the decomposition (given by Lemma 8.6) of a_{λ}|_{g,even} we get a linear recurrence relation for a_{λ}|_{g,even}. The characteristic polynomial of this linear recurrence relation equals (5.1).

Proof: We know that the decomposition of a_{λ}|_{g} is independent of characteristic, and since the left hand side of the equation in Theorem 9.12 is the same as the left hand side of the equation of Theorem 4.14 this theorem follows in the same way as Theorem 5.2. □

10. Results for weight up to 7 in even characteristic

In this section we compute, for any number g and any finite field k of even characteristic, all a_{λ}|_{g,even} of weight at most 7. First we will exploit the similarities of Theorems 4.12 and 9.11.

Lemma 10.1. If g ≥ n − 2 then b̂_{2g+2} = q^{g+1}b̂_{g+1}.

Proof: Fix a pair (n; r) ∈ N_{m}. Lemma 4.13 tells us that b̂_{j} = qb̂_{j−1} +d_{|n|−1−j}, so if j ≥ |n| then b̂_{j} = qb̂_{j−1} and thus b̂_{j} = q^{j+1−|n|}b̂_{|n|−1}. □

Remark 10.2. If r_{i} = 1 for some i and g ≥ (|n| − 3)/2, then the recursive relations of Theorems 9.11 and 4.12 are equal. On the other hand, if r_{i} = 2 for all i we see from Lemma 10.1 that the recursive relations of Theorems 9.11 and 4.12 are equal if g ≥ |n| − 2.

Theorem 10.3. For weight less than or equal to 5, a_{λ}|_{g,even} = a_{λ}|_{g,odd} as functions (in this case polynomials) in q.

Proof: Consider any a_{λ}|_{g} with |λ| ≤ 5. By Lemma 3.12 it suffices to show that u_{g} is independent of characteristic when (n; r) ∈ N_{m} is such that ∑_{i=1}^{m} n_{i}r_{i} ≤ 5. Clearly u_{−1} = J is always independent of characteristic. Clearly, Lemma 6.1 also holds in even characteristic. We can therefore assume that r_{i} = 2 for all i in the case of genus 0. But if r_{i} = 2 for all i then |n| ≤ 2 and hence, by Remark 10.2, u_{0} will be independent of characteristic. This takes care of the base cases of the recurrence relations for u_{g} when g ≥ 1, given by Theorems 4.12 and 9.11. Again by Remark 10.2 we see that (both in the case when r_{i} = 2 for all i, and when r_{i} = 1 for some i) when g ≥ 1 these recurrence relations are the same. We can therefore conclude that u_{g} is independent of characteristic for all g. □

We will now compute a_{λ}|_{g,even} for weight 6 in the same way as in Section 7.3. To compute u_{g} of degree at most 5 using Theorem 9.11 we need to find the base case u_{0}. But when the genus is 0 we can use Lemma 6.1 (which also holds in even characteristic) to reduce to the case that r_{i} = 2 for all i, which is always computable using Theorem 9.11. What is left is the general case of the decomposition of a_{λ}|_{g,even}. We then need the base cases of genus 0 and 1. Again, the genus 0 part is no problem. The computation of a_{λ}|_{1} in [1] is independent of characteristic. We can therefore compute the genus 1 part (compare Section 7.3).

Remark 10.4. As in the case of odd characteristic, for all g and all λ such that |λ| ≤ 7, a_{λ}|_{g,even} is a polynomial when considered as a function in the number q (compare Remark 7.11) of elements of the finite field k of even characteristic. In Theorem 10.3 we saw that the polynomial functions a_{λ}|_{g,odd} and a_{λ}|_{g,even} are equal (for a fixed g), if |λ| ≤ 5. But for weight 6 there are λ such that the two polynomials are different, this occurs for the first time for genus 3, see Example 10.6.

Example 10.5. Let us compute u_{g,even} when (n; r) = ((1, 1); (2, 2, 2)). We see that u_{−1} = 1 and Theorem 9.11 gives u_{0} = q^{2} − 3q + 2. This result is different from the 1 in the case of odd characteristic, see Example 4.16. Continued use of Theorem 9.11 gives u_{1} = q^{4} − 3q^{3} + 5q^{2} − 6q + 3 and then Theorem 9.12 gives

u_{g} = 2u_{g−1} − u_{g−2} + q^{2g−1}(q − 1)^{3} for g ≥ 2.

Solving this leaves us with

u_{g,even}^{((1,1,1);(2,2,2))} = \frac{(q − 1)(q^{2g+3} + g(q^{2} − 1) − 3q − 2)}{(q + 1)^{2}}.

Example 10.6. The result for a_{[1^{6}]}|_{g,even} is

a_{[1^{6}]}|_{g,even} = a_{[1^{6}]}|_{g,odd} − \frac{5}{8} g(g − 1)(g − 2)((g − 3)(q − 1) − 4).

Documenta Mathematica 14 (2009) 259–296

Example 10.7. The result for a_{[1^{2},4]}|_{g,even} is

a_{[1^{2},4]}|_{g,even} = a_{[1^{2},4]}|_{g,odd} − \frac{1}{4} \begin{cases}g(q − 1) & if g ≡ 0 mod 4; \\ (g − 1)(q − 1) & if g ≡ 1 mod 4; \\ (g − 2)(q − 1) & if g ≡ 2 mod 4; \\ (g − 3)(q − 1) − 4 & if g ≡ 3 mod 4. \end{cases}

11. Cohomological results

11.1. Cohomological results for H_{g,n}. Define the local system 𝕍 := R^{1}π_{∗}(ℚ_{ℓ}) where π : M_{g,1} → M_{g} is the universal curve. For every partition (note that in this section we use a different notation for partitions) λ = (λ_{1} ≥ … ≥ λ_{g} ≥ 0) there is an irreducible representation of GSp(2g) with highest weight (λ_{1} − λ_{2})γ_{1} + … + λ_{g}γ_{g} − |λ|η, where the γ_{i} are suitable fundamental roots and η is the multiplier representation, and we define 𝕍_{λ} to be the corresponding local system. Let us also denote by 𝕍_{λ} its restriction to H_{g}. In Lemma 13.5 below we will see that making an 𝕊ñ-equivariant count of points of H_{g,ñ} over a finite field k, for all ñ ≤ n, is equivalent to computing the trace of Frobenius on the compactly supported ℓ-adic Euler characteristic e_{c}(H_{g} ⊗ k̄, 𝕍_{λ}), for every λ with |λ| ≤ n (where ℓ ∤ |k|). For more details, see [14] and [15]. Thus, we can use the results of Section 7 together with Theorem 3.2 in [1] to compute the ℓ-adic Euler characteristic e_{c}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}) in K_{0}(Gal_{ℚ}), the Grothendieck group of Gal(ℚ̄/ℚ)-representations, for every λ with |λ| ≤ 7. Specifically, Theorem 3.2 in [1] tells us that if there is a polynomial P such that Tr(F, e_{c}(H_{g} ⊗ k̄, 𝕍_{λ})) = P (q) for all finite fields k, possibly with the exception of a finite number of characteristics, then e_{c}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}) = P (q), where q is the class of ℚ_{ℓ}(−1) in K_{0}(Gal_{ℚ}). By excluding even characteristic, Section 7 (see Remark 7.11) and Lemma 13.5 shows that there is indeed such a polynomial for all g and all |λ| ≤ 7.

Example 11.1. For g = 8 and λ = (5, 1) we have

e_{c}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}) = 5q^{5} − 28q^{4} + 4q^{3} + 96q^{2} − 34q − 88.

11.2. Cohomological results for \overline{M}_{2,n} and M_{2,n}. Using the stratification of \overline{M}_{g,n} we can make an 𝕊_{n}-equivariant count of its number of points using the 𝕊_{n}-equivariant counts of the points of M_{g̃,ñ} for all g̃ ≤ g and ñ ≤ n+2(g−g̃) (see [13, Thm 8.13] and also [2]). Since all curves of genus 2 are hyperelliptic, M_{2,n} is equal to H_{2,n}. Above, we have made 𝕊_{n}-equivariant counts of H_{2,n} for n ≤ 7 and they were all found to be polynomial in q. These 𝕊_{n}-equivariant counts can now be complemented with ones of M_{1,n} for n ≤ 9 (see [1, Section 15]) and of M_{0,n} for n ≤ 11 (see [18, Prop 2.7]), which are also found to be polynomial in q. We can then apply Theorem 3.4 in [2] to conclude, for all n ≤ 7, the 𝕊_{n}-equivariant Gal_{ℚ} (resp. Hodge) structure of the ℓ-adic (resp. Betti) cohomology of \overline{M}_{2,n}.

In the theorems below we give the 𝕊_{n}-equivariant Hodge Euler characteristic (which by purity is sufficient to conclude the Hodge structure) in terms of the Schur polynomials and L, the class of the Tate Hodge structure of weight 2 in K_{0}(HS_{ℚ}), the Grothendieck group of rational Hodge structures. That is, the action of 𝕊_{n} on \overline{M}_{2,n} induces an action on its cohomology, and hence H^{i}(\overline{M}_{2,n} ⊗ ℂ, ℚ) may be written as a direct sum of H_{λ}^{i}(\overline{M}_{2,n} ⊗ ℂ, ℚ), which correspond to the irreducible representations of 𝕊_{n} indexed by λ ⊢ n and with characters χ_{λ}. In terms of this, the coefficient of the Schur polynomial s_{λ} is equal to 1/χ_{λ}(id) · ∑_{i}(−1)^{i}[H_{λ}^{i}(\overline{M}_{2,n} ⊗ ℂ, ℚ)]. The results for n ≤ 3 were previously known by the work of Getzler in [14, Section 8].

Theorem 11.2. The 𝕊_{n}-equivariant Hodge Euler characteristic of \overline{M}_{2,4} is equal to

(L^{7} + 8L^{6} + 33L^{5} + 67L^{4} + 67L^{3} + 33L^{2} + 8L + 1)s_{4}
+(4L^{6} + 26L^{5} + 60L^{4} + 60L^{3} + 26L^{2} + 4L)s_{31}
+(2L^{6} + 12L^{5} + 28L^{4} + 28L^{3} + 12L^{2} + 2L)s_{2^{2}}
+(3L^{5} + 10L^{4} + 10L^{3} + 3L^{2})s_{21^{2}}

Theorem 11.3. The 𝕊_{n}-equivariant Hodge Euler characteristic of \overline{M}_{2,5} is equal to

(L^{8} + 9L^{7} + 49L^{6} + 128L^{5} + 181L^{4} + 128L^{3} + 49L^{2} + 9L + 1)s_{5}
+(6L^{7} + 48L^{6} + 156L^{5} + 227L^{4} + 156L^{3} + 48L^{2} + 6L)s_{41}
+(3L^{7} + 31L^{6} + 106L^{5} + 159L^{4} + 106L^{3} + 31L^{2} + 3L)s_{32}
+(8L^{6} + 42L^{5} + 65L^{4} + 42L^{3} + 8L^{2})s_{31^{2}}
+(6L^{6} + 26L^{5} + 43L^{4} + 26L^{3} + 6L^{2})s_{2^{2}1}
+(L^{5} + 3L^{4} + L^{3})s_{21^{3}}

Theorem 11.4. The 𝕊_{n}-equivariant Hodge Euler characteristic of \overline{M}_{2,6} is equal to

(L^{9} + 11L^{8} + 68L^{7} + 229L^{6} + 420L^{5} + 420L^{4} + 229L^{3} + 68L^{2} + 11L + 1)s_{6}

+(7L^{8} + 75L^{7} + 317L^{6} + 641L^{5} + 641L^{4} + 317L^{3} + 75L^{2} + 7L)s_{51}
+(5L^{8} + 62L^{7} + 292L^{6} + 615L^{5} + 615L^{4} + 292L^{3} + 62L^{2} + 5L)s_{42}
+(L^{8} + 21L^{7} + 108L^{6} + 236L^{5} + 236L^{4} + 108L^{3} + 21L^{2} + L)s_{3^{2}}
+(17L^{7} + 118L^{6} + 278L^{5} + 278L^{4} + 118L^{3} + 17L^{2})s_{41^{2}}
+(16L^{7} + 115L^{6} + 277L^{5} + 277L^{4} + 115L^{3} + 16L^{2})s_{321}

+(3L^{7} + 22L^{6} + 53L^{5} + 53L^{4} + 22L^{3} + 3L^{2})s_{2^{3}}

+(9L^{6} + 29L^{5} + 29L^{4} + 9L^{3})s_{31^{3}}
+(6L^{6} + 21L^{5} + 21L^{4} + 6L^{3})s_{2^{2}1^{2}}

Theorem 11.5. The 𝕊_{n}-equivariant Hodge Euler characteristic of \overline{M}_{2,7} is equal to

(L^{10} + 12L^{9} + 90L^{8} + 363L^{7} + 854L^{6} + 1125L^{5} + 854L^{4} + 363L^{3} + 90L^{2} + …)s_{7}

+(9L^{9} + 109L^{8} + 580L^{7} + 1529L^{6} + 2109L^{5} + 1529L^{4} + 580L^{3} + 109L^{2} + 9L)s_{61}

+(6L^{9} + 100L^{8} + 606L^{7} + 1728L^{6} + 2430L^{5} + 1728L^{4} + 606L^{3} + 100L^{2} + 6L)s_{52}

+(3L^{9} + 58L^{8} + 389L^{7} + 1153L^{6} + 1647L^{5} + 1153L^{4} + 389L^{3} + 58L^{2} + 3L)s_{43}

+(28L^{8} + 258L^{7} + 831L^{6} + 1221L^{5} + 831L^{4} + 258L^{3} + 28L^{2})s_{51^{2}}
+(34L^{8} + 331L^{7} + 1133L^{6} + 1675L^{5} + 1133L^{4} + 331L^{3} + 34L^{2})s_{421}
+(12L^{8} + 140L^{7} + 489L^{6} + 738L^{5} + 489L^{4} + 140L^{3} + 12L^{2})s_{3^{2}1}
+(8L^{8} + 91L^{7} + 335L^{6} + 502L^{5} + 335L^{4} + 91L^{3} + 8L^{2})s_{32^{2}}
+(28L^{7} + 143L^{6} + 228L^{5} + 143L^{4} + 28L^{3})s_{41^{3}}
+(34L^{7} + 170L^{6} + 275L^{5} + 170L^{4} + 34L^{3})s_{321^{2}}
+(10L^{7} + 47L^{6} + 77L^{5} + 47L^{4} + 10L^{3})s_{2^{3}1}
+(4L^{6} + 7L^{5} + 4L^{4})s_{31^{4}}
+(2L^{6} + 6L^{5} + 2L^{4})s_{2^{2}1^{3}}

In Table 1 we present the nonequivariant information (remember that all cohomology is Tate) in the form of Betti numbers of \overline{M}_{2,n} for all n ≤ 7. Notice that the table only contains as many numbers as we need to be able to fill in the missing ones using Poincaré duality. These results agree with Table 2 of ordinary Euler characteristics for \overline{M}_{2,n} for n ≤ 6 found in [4].

Table 1. Dimensions of H^{i}(\overline{M}_{2,n} ⊗ ℂ, ℚ) for n ≤ 7.
H^{0} H^{2} H^{4}  H^{6}  H^{8}  H^{10}
\overline{M}_{2}  1  2
\overline{M}_{2,1} 1  3  5
\overline{M}_{2,2} 1  6  14
\overline{M}_{2,3} 1  12  44  67
\overline{M}_{2,4} 1  24 144  333
\overline{M}_{2,5} 1  48 474 1668  2501
\overline{M}_{2,6} 1  96 1547 8256 18296
\overline{M}_{2,7} 1 192 4986 39969 129342 189289

The theorem used above also gives the corresponding results for M_{2,n} for n ≤ 7, which we will present in terms of local systems 𝕍_{λ} defined as above, but starting from 𝕍 := R^{1}π_{∗}ℚ. See [14, Section 8] for the results on e_{c}(M_{2} ⊗ ℂ, 𝕍_{λ}), for all λ of weight at most 3.

Theorem 11.6. The Hodge Euler characteristics of the local systems 𝕍_{λ} on M′_{2} := M_{2} ⊗ ℂ of weight 4 or 6 are equal to

e_{c}(M′_{2}, 𝕍_{(4,0)}) = 0, e_{c}(M′_{2}, 𝕍_{(3,1)}) = L^{2} − 1, e_{c}(M′_{2}, 𝕍_{(2,2)}) = −L^{4},

e_{c}(M′_{2}, 𝕍_{(6,0)}) = −1, e_{c}(M′_{2}, 𝕍_{(5,1)}) = L^{2} − L − 1,

e_{c}(M′_{2}, 𝕍_{(4,2)}) = L^{3}, e_{c}(M′_{2}, 𝕍_{(3,3)}) = −L − 1.

12. Appendix: Introducing b_{i}, c_{i} and r_{i}

This section will give an interpretation of the information carried by the u_{g}’s. It will be in terms of counts of hyperelliptic curves together with prescribed inverse images of points on P^{1} under their unique degree 2 morphism.

Definition 12.1. Let C_{φ} be a curve defined over k together with a separable degree 2 morphism φ over k from C to P^{1}. We then define

b_{i}(C_{φ}) := |{α ∈ A(i) : |φ^{−1}(α)| = 2, φ^{−1}(α) ⊆ C(k_{i})}|,
c_{i}(C_{φ}) := |{α ∈ A(i) : |φ^{−1}(α)| = 2, φ^{−1}(α) ⊈ C(k_{i})}|

and put r_{i}(C_{φ}) := b_{i}(C_{φ}) + c_{i}(C_{φ}).

The number of ramification points of f that lie in A(i) is then equal to |A(i)| − r_{i}(C_{φ}). Let λ_{i} denote the partition of i consisting of one element. We then find that

|C_{φ}(λ_{i})| = |A(i)| + b_{i}(C_{φ}) − c_{i}(C_{φ}) + \begin{cases}2c_{i/2}(C_{φ}) & if i is even; \\ 0 & if i is odd. \end{cases}

and thus

a_{n}(C_{φ}) = ∑_{i|n: 2i∤n}(c_{i}(C_{φ}) − b_{i}(C_{φ})) + ∑_{i:2i|n}(−b_{i}(C_{φ}) − c_{i}(C_{φ})).

Definition 12.2. For partitions μ and ν, g ≥ 2 and odd characteristic, define

b_{μ}c_{ν}|_{g} := ∑_{[C_{f}]∈H_{g}(k)/≅_{k}} \frac{1}{|Aut_{k}(C_{f})|} ∏_{i=1}^{l(μ)} b_{i}(C_{f})^{μ_{i}} ∏_{j=1}^{l(ν)} c_{j}(C_{f})^{ν_{j}}.

The number |μ| + |ν| will be called the weight of this expression.

Remark 12.3. We can, in the obvious way, also define a_{λ}b_{μ}c_{ν}|_{g}, but from the relation between a_{i}(C_{f}), b_{i}(C_{f}) and c_{i}(C_{f}) we see that this gives no new phenomena.

Directly from the definitions we get the following lemma.

Lemma 12.4. Let the characteristic be odd and let f be an element of P_{g}. We then have

b_{i}(C_{f}) = \frac{1}{2} ∑_{α∈A(i)}(χ_{2,i}(f(α))^{2} + χ_{2,i}(f(α)))

and

c_{i}(C_{f}) = \frac{1}{2} ∑_{α∈A(i)}(χ_{2,i}(f(α))^{2} − χ_{2,i}(f(α))).

If the characteristic is odd we then use the same arguments as in Section 3 to conclude that

b_{μ}c_{ν}|_{g} = \frac{I}{2^{|μ|+|ν|}} ∑_{f∈P_{g}} ∏_{i=1}^{l(μ)}( ∑_{α∈A(i)} χ_{2,i}(f(α)) + χ_{2,i}(f(α))^{2})^{μ_{i}}·

· ∏_{j=1}^{l(ν)}( ∑_{α∈A(j)} χ_{2,j}(f(α)) − χ_{2,j}(f(α))^{2})^{ν_{j}}.

Note that this expression is defined for all g ≥ −1. It can be decomposed in terms of u_{g}’s (that is, we can find a result corresponding to Lemma 3.8) for tuples (n; r) ∈ N_{m} such that

(12.1)  |n| ≤ |μ| + |ν|.

Remark 12.5. The corresponding results clearly hold for elements (h, f) in P_{g} in even characteristic and the decomposition of b_{μ}c_{ν}|_{g} is independent of characteristic.

Example 12.6. For each N we have the decomposition:

b_{[N]}|_{g} = \frac{1}{2}(u_{g}^{((N);(2))} + u_{g}^{((N);(1))}) and c_{[N]}|_{g} = \frac{1}{2}(u_{g}^{((N);(2))} − u_{g}^{((N);(1))}).

Example 12.7. Let us decompose b_{[1^{2}]}c_{[2]}|_{g} into u_{g}’s:

b_{[1^{2}]}c_{[2]}|_{g} = \frac{1}{8}(u_{g}^{((2,1,1);(2,2,2))} + u_{g}^{((2,1,1);(2,1,1))} + 2u_{g}^{((2,1);(2,2))}

− u_{g}^{((2,1,1);(1,2,2))} − u_{g}^{((2,1,1);(1,1,1))} − 2u_{g}^{((2,1);(1,2))}).

In this expression we have removed the u_{g}’s for which ∑_{i=1}^{m} r_{i}n_{i} is odd, since they are always equal to 0.

Lemma 12.8. For each N, the following information is equivalent: (1) all u_{g}’s of degree at most N;

(2) all b_{μ}c_{ν}|_{g} of weight at most N.

Proof: From property (12.1) of the decomposition of b_{μ}c_{ν}|_{g} into u_{g}’s we directly find that if we know (1) we can compute (2). For the other direction we note on the one hand that

(12.2)  I ∑_{f∈P_{g}} ∏_{i=1}^{j}(b_{i}(C_{f}) − c_{i}(C_{f}))^{s_{i}}(b_{i}(C_{f}) + c_{i}(C_{f}))^{t_{i}}

can be formulated in terms of b_{μ}c_{ν}|_{g}’s of weight at most

S := ∑_{i=1}^{j} i (s_{i} + t_{i}).

If we on the other hand decompose (12.2) into u_{g}’s we find that there is a unique u_{g} of degree S. The corresponding pair (n; r) contains, for each i, precisely s_{i} entries of the form i^{1} and t_{i} entries of the form i^{2}. Every u_{g} of degree S can be created in this way and hence if we know (2) we can compute (1). □

Remark 12.9. From the definitions of a_{i}(C_{f}) and r_{i}(C_{f}) we see that knowing (1) and (2) in Lemma 12.8 is also equivalent to knowing

(3) all a_{λ}r_{ξ}|_{g} of weight at most N, where a_{λ}r_{ξ}|_{g} is defined in the obvious way. Moreover, a_{λ}r_{ξ}|_{g} = 0 if |λ| is odd.

13. Appendix: The stable part of the counts

Remark 13.1. All results in this section are independent of characteristic.

Definition 13.2 ([8, Def. 1.2.1, 1.2.2]). Let F be a constructible (ℓ-adic) sheaf on a scheme X of finite type over ℤ. The sheaf F is said to be pure of weight m if, for every closed point x in X and eigenvalue α of Frobenius F (relative to k = k(x)) acting on Fx̄, α is an algebraic integer of weight equal to m, i.e., such that all its conjugates have absolute value equal to q^{m/2}. The sheaf F is said to be mixed of weight ≤ m if there exists a filtration 0 = F_{−1} ⊂ F_{0} ⊂ … ⊂ F_{m} = F of constructible subsheaves such that, for all j, F_{j}/F_{j−1} is pure of weight j.

Theorem 13.3 ([8, Cor. 3.3.3, 3.3.4]). Let X ⟶^{f} ℤ be a scheme of finite type, and F a constructible sheaf mixed of weight ≤ m. Then R^{i}f_{!}F is mixed of weight ≤ m + i. Thus, for every finite field k, there is a filtration 0 = W_{−1} ⊂ W_{0} ⊂ … ⊂ W_{i+m} = H_{c}^{i}(Xk̄, F) of Gal(k̄/k)-representations such that, for all j, W_{j}/W_{j−1} is pure of weight j.

Definition 13.4. Let K_{0}(Gal_{k}) be the Grothendieck group of Gal(k̄/k)-repre-sentations. In this category, and with the notation of Theorem 13.3, we have [H_{c}^{i}(Xk̄, F)] = ∑_{j=0}^{i+m}[W_{j}/W_{j−1}]. For any w ≥ 0, let us define [H_{c}^{i}(Xk̄, F)]^{w} := ∑_{j=w}^{i+m}[W_{j}/W_{j−1}] and e_{c}^{w}(Xk̄, F) := ∑_{i≥0}(−1)^{i}[H_{c}^{i}(Xk̄, F)]^{w} in K_{0}(Gal_{k}). We make the corresponding definition of e_{c}^{w}(X\overline{ℚ}, F) in K_{0}(Gal_{ℚ}).

Recall the definition in Section 11.1, for a prime ℓ ∤ q, of the ℓ-adic local system 𝕍_{λ} on H_{g}. If τ is the canonical morphism from H_{g}⊗k̄ to H_{g}, we put 𝕍′_{λ} = τ_{∗}𝕍_{λ}. This is a constructible sheaf pure of weight |λ|. In this section we will see that if g and w are large enough we can compute the trace of Frobenius on e_{c}^{w}(H_{g} ⊗ k̄, 𝕍_{λ}), which by definition (cf. Section 2 in [3]) is equal to e_{c}^{w}(H_{g}, 𝕍′_{λ}). We first make the connection to 𝕊_{n}-equivariant counts of points of H_{g,n} explicit.

Lemma 13.5. Let the symmetric polynomial s_{<λ>} be the Schur polynomial in the symplectic case (see [11, A.45]), and p_{λ} the power sum. If s_{<λ>} = ∑_{|μ|≤|λ|} m_{μ} p_{μ} then

(13.1)  Tr(F, e_{c}(H_{g} ⊗ k̄, 𝕍′_{λ})) = ∑_{|μ|≤|λ|} m_{μ} q^{\frac{1}{2} (|λ|−|μ|)} a_{μ}|_{g}.

From Theorems 4.14 and 9.12 we see that only the u_{g}’s with all r_{i} = 2 have inhomogeneous recurrence relations. Theorem 5.2 dealt with the homogeneous part of the linear recurrence relations for a_{λ}|_{g}. The following lemma, which is a direct consequence of Theorems 4.14, 9.12 and 5.2, deals with the “inhomogeneities”.

Lemma 13.6. Denote by t_{n} the coefficient of u_{g}^{(n;(2,...,2))} in the decomposition of a_{λ}|_{g} (given in Construction-Lemma 3.8). Each value of |n| for a pair (n; (2, …, 2)) appearing in this decomposition of a_{λ}|_{g} is at most equal to |λ|/2. Define the polynomial

f_{n}(x) := (∏_{i=1}^{m}(x^{n_{i}} − 1))/(x − 1).

For g ≥ 0, let R_{λ}(q)|_{g} be the sum, over the pairs (n; (2, …, 2)) that occur in the decomposition of a_{λ}|_{g}, of the polynomial quotients of,

(13.2)  t_{n} q^{2g+|n|} J (q − 1) f_{n}(q) by f_{n}(q^{2}),

which is of degree at most (|λ|+4g−2)/2. The polynomial R_{λ}(q)|_{g} is a particular solution to the recurrence relation, described in Section 5, for a_{λ}|_{g}.

Since the power sums form a rational basis of the ring of symmetric polynomials, equation (13.1) and Theorem 13.3 show that a_{λ}|_{g} is of the form ∑_{j} z_{j}α_{j} for a finite set of rational numbers z_{j} and distinct algebraic integers α_{i} of weight at most |λ| + 4g − 2 (note that 2g − 1 is the dimension of H_{g}). If our base field k is replaced by an extension k_{m} of degree m then a_{λ}|_{g} is equal to ∑_{j} z_{j}α_{j}^{m}. For g ≥ |λ| − 1, the linear recurrence relation for a_{λ}|_{g} (see Section 5) shows that it can be written as the particular solution R_{λ}(q)|_{g} plus the homogeneous part, an integer sum of a_{λ}|g̃ − R_{λ}(q)|g̃ for g̃ ≤ |λ| − 2. We then see that if g ≥ |λ| − 1 and w = 5 |λ|−9, the homogeneous part of the solution to the linear recurrence relation for a_{λ}|_{g} does not contribute to Tr(F, e_{c}^{w}(H_{g} ⊗ k̄, 𝕍′_{λ})). To conclude this we used the fact that ∑_{i} z_{i}α_{i}^{m} = 0 for all m implies that z_{i} = 0 for all i, where the z_{i} and α_{i} are complex numbers and the α_{i} are distinct and nonzero. We can now summarize using Theorem 3.2 in [1].

Definition 13.7. For a polynomial f(x) = ∑_{i} f_{i}x^{i} put f^{w}(x) := ∑_{i≥w} f_{i}x^{i}.

Theorem 13.8. Let q denote the class of ℚ_{ℓ}(−1) in K_{0}(Gal_{ℚ}). For g ≥ |λ| − 1 and w = 5 |λ| − 9 we have an equality in K_{0}(Gal_{ℚ}),

e_{c}^{w}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}) = ∑_{|μ|≤|λ|} m_{μ} q^{\frac{1}{2} (|λ|−|μ|)} R_{μ}^{w−|λ|+|μ|}(q)|_{g}.

Example 13.9. In the case λ = (4, 2, 2), for w = 31 and g ≥ 7, we find that Tr(F, e_{c}^{w}(H_{g} ⊗ k̄, 𝕍_{λ})) is equal to f_{g}^{w}(q), where f_{g} is the polynomial quotient of q^{2g+4}(3q^{2} + 3q + 2) by (q^{2} + 1)^{2}(q + 1)^{3}.

Remark 13.10. By Poincaré duality (cf. Section 2 in [3]) we find that there is a filtration 0 = W′_{i+|λ|−1} ⊂ W′_{i+|λ|} ⊂ … ⊂ W′_{2(2g−1+|λ|)} = H^{i}(H_{g} ⊗ k̄, 𝕍_{λ}) of Gal(k̄/k)-representations such that W′_{j}/W′_{j−1} is pure of weight j. Let us define [H^{i}(H_{g} ⊗ k̄, 𝕍_{λ})]_{w} := ∑_{j=i+|λ|}^{w}[W′_{j}/W′_{j−1}] and e_{w}(H_{g} ⊗ k̄, 𝕍_{λ}) := ∑_{i≥0}(−1)^{i}[H^{i}(H_{g} ⊗ k̄, 𝕍_{λ})]_{w} in K_{0}(Gal_{k}) and similarily e_{w}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}). Theorem 13.8 shows that, for g ≥ g̃ ≥ |λ| − 1 and w = 4g̃ − 3 |λ| + 7, one has that e_{w}(H_{g} ⊗ \overline{ℚ}, 𝕍_{λ}) is stable, in the sense that it is independent of g.

Computations for λ’s of low weight lead us to make a conjecture, which is true for |λ| ≤ 30.

Conjecture 13.11. For g ≥ |λ|−1 and w = 5 |λ|−9, we have e_{c}^{w}(H_{g}⊗\overline{ℚ}, 𝕍_{λ}) = 0 for all λ such that λ_{1} > |λ|/2.

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Jonas Bergström

Korteweg-de Vries Instituut Universiteit van Amsterdam Postbus 94248 1090 GE, Amsterdam The Netherlands o.l.j.bergstrom@uva.nl