Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

We consider the moduli space Hg,n of n-pointed smooth hyperelliptic curves of genus g. In order to get cohomological information we wish to make Sn-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 Sn-equivariant counts of Hg,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 Hg,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 Sn-equivariant Galois (resp. Hodge) structure of their l-adic (resp. Betti) cohomology. 2000 Mathematics Subject Classification: 14H10, 11G20


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 Documenta Mathematica 14 (2009)   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 ⊗ 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 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 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].
⋆ 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.⋆ 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.⋆ 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.⋆ 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.⋆ It is shown how to compute u 0 for any pair (n; r).⋆ 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. ⋆ 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.⋆ 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.⋆ 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).⋆ The counts of points of the previous sections are used to get cohomological information.This is, in particular, applied to M 2,n for n ≤ 7. ⋆ 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).⋆ In the second appendix, we find that for sufficiently large g we can compute the Euler characteristic, with Gal(Q/Q)-structure, of the part Documenta Mathematica 14 (2009) 259-296 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.

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 ⊗ k 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 k 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 k.For any pointed curve X that is a representative of a point in H F σ g,n , 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 k-isomorphism classes of the category R σ is equal to We then have the following equality (see [12] or [17]): This enables us to go from k-isomorphism classes to k-isomorphism classes: For any curve C over k, define C σ to be the set of n-tuples of distinct points Notation 2.1.A partition λ of an integer m consists of a sequence of nonnegative integers λ 1 , . . ., λ ν such that |λ| := 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 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 Documenta Mathematica 14 (2009) 259-296 directly follows that (2.1) 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 We will compute slightly different numbers than |H F σ g,n |, but which contain equivalent information.Let C be a curve defined over k.The Lefschetz trace formula tells us that for all m ≥ 1, If we consider equations (2.1) and (2.2) in view of equation ( 2.3) we find that 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 The numbers which we will pursue will be the following.Definition 2.2.For g ≥ 2 and any partition λ define (2.4) This expression will be said to have weight |λ|.Let us also define an expression of weight 0.

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 Documenta Mathematica 14 (2009) 259-296 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 1 2 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 op 2 (k) × k * /D where and where an element This defines a left group action of G on P g , where γ ∈ G takes f ∈ P g to f ∈ P g , with 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.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 This can up to sign be rewritten as where S := ν i=1 P 1 (k i ) λi , 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.
Let us also define u (n;r) g,α .
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 mi for each 1 ≤ i ≤ s, such that for any finite field k, g .

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,δi = 1 be the divisors of i.
⋆ If x ∈ T i,j then: Define n to be equal to the tuple 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 The equality is clear in view of the following three simple properties of the quadratic character.
⋆ Finally, for any α ∈ P 1 and any s, we have 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 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| := m i=1 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| = ν i=1 δi j=1 κ i,j d i,j .Since λ i = δi j=1 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 m i=1 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 m i=1 r i n i = ν i=1 δi j=1 κi,j k=1 ρ i,j,k d i,j .Let us prove the lemma by induction on m, starting with the case that m = ν i=1 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 = δi j=1 κ i,j , it follows that m i=1 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; r) with m > k.Since m < ν i=1 λ i we know that there exists numbers i 0 , j 0 , k 0 such that |Q i0,j0,k0 | ≥ 2. Let us fix an element x ∈ Q i0,j0,k0 and define a new pair (n ′ ; r ′ ) associated to the partitions {T ′ i,j } and {Q ′ i,j,k } by putting: ,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 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: Example 3.14.The decomposition of a [1 4 ,2] | g , starting with the general case: 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 1 2 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 1 2 ), 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 1 2 .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: 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 1 2 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

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We therefore have the equality The Riemann hypothesis tells us that |a r (C)| ≤ 2g √ 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 √ 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 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: 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 .

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 Multiplying with the element t gives a fixed point free action on the set P g and therefore This computation and Lemmas 3.8 and 3.12 proves the following lemma.
Thus, the only interesting cases are those for which m i=1 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 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 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 r i n i is even.
Definition 4.4.For any α = (α 1 , . . ., α m ) ∈ A ′ (n), let b j = b n j be the number of monic polynomials l of degree j such that l(α i ) is nonzero for all i.Let us also put bj = bn j := j i=0 b n i .Lemma 4.5.For each j ≥ 0 and n ∈ N m ≥1 , we have the equality 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 m1 , . . ., α mi .
Notation 4.6.For any α ∈ A ′ (n), let p αi denote the minimal polynomial of α i and put p α := m i=1 p αi .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 The lemma now follows from the Chinese remainder theorem, which tells us that the morphism k 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 Summing this equality over all s between −1 and g gives If r i = 2 for all i, then it follows from equation (4.3) that Summing this equality over all s between −1 and g gives In Ûg,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 Using Lemma 4.7 we can reformulate this equality as
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) If m i=1 r i n i is even, equation ( 4.3) and the definition of h(∞) shows that (4.9) U g,α = I (q − 1) If r i = 2 for all i, then equation (4.9) tells us that Remark 4.9.Fix an α ∈ A(n).If there is an element . 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.Proof: Fix any tuple n = (n 1 , . . ., n m ) and put n := |n|.If we let then the right hand side is equal to the right hand side of equation (4.4), and hence Say that b n j = j i=0 c n j,i q i and bn j = j i=0 ĉn j,i q i .If i ≤ j then equation (4.4) implies that c n j,i = c n n,n+i−j and hence ĉn j,i = j s=0 c n n,n+i−s .By equation (4.12) we know that q − 1 divides b n n , and Theorem 4.12.For any pair 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) = |n|−1 The first claim is shown in the proof of Lemma 4.10.Using the notation of that proof we find that bj − q bj−1 = Note that d n−1−j only depends upon n and not on q.

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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. 5 /(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 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 Solving this recurrence relation gives Proof: Fix any pair (n; r) in the decomposition of a λ | g and put n = |n|.Lemma 4.13 tells us that bj − q bj−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 ( m i=1 (X ni − 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 Documenta Mathematica 14 (2009) 259-296 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 where A 0 , A 1 , A 2 , A 3 and B 0 do not depend upon g.

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.

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 bj = j i=0 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.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 ≥  2)) g = (−1) g − q 2g for g ≥ 0.
Example 7.4.The result for a 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 7.8.The result for a 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 Documenta Mathematica 14 (2009) 259-296 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 7.10.The result for a [6] | g,odd is 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 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

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, Documenta Mathematica 14 (2009) 259-296 where h and f are polynomials defined over k that fulfill the following conditions: 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 1 2 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 The k-isomorphisms between curves corresponding to elements of P g are then precisely the ones induced by elements of G g by letting 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 ( h, f ) ∈ P g , with ( h, f ) = (φ g (Λ, e)(h), e −1 φ 2g (Λ, e)(f ) + l φ g (Λ, e)(h) + l 2 ).
Proof: Follows in the same way as Lemma 3.4.Notation 8.4.Let us put In the same way as in the case of odd characteristic we get the equality 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.u (n;r) g,α .
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.
⋆ Finally, for any α ∈ P 1 and any s, τ ,s h(α), f (α) 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.
Documenta Mathematica 14 (2009) 259-296 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 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 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: ⋆ 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 +f h ′ 2 ).Let l be equal to f q n /2 .Working modulo (x−α) 2 we then get 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 .

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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 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: Here we used that t| gcd(h 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 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 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 Since (h 2 , f 2 ) ∈ P i2 we see that r must be constant.Hence every irreducible Documenta Mathematica 14 (2009) 259-296 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 m1 := m 1 /m, m2 := m 2 /m and l := l/m.We are then in the same situation as above, that is m1 h 1 = m2 h 2 and m2 1 f 1 = m2 2 f 2 + m2 h 2 l + l2 .Thus, if r is an irreducible polynomial such that r| m1 but r ∤ m2 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
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 If so, take any such polynomials m i , l i and define . 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 )] gj for which deg(f j+1 ) is minimal.Say that f j+1 = f j + h j l j + l 2 j 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 gj+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 2 j+1 |f j+1 + h j l j+1 + l 2 j+1 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 2 j+1 ) ≤ 2g j+1 then this would contradict the minimality of deg(f j+1 ).We conclude that (h j , f j+1 ) ∈ P gj+1 .Finally we see that if we put mr := r−1 i=1 m i and l := Using the lemmas above we will be able to write Ûg in terms of u i for i between −1 and g.After this we will determine Ûg 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.
Lemma 9.10.For any element (h, f ) ∈ P ′ i and any monic polynomial m of degree g − i, 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, 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 If we choose g such that 2g + 2 ≥ |n| − 1, h 0 ∈ R g+1 and we put p α (x) := x p α, then we find in the same way as for equation (9.6) that (9.9) 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) Ûg,α = I (q − 1)q g+1 b ñ g+1 if ∀i : Documenta Mathematica 14 (2009) 259-296 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 m i=1 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.

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 S ñ-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 ⊗ k, 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 ⊗ Q, V λ ) in K 0 (Gal Q ), the Grothendieck group of Gal( Q/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 ⊗ k, V λ )) = P (q) for all finite fields k, possibly with the exception of a finite number of characteristics, then e c (H g ⊗ 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.

11.2.
Cohomological results for M 2,n and M 2,n .Using the stratification of 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 g,ñ for all g ≤ g and ñ ≤ n+2(g−g) (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 M 2,n .

Documenta Mathematica 14 (2009) 259-296
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 M 2,n induces an action on its cohomology, and hence H i (M 2,n ⊗ C, Q) may be written as a direct sum of H i λ (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) 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 M 2,4 is equal to Theorem 11.3.The S n -equivariant Hodge Euler characteristic of M 2,5 is equal to Theorem 11.4.The S n -equivariant Hodge Euler characteristic of M 2,6 is equal to Documenta Mathematica 14 (2009) 259-296 Theorem 11.5.The Sn-equivariant Hodge Euler characteristic of M2,7 is equal to (L 10 + 12L 9 + 90L 8 + 363L 7 + 854L 6 + 1125L 5 + 854L 4 + 363L 3 + 90L 2 + . ..)s7 In Table 1 we present the nonequivariant information (remember that all cohomology is Tate) in the form of Betti numbers of 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 M 2,n for n ≤ 6 found in [4].
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 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 and thus Definition 12.2.For partitions µ and ν, g ≥ 2 and odd characteristic, define 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 If the characteristic is odd we then use the same arguments as in Section 3 to conclude that 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: ).
In this expression we have removed the u g 's for which m i=1 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 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 F x, 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 i c (Xk, F ) of Gal( k/k)-representations such that, for all j, W j /W j−1 is pure of weight j. ] and e w c (Xk, F ) := i≥0 (−1) i [H i c (Xk, F )] w in K 0 (Gal k ).We make the corresponding definition of e w c (X 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 ⊗ k 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 w c (H g ⊗ k, V λ ), which by definition (cf.Section 2 in [3]) is equal to e w c (H g , V ′ λ ).We first make the connection to S n -equivariant counts of points of H g,n explicit.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 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 m j .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 | g − R λ (q)| g for g ≤ |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 w c (H g ⊗ k, V ′ λ ) .To conclude this we used the fact that i z i α m i = 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 ), Example 13.9.In the case λ = (4, 2, 2), for w = 31 and g ≥ 7, we find that Tr F, e w c (H g ⊗ k, V λ ) is equal to f w g (q), where f g is the polynomial quotient of q 2g+4 (3q 2 + 3q + 2) by (q 2 + 1) 2 (q + 1) 3 .

5 .Theorem 5 . 2 .
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).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

Example 12 . 7 .
Let us decompose b [1 2 ] c [2] | g into u g 's: b [1 2 ] c [2] C f ) − c i (C f ) si b i (C f ) + c i (C f ) ti can be formulated in terms of b µ c ν | g 's of weight at most S := j i=1 i (s i + t i ).Documenta Mathematica 14 (2009) 259-296

Definition 13 . 4 .
Let K 0 (Gal k ) be the Grothendieck group of Gal( k/k)-representations.In this category, and with the notation of Theorem 13.3, we have [H i c (Xk, F )] = i+m j=0 [W j /W j−1 ].For any w ≥ 0, let us define [H i c (Xk, F )] w := i+m j=w [W j /W j−1
. 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 Let τ m be the function that takes (a, b) ∈ k 2 m 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.
[14,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.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