Cohomology of moduli spaces via a result of Chenevier and Lannes

. We use a classiﬁcation result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with ℓ -adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of M 3 ,n and M 3 ,n for n ≤ 14 and of local systems V λ on A 3 for | λ | ≤ 16 .


Introduction
The central role in this paper is played by a classification result of Chenevier and Lannes [CL19].For n ≥ 1, they classify the level 1 algebraic cuspidal automorphic representations of PGL n of motivic weight w ≤ 22.There are 11 of them, given in Theorem 2.1.
As a part of the web of conjectures in the Langlands program, there is the so-called Langlands correspondence, which says, loosely speaking, that there should be a bijection between algebraic automorphic representations and irreducible geometric Galois representations.The assumption that such a bijection exists, formulated explicitly for weight at most 22 in Conjecture 3.2, has very concrete geometric consequences, as exemplified by Theorem 3.5.Namely, the cohomology groups of Deligne-Mumford stacks that are smooth and proper over Z of relative dimension at most 22 would all consist of Galois representations corresponding to the automorphic representations given by Chenevier and Lannes.
We stress the strong implications this has for several moduli spaces of central interest in algebraic geometry.For example, assuming Conjecture 3.2, we determine the cohomology groups of the moduli spaces M 3,n of stable n-pointed curves of genus 3 for all n ≤ 14, as Galois representations endowed with an S n -action (up to semisimplification).Similarly, we determine the S n -equivariant Euler characteristic of M 3,n for all n ≤ 14 as an element of the Grothendieck group of Galois representations, and we also determine the Euler characteristics of all standard symplectic local systems of weight at most 16 on the moduli space A 3 of principally polarized abelian threefolds.For this, we use the fact that the integervalued Euler characteristics are known, and we also use computer counts over small finite fields (with at most 25 elements) of all smooth curves of genus 3 with their zeta functions.The results are available at https://github.com/jonasbergstroem/M3A3interp.This adds to a series of recent results about the cohomology of moduli spaces of pointed curves; see for instance [BFP22,CL22,CLP23].

A result of Chenevier and Lannes
Let G be a semisimple group scheme over Z. Recall that the space A 2 (G) of square integrable automorphic forms for G may be seen as the space of square integrable functions on G(Q)\G(A)/G( Ẑ) for a natural Radon measure.
Further strong results along these lines, as well as a simplification of the proof of Theorem 2.1, have been obtained by Chenevier and Taïbi [CT20].See also the earlier work by Chenevier and Renard [CR15].

Galois representations and cohomology
Fix a prime ℓ.By a Galois representation, we will mean an ℓ-adic representation of Gal(Q/Q).A Galois representation will be called geometric if it is unramified for all primes p ℓ and crystalline at p = ℓ.The cyclotomic character will be denoted by L. It is of dimension 1, Hodge-Tate weight 1, and motivic weight 2. For an irreducible geometric Galois representation V of dimension N , we have that ∧ N V = L i for some i, and we define the motivic weight w(V ) to be 2i/N .For a semisimple V , we let w(V ) denote the maximum of the weights of its irreducible pieces.Let F p denote a Frobenius element at p.
For a Hecke eigenform f in S k (SL 2 (Z)) (or in S j,k (Sp 4 (Z))), let a p (f ) denote the eigenvalue of the Hecke operator T p .By [Del71] (respectively, [Lau05] and [Wei05]), there is a 2-dimensional (respectively, 4-dimensional) semisimple geometric Galois representation V f such that and with Hodge-Tate weights 0, w(V ) = k −1 (respectively, 0, k −2, j +k −1, w(V ) = j +2k −3).It is irreducible, except when j = 0 and f is a Saito-Kurokawa lift.Also, Sym 2 V f , for an eigenform f in S k (SL 2 (Z)), will be geometric and irreducible; see for instance [Ram09,Lemma 4.7].Hence, to each of the 11 automorphic representations given in Theorem 2.1, there is naturally associated a Galois representation φ i for 1 ≤ i ≤ 11.Note that all these representations are self-dual in the sense that This means in particular that which aligns with the notation used in [BFvdG14] (note that N = 1 in all these cases).
As a part of the web of conjectures in the Langlands program, there is the so-called Langlands correspondence, which says, loosely speaking, that there should be a bijection between algebraic automorphic representations and irreducible geometric Galois representations.In our case the Galois representations corresponding to the automorphic representations have been constructed.Moreover, a semisimple Galois representation is determined by the traces of Frobenius for almost all primes.Hence, what we need is something similar to [Tay04, Conjecture 3.5], saying that to an irreducible geometric Galois representation there corresponds an algebraic automorphic representation such that the traces of Frobenius equal the Hecke eigenvalues for all primes.On the basis of such a conjecture, we are in turn led to conjecture the following.Note that this conjecture is in no way original.Conjecture 3.2.Every semisimple geometric Galois representation of motivic weight at most 22, and with non-negative Hodge-Tate weights, is a direct sum of representations φ j for 1 ≤ j ≤ 11 tensored with L k for some k ≥ 0.
Let X be a separated Deligne-Mumford stack of finite type over Z of relative dimension d together with an action of the symmetric group S n .For a field k, let H i c (X ⊗ k) denote the i th compactly supported ℓ-adic étale cohomology group.It is an ℓ-adic representation of Gal( k/k) × S n .We denote the category of such by Gal S n k and the corresponding Grothendieck group by K 0 (Gal S n k ).Denote by Gal k and K 0 (Gal k ) the versions without the S n -action.We identify an irreducible representation of S n with a partition µ of n in the usual way, and we denote the corresponding Schur polynomial by s µ .So we have a decomposition Note that for any closed substack Z ⊂ X , defined over k and invariant under S n , it holds that The analogous statements hold for the integer-valued Euler characteristics E c and E c,µ .When X is smooth, the semisimplification of the Galois representation H i c (X ⊗ Q) will be of motivic weight between 0 and i.If X is also proper, then it will be geometric and pure of weight i (meaning that all of its irreducible pieces will have weight i).It follows that e c (X ⊗ Q) as an element of K 0 (Gal Q ) for all i.Moreover, by Poincaré duality, we then have that For a finite field k = F q with q = p r and ℓ p, we define the (geometric) Frobenius F q ∈ Gal( k/k) to be the inverse of x → x q .We choose a Frobenius element (by abuse of notation, we will use the same letter) F q ∈ Gal(Q/Q), using an element of the Galois group of the p-adic completion of Q that is mapped to the Frobenius element F q ∈ Gal( k/k).If X is smooth and proper, then for all q such that p ℓ, these Frobenii satisfy (by [vdBE05, Proposition 3.1]) and these traces will be elements of Z.Note that the traces of F q for (almost) all primes q will determine e c,µ (X ⊗ Q) as an element of K 0 (Gal Q ), by a Chebotarov density argument.
Example 3.4.The following 34 Galois representations generate Proof.For every i ≥ 0, the cohomology group H i c,µ (X ⊗ Q) with X a smooth and proper Deligne-Mumford stack over Z is geometric and pure of weight i; see [Tay04, Section 1, p. 79] and also [vdBE05] for the generalization to Deligne-Mumford stacks.The result for 0 ≤ i ≤ d now follows from Conjecture 3.2.□ Remark 3.6.By (3.2) it follows that the L k φ j that occur in Remark 3.8.Fix any i ≥ 0, and assume that

Moduli spaces of curves
For any g, n, with n ≥ 3 if g = 0 and n ≥ 1 if g = 1, let M g,n and M g,n denote the moduli spaces of smooth, respectively stable, n-pointed curves of genus g.These are smooth Deligne-Mumford stacks over Z of dimension 3g − 3 + n, and M g,n is also proper over Z.They come with a natural action of the symmetric group S n , permuting the n marked points.
The complement of M g,n in M g,n , denoted by ∂M g,n , can be written as a disjoint union of locally closed subsets (4.1) We will analyze which elements of K 0 (Gal Q ) can appear in e c,µ (M g,n ⊗ Q) using a formula by Getzler and Kapranov, namely [GK98, Theorem 8.13].We will follow the notation of [GK98], but with a slight generalization.Put G = Gal k for some field k.We will consider a stable S-module V = {V ((g, n)) : g, n ≥ 0}, see [GK98, Section 2.1], but where the graded pieces V ((g, n)) i will be finite-dimensional ℓadic G × S n -representations (instead of just S n -representations).These pieces will then be direct sums of G × S n -representations of the form W ⊠ V .Let [W ] denote the element of the Grothendieck group K 0 (G) of finite-dimensional G-representations corresponding to a G-representation W .For an element W ′ = j a j [W j ] ∈ K 0 (G), with a j ∈ Z and G-representations W j with characters χ W j , define the character χ W ′ = j a j χ W j .Note that this gives a bijection between elements of K 0 (G) and their characters.
Let Λ be the ring of symmetric power series; see [GK98, Section 7.1].We define the character ch n (W ⊠ V ) to be χ W • ch n (V ), compare with [GK98, Section 7], which is an element of Λ with coefficients in K 0 (G).With this definition, Ch(V ), see [GK98, Section 8.2], will be an element of Λ((h)) with coefficients in K 0 (G).We then define MV as in [GK98, Section 2.17].
Let p i denote the i th power sum in Λ.For a G-representation W , we then put Theorem 8.13 of [GK98] then follows, with the same proof, and so Putting V ((g, n)) i = H i c (M g,n ⊗ Q), which are finite-dimensional ℓ-adic G × S n -representations, we get a stable S-module V .For a partition µ ⊢ n, we let s µ denote the corresponding Schur polynomial.We now have, see [GK98, Section 6.2], Proposition 4.1.Assume that Conjecture 3.2 holds.For any g, n such that 3g − 3 + n ≤ 22 and partition µ ⊢ n, e c,µ Proof.We use induction on the dimension.The statement clearly holds for the zero-dimensional base case M 0,3 = M 0,3 .Take any g, n such that d := 3g − 3 + n ≤ 22, and assume that e c,µ (M g, ñ ⊗ Q) ∈ Ψ 3 g−3+ ñ for any g, ñ such that 3 g − 3 + ñ < d.Using the Getzler-Kapranov formula, we find that e c,µ (∂M g,n ⊗ Q) is in Ψ d−1 , noting that no non-trivial tensor product of representations φ i with 2 ≤ i ≤ 11 can appear, for dimension reasons.
Applying Theorem 3.5 to M g,n , we see that if L k φ j appears in H i c (M g,n ⊗ Q) for i ≤ d with w = w(φ j ), then its Poincaré dual (3.2) will have weight 2d − i = w + 2(d − w − k) and be of the form L d−w−k φ j .Hence We conclude by applying (3.1).□
Remark 4.3.Also note that the cohomology groups H i c (A g ⊗ Q, V λ ) will have motivic weight between 0 and |λ| + i; see [Del80, Corollaire 3.3.4].Definition 4.4.Let Ψ λ denote the submonoid of K 0 (Gal Q ) generated by L k φ j for all k ≥ 0 and 1 ≤ j ≤ 11 such that 2k + w(φ j ) ≤ g(g + 1) + |λ| and such that k + w j is in W λ , for any Hodge-Tate weight w j of φ j .

The Torelli map
The moduli spaces we are considering are related via the Torelli map t g : M g → A g for g ≥ 2 and t 1 : M 1,1 → A 1 .This morphism between stacks has degree 1 if g ≤ 2 (it is an isomorphism in genus 1) and degree 2 if g ≥ 3, ramified along the hyperelliptic locus , where these Euler characteristics (and E c ) are defined as in Section 3. If |λ| is odd, then due to the presence of the automorphism −1 on all Jacobians.But for g ≥ 3, there is no reason for e c (M g ⊗ Q, t * g V λ ) to be zero.The first non-zero example is Remark 4.6.In fact, e c (M g ⊗ Q, t * g V λ ) will contain Galois representations that are not present in any e c (A h ⊗ Q, V λ ′ ) with h = g and |λ ′ | ≤ |λ|, or h < g.These could be called Teichmüller motives.Conjecturally, the first two such instances occur for g = 3 and λ = (11, 3, 3) or (7, 7, 3) and correspond to two of the seven automorphic representations of weight 23 in [CT20, Theorem 3], as described in [CFvdG20].Proof sketch.This is well known.The Leray spectral sequence associated to the forgetful morphism π : M g,n → M g gives a relation between the cohomology of M g,n and the cohomology of the higher direct images R i π * Q ℓ .The latter sheaves can be expressed in terms of local systems t * g V λ for λ with |λ| ≤ n.Taking Euler characteristics, this connection becomes a representation-theoretic question of relating S n -representations to GSp 2g -representations.This can be formulated in symmetric polynomials as finding an expression of the (usual) Schur polynomial s µ in terms of symplectic Schur polynomials s <λ> (see the definition in [FH91, Appendix A.45], which should be homogenized using an extra variable q).This can always be done, and there are elements a ′ µ,λ ∈ Z[q] such that and replacing q with L in a ′ µ,λ gives a µ,λ .There are also b ′ λ,µ ∈ Z[q] such that and replacing q with L in b ′ λ,µ , we get b λ,µ .□

The boundary of M 3,n
Computing e c,µ (M 0,n ⊗ Q) for any n and partition µ is not difficult; for details see for instance [Get95].Using Lemma 4.7 and the formula for e c (A 1 ⊗ Q, V a ) described in Section 4.2, one can compute the Euler characteristic e c,µ (M 1,n ⊗ Q) for any n and partition µ; see [Get99].
There is a stratification (4.6) Again, using the formula for e c (A 1 ⊗ Q, V a ), we can compute e c ((A ×2 1 /S 2 ) ⊗ Q, V a,b ) for any local system V a,b .For details on this computation, see for instance [Pet13].Now, using the formula for e c (A 2 ⊗ Q, V a,b ) mentioned in Section 4.2 and Lemma 4.7, we can also compute e c,µ (M 2,n ⊗ Q) for any n and partition µ.
Remark 4.10.For any n ≤ 9 and partition µ, e c,µ (M 2,n ⊗ Q) is a polynomial in L. Adding the contribution from the boundary, we see that the same statement holds for M 2,n .(Note that S [12] in H 11 c (M 1,11 ) comes with the alternating representation.)But, one finds L S [12] both in e c,[1 10 ] (M 2,10 ⊗ Q) and in Together, this gives us all the pieces to compute e c,µ (∂M 3,n ⊗ Q) for any n and partition µ using the Getzler-Kapranov formula described in Section 4.1.

The case A 3
In [BFvdG14, Conjecture 7.1] there is a conjectural formula, corroborated by extensive computations, for e c (A 3 ⊗Q, V a,b,c ) with "main term" S[a−b, b−c, c+4], described below.For any eigenform f ∈ S j,k,l (Sp 6 (Z)), we conjecture in [BFvdG14] that there is a corresponding 8-dimensional Galois representation V f similarly to the cases of SL 2 (Z) and Sp 4 (Z).If f 1 , . . ., f N is any basis of eigenforms of S j,k,l (Sp 6 (Z)), then we let S[j, k, l] denote the direct sum of the Galois representations . This is conjectured to be a lift with Galois representation Proof.There is a stratification (4.8) Assume that a + b + c + 6 ≤ 22.It follows from Equation (4.5), Proposition 4.1, and Lemma 4.7 that e c (t 3 (M 3 ) ⊗ Q, V a,b,c ) is in Ψ a+b+c+6 .Using the same results, we find that e c (( ) and e c ((A ×3 1 /S 3 )⊗Q, V a,b,c ) are in Ψ a+b+c+6 .The result now follows from the stratification (4.8) and Theorem 4.2 together with Remark 4.3.□

Finding linear relations
For any partition µ of n ≤ 16, we know (assuming Conjecture 3.2) by Theorem 3.5 and Poincaré duality that there are integers c µ k,j such that Similarly, for every λ such that |λ| ≤ 16, there are integers d λ k,j such that (5.2) where the sum is over k, j such that L k φ j ∈ Ψ λ .Taking dimensions we have

Integer-valued Euler characteristic
The integer-valued Euler characteristics E c (M 3 ⊗ Q, V λ ) and E c (A 3 ⊗ Q, V λ ) can be computed for any λ using an algorithm found in [BvdG08]; see Tables 3, 4 and 6 in op.cit.for some values.Using an analogue of Lemma 4.7, we can then compute E c,µ (M 3,n ⊗ Q) for any n.By the result in Section 4.5, we can, for any n and partition µ, compute e c,µ (∂M 3,n ⊗ Q) and E c,µ (∂M 3,n ⊗ Q). (Note that by the work of Gorsky, E c,µ (M g,n ⊗ Q) can in fact be computed for any g, n, and µ; see [Gor14].) With this we can compute , which gives a linear relation between the c µ k,j (respectively, the d λ k,j ) for all n ≤ 16 (respectively, |λ| ≤ 16).

Traces of Frobenius
Using the computer we have found the necessary information about curves of genus at most 3 over finite fields F q with q ≤ 25, together with their zeta functions, in order to apply the Lefschetz trace formula to compute Tr(F q , e c,µ (M 3,n ⊗ F q )) and Tr(F q , e c (A 3 ⊗ F q , V λ )); see [BFvdG14, Section 8] for more details.

Euler characteristics of local systems on
for some integers α 1 , . . ., α 12 .Let us write the first three of the linear relations we found as in Section 5: If we continue and take the first 12 linear equations and solve, we find that Proof.For all λ such that |λ| ≤ 16, Ψ λ has rank r λ at most 12.In Section 5 we found 15 linear relations.The r λ first relations, coming from the integer-valued Euler characteristic together with the traces of Frobenius (ordered by the size of q), for the r λ unknowns d λ k,j in (5.2) turn out to be linearly independent.These can therefore be used to determine e c (A 3 ⊗ Q, V λ ).The results agree with the ones given in [BFvdG14, Section 10.2.2, pp.120-121].By [Taï17]

The cohomology of M 3,n for n ≤ 14
Compared to the situation for local systems on A 3 , we have less information about the Hodge-Tate weights of the cohomology groups of M 3,n .
Theorem 7.1.Let X be a smooth and proper Deligne-Mumford stack over Q ℓ of dimension d with unirational coarse moduli space.For any m 0, 2d, the ℓ-adic Gal(Q ℓ /Q ℓ )-representation H m c (X ⊗ Q ℓ ) is crystalline and cannot contain the Hodge-Tate weight m.
Proof.We follow the notation of [vdB08] and let D dR denote the de Rham Fontaine-functor.By [Fal89,Tsu02], generalized in [vdB08,Corollary 8.12] to Deligne-Mumford stacks, we have an isomorphism of filtered vector spaces is a birational invariant, and since it vanishes for any projective space, the result follows.
Proof.It is easy to see that the coarse moduli space of M 3,n ⊗ Q is rational for n = 14; hence it is unirational for n < 14.Using Theorem 7.1 and Conjecture 3.2, we conclude that H m c (M 3,n ⊗ Q) is in Φ ′ m for any 0 < m ≤ 6 + n.
Since M g,n is irreducible, we know that H 0 c M g,n ⊗ Q is 1-dimensional, and it will furthermore have a trivial action by Gal(Q/Q).In [AC98, Theorem 2.2] there is a formula for the dimension (using duality) of H 2 c M g,n ⊗ Q , and it is shown that this cohomology group is tautological, so it will consist of a direct sum of Galois representations L.
For any n ≤ 13, Φ ′ ≤6+n has rank r n ≤ 17.Fix any partition µ of n.Since we know the coefficients of 1 and L, we have r n − 2 unknowns c µ k,j .In Section 5 we found 15 linear relations for the r n unknowns c µ k,j , which turn out to be linearly independent.These therefore determine e c,µ (M 3,n ⊗ Q).Since we can compute e c,µ (∂M 3,n ⊗ Q) for any n, see Section 4.5, we can also determine e c,µ (M 3,n ⊗ Q) for n ≤ 13.
Since Φ ′ ≤20 has rank 18, we are missing one relation for n = 14.But by Theorem 6.2 we know e c (A 3 , V λ ) for any λ such that |λ| = 14.Using the stratification (4.8) together with the information about genus 1 and 2, we can compute e c (t 3 (M 3 ) ⊗ Q, V λ ) for any λ such that |λ| = 14.Using Lemma 4.7 and the above, we can compute e c (M 3 ⊗ Q, t * 3 V λ ) for any λ such that |λ| ≤ 14.Using Lemma 4.7 again, we can compute e c,µ (M 3,n ⊗ Q) for all n ≤ 14 and partitions µ.By Section 4.5 we can also compute e c,µ (∂M 3,n ⊗ Q) for all n ≤ 14 and partitions µ. □ Remark 7.4.The Euler characteristics e c,µ (M 3,n ⊗ Q) and e c,µ (M 3,n ⊗ Q) are determined unconditionally for n ≤ 7; see [Ber08] together with the results of Section 4.5.In [CL22] it is proven that e c,µ (M 3,n ⊗ Q) is a polynomial in L for n up to 8.So the result for e c,µ (M 3,n ⊗ Q) and e c,µ (M 3,n ⊗ Q) determined above also holds unconditionally for n = 8.In turn, this means that the Euler characteristics e c (A 3 ⊗ Q, V λ ) are determined unconditionally for all λ with |λ| = 8 (cf.Remark 6.3).
Remark 7.5.For 8 ≤ n ≤ 14, the Euler characteristic e c (M 3,n ⊗ Q) is not a polynomial in L. The same holds for e c (M 3,n ⊗ Q) for 10 ≤ n ≤ 14.Note that this result is unconditional, as can be seen using the interpolation in the proof of Theorem 7.3.