2.1 Dimension and Hodge structure

We start by explaining how to extract $H^{11}(\overline {\mathcal {M}}_{1,n})$ with its Hodge structure or Galois representation from [Reference Getzler14]. Getzler gives generating functions that determine the cohomology groups $H^{*}(\overline {\mathcal {M}}_{1,n})$ with their $\mathbb {S}_n$ -actions. These formulas simplify substantially when forgetting the $\mathbb {S}_n$ -action, so we begin by using Getzler’s formula to extract $H^{11}(\overline {\mathcal {M}}_{1,n})$ nonequivariantly. Below, as in [Reference Getzler14], we write $\mathsf {L}$ for the Tate motive and $\mathsf {S}_{2k+2}$ for the Hodge structure associated to the space of cusp forms of weight $2k+2$ (see [Reference Getzler14, p. 489] for definition). We note that $\mathsf {S}_{2k+2} = 0$ for $k \leq 4$ .

Proof. Let $\mathsf {e}^{\mathbb {S}_n}(\mathcal {M})$ denote the $\mathbb {S}_n$ -equivariant Euler characteristic of $\mathcal {M}$ in the Grothendieck ring of equivariant mixed Hodge structures. Getzler defines two families of generating functions $\mathbf {a}_i = \sum \mathsf {e}^{\mathbb {S}_n}(\mathcal {M}_{i,n})$ and $\mathbf {b}_i = \sum \mathsf {e}^{\mathbb {S}_n}(\overline {\mathcal {M}}_{i,n})$ . For $i = 0$ or $1$ , these generating functions are power series in the Hodge structures $\mathsf {L}$ and $\mathsf {S}_{2k+2}$ whose coefficients are symmetric functions.

To get to the ordinary Euler characteristic generating function $\mathsf {e}(\mathcal {M})$ from $\mathsf {e}^{\mathbb {S}_n}(\mathcal {M})$ , we apply Getzler’s $\mathrm {rk}$ functor, which is defined by setting the power sums to $p_1 = x$ and $p_n = 0$ for $n> 0$ . It sends $\mathsf {e}^{\mathbb {S}_n}(\overline {\mathcal {M}}_{i,n})$ to $\mathsf {e}(\overline {\mathcal {M}}_{i,n}) \frac {x^n}{n!}$ [Reference Getzler14, p. 484]. We shall write $a_i = \mathrm {rk}(\mathbf {a}_i)$ and $b_i = \mathrm {rk}(\mathbf {b}_i)$ , which are power series in $\mathsf {L}$ and $\mathsf {S}_{2k+2}$ when $i = 0$ or $1$ . For example,

$$\begin{align*}b_0 = \mathrm{rk}(\mathbf{b}_0) = \sum_{n \geq 3} \mathsf{e}(\overline{\mathcal{M}}_{0,n}) \frac{x^n}{n!}. \end{align*}$$

Important for us is that

$$\begin{align*}1 + x + b_0' = e^x + \text{terms divisible by }\mathsf{L}.\end{align*}$$

For $b_1$ , we are interested in the coefficient of $\frac {x^n}{n!}\mathsf {S}_{12}$ , which is equal to the negative of the multiplicity of $\mathsf {S}_{12}$ in $H^{11}(\overline {\mathcal {M}}_{1,n})$ by construction. Applying $\mathrm {rk}$ to [Reference Getzler14, Theorem 2.5] relates $b_1$ to $a_1$ . Note that the symbol $\circ $ in [Reference Getzler14, Theorem 2.5] denotes plethysm of symmetric functions; applying $\mathrm {rk}$ turns this plethysm into composition of functions, as can be seen from the properties characterizing plethysm in [Reference Getzler13, Section 5.2]. In particular, we find

(2.1) $$ \begin{align} b_1 = a_1(x + b_0') + \text{terms built from }a_0. \end{align} $$

The terms built from $a_0$ cannot contribute to the coefficient of $\mathsf {S}_{12}$ . (We note that there is a small error in [Reference Getzler14, Theorem 2.5], which is corrected in [Reference Consani and Faber9, p. 306], but it occurs in these terms built from $a_0$ , and thus will not affect the outcome of our calculation.)

To expand the right-hand side of equation (2.1), apply $\mathrm {rk}$ to the equation for $\mathbf {a}_1$ in [Reference Getzler14, p. 489], and then plug in $x + b_0'$ for x (so substitute $p_1 = x+b_0'$ and $p_n = 0$ for $n> 1$ ):

$$ \begin{align*} \begin{split} a_1 (x + b_0') &= \mathrm{res}_0{} \Bigg[ \frac{(1 + x + b_0')^{1 - \omega -\mathsf{L}/\omega + \mathsf{L}}-1}{1 - \omega -\mathsf{L}/\omega + \mathsf{L}} \\ & \qquad\qquad\times \left(\sum_{k=1}^\infty \left(\frac{\mathsf{S}_{2k+2} + 1}{\mathsf{L}^{2k+1}}\right) \omega^{2k} - 1\right)(\omega - \mathsf{L}/\omega )d\omega \Bigg] {}. \end{split} \end{align*} $$

In the middle parenthesized term, $\mathsf {S}_{12}$ is multiplied by $\omega ^{10}/\mathsf {L}^{11}$ . Since we need to take the residue at $0$ with respect to $\omega $ , the coefficient of $\mathsf {S}_{12}$ appears when the other terms combine to give $\mathsf {L}^{11}/\omega ^{11}$ . To get $\mathsf {L}^{11}/\omega ^{11}$ , we must use the $-\mathsf {L}/\omega $ piece of the $(\omega - \mathsf {L}/\omega )$ term. Similarly, when we expand the first term, only the powers of $\mathsf {L}/\omega $ are relevant. From this, we see that the coefficient of $\mathsf {S}_{12}$ in the above display is the negative of the coefficient of $y^{10}$ in

$$\begin{align*}\frac{(1 + x + b_0')^{1 + y}}{1+y} = \frac{e^{x(1+y)}}{1 + y} + \langle \mathsf{L} \rangle = \sum_{n = 1}^\infty \frac{x^n}{n!}(y+1)^{n-1} + \langle \mathsf{L} \rangle. \end{align*}$$

In conclusion, $H^{11}(\overline {\mathcal {M}}_{1,n})$ consists of ${n-1 \choose 10}$ copies of $\mathsf {S}_{12}$ .

Getzler’s formulas also encode the $\mathbb {S}_n$ -action on $H^{11}(\overline {\mathcal {M}}_{1,n})$ . We recover this information in a different way, by describing generators on which the $\mathbb {S}_n$ -action is evident, as follows.

2.2 Generators and their pullbacks

To begin, in the case $n = 11$ , Lemma 2.1 tells us

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{1,11})\otimes \mathbb{C} = H^{11,0}(\overline{\mathcal{M}}_{1,11}) \oplus H^{0, 11}(\overline{\mathcal{M}}_{1,11}).\end{align*}$$

The weight $12$ cusp form of $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ gives rise to a distinguished generator $\omega \in H^{11,0}(\overline {\mathcal {M}}_{1,11})$ ; see [Reference Faber and Pandharipande12, p. 14] for an explicit geometric construction. It is evident from this construction (or from [Reference Getzler14]) that $\mathbb {S}_{11}$ acts by the sign representation.

We now describe a natural collection of forms in $H^{11,0}(\overline {\mathcal {M}}_{1,n})$ , which we will soon see are generators. These forms come from pulling back the distinguished generator of $H^{11,0}(\overline {\mathcal {M}}_{1,11})$ under the various forgetful maps $\overline {\mathcal {M}}_{1,n} \to \overline {\mathcal {M}}_{1,11}$ . Precisely, given an ordered subset $A \subset \{1, \ldots , n\}$ with $|A| = 11$ , write $f_A\colon \overline {\mathcal {M}}_{1,n} \to \overline {\mathcal {M}}_{1,A} \cong \overline {\mathcal {M}}_{1,11}$ for the projection map and define $\omega _A := f_A^*\omega \in H^{11,0}(\overline {\mathcal {M}}_{1,n})$ .

The pullbacks of these forms to boundary divisors follow a simple rule. By the Künneth formula, the only boundary divisors of $\overline {\mathcal {M}}_{1,n}$ with nonzero $H^{11}$ are those of the form

(2.2) $$ \begin{align} D_B = \overline{\mathcal{M}}_{1,B \cup p} \times \overline{\mathcal{M}}_{0, B^c \cup q}, \end{align} $$

where $|B| \geq 10$ and $\iota _B\colon D_B \to \overline {\mathcal {M}}_{1,n}$ is the map that glues p to q. In this case, projection onto the first factor $\mathrm {pr}_1\colon D_B \to \overline {\mathcal {M}}_{1,B \cup p}$ induces an isomorphism

(2.3) $$ \begin{align} \mathrm{pr}_1^*\colon H^{11}(\overline{\mathcal{M}}_{1,B \cup p}) \xrightarrow{\sim} H^{11}(D_B). \end{align} $$

Given ordered subsets A and B of $\{1, \ldots , n\}$ with $|A| = 11$ and $|A \cap B| = 10$ , there is a unique element $i \in A$ such that $i \notin B$ . Let $\epsilon (A)$ denote the ordered set obtained from A by replacing i with p, so $\epsilon (A)$ is a subset of B. If $|A \cap B| = 11$ so that A is already contained in B, then we set $\epsilon (A) = A$ .

Lemma 2.2. Given a boundary divisor $\iota _B\colon D_B \to \overline {\mathcal {M}}_{1,n}$ , and $A \subset \{1, \ldots ,n\}$ with $|A| = 11$ , we have

$$\begin{align*}\iota_B^*\omega_A = \begin{cases} \mathrm{pr}_1^*\omega_{\epsilon(A)} & \text{if }|A \cap B| \geq 10 \\ 0 & \text{otherwise.} \end{cases} \end{align*}$$

Proof. First, suppose $|A \cap B| \leq 9$ , so $|A \cap B^c|\geq 2$ . Then there is a commutative diagram

(2.4)

where the horizontal maps glue p to q and the vertical maps forget markings not in A. In this case, the image of $f_A\circ \iota _B$ is a proper boundary divisor in $\overline {\mathcal {M}}_{1,A}$ , which has no holomorphic $11$ -forms. Hence, the pullback of the generator of $H^{11,0}(\overline {\mathcal {M}}_{1,A})$ to $D_B$ vanishes.

Now, suppose $|A \cap B| \geq 10$ , so $|A \cap B^c| \leq 1$ . Then, the lower left hand term in diagram (2.4) must be replaced by $\overline {\mathcal {M}}_{(A \cap B) \cup p}$ . Thus, there is another commutative diagram

(2.5)

where if $|A \cap B| = 10$ , we identify p with the unique symbol of A not contained in B.

2.3 Relations and the $\mathbb {S}_n$ -action

The group $\mathbb {S}_n$ acts on the subsets $A \subset \{1, \ldots , n\}$ and correspondingly on the subspace of $H^{11,0}(\overline {\mathcal {M}}_{1,n})$ generated by the $\omega _A$ . Note that, for any permutation $\sigma $ in the subgroup of $\mathbb {S}_n$ fixing A, we have $\omega _{\sigma (A)} = \operatorname {\mathrm {sign}}(\sigma ) \omega _{A}$ .

To identify our representation, we briefly recall some of the combinatorial objects that arise in the representation theory of $\mathbb {S}_n$ .

A tabloid is an equivalence class of tableaux, which identifies tableaux up to reordering rows. Given a tableau T, we write $\{T\}$ for the corresponding tabloid. Given a partition $\lambda $ of n, we denote by $M_{\lambda }$ the vector space with basis given by tabloids of shape $\lambda $ . The Specht module generator associated to a tableau T is the vector

(2.6) $$ \begin{align} \sum_{\sigma \in \mathrm{C}_T} \operatorname{\mathrm{sign}}(\sigma) \{\sigma(T)\} \in M_{\lambda}, \end{align} $$

where $\mathrm {C}_T \subset \mathbb {S}_n$ is the subgroup that preserves the columns of T setwise. The subspace of $M_{\lambda }$ generated by the vectors (2.6) as T runs over all tableaux is an irreducible representation called the Specht module $V_\lambda $ . The Specht module generators associated to the standard tableaux on $\lambda $ form a basis for $V_{\lambda }$ .

To each ordered subset $A \subset \{1, \ldots , n\}$ of size $11$ , we associate a tableau $T_A$ of shape $(n-10,1^{10})$ which has the symbols of A in order down the first column and the rest of the first row filled in increasing order.

Proof. The dimension of the Specht module $V_{n-10,1^{10}}$ is the number of standard tableaux of shape $(n-10,1^{10})$ , which is ${n-1 \choose 10}$ . Therefore, by Lemma 2.1, it will suffice to give a $\mathbb {S}_n$ -equivariant map from the subspace of $H^{11,0}(\overline {\mathcal {M}}_{1,n})$ generated by $\{\omega _A\}$ to $V_{n-10,1^{10}}$ that takes $\omega _A$ to the Specht module generator associated to A.

We are going to study the image of subspace generated by $\{\omega _A\}$ under the pullback map

(2.7) $$ \begin{align} H^{11,0}(\overline{\mathcal{M}}_{1,n}) \to \bigoplus_{\substack{|B| = 10}} H^{11,0}(D_{B}), \end{align} $$

where B is an ordered subset of $\{1, \ldots , n\}$ and $D_B$ is as in equation (2.2). Let $W_B$ be the element in the target of the map (2.7) which has component $\mathrm {pr}_1^*\omega \in H^{11,0}(D_B)$ and $0$ in all other components. The collection of ordered subsets B of size $10$ is in bijection with tabloids on $(n-10,1^{10})$ , where $B^c$ fills the row and B (in order) fills the column. Thus, the right-hand side is identified with the vector space of tabloids $M_{n-10,1^{10}}$ ; given a tabloid $\{T\}$ corresponding to B we write $W_{\{T\}} = W_B$ .

Fix $A \subset \{1, \ldots , n\}$ with $|A| = 11$ . If $B \subset A$ , with $|B| = 10$ , then A is a permutation of $i \cup B$ for some $i \notin B$ . Let $\sigma _{A \to B}\colon i \cup B \to A$ denote the corresponding permutation. By Lemma 2.2, the image of $\omega _A$ under the map (2.7) is

$$\begin{align*}\omega_A \mapsto \sum_{B \subset A} \operatorname{\mathrm{sign}}(\sigma_{B \to A}) W_B = \sum_{\sigma \in S_A} \operatorname{\mathrm{sign}}(\sigma) W_{\{\sigma(T_A)\}},\end{align*}$$

which is the Specht module generator defined in the expression (2.6), and the proposition follows.

Remark 2.6. Dan Petersen suggested an alternate method to obtain several of the results in this section, which avoids the manipulations with generating functions in Lemma 2.1. We sketch his argument here. Let $f^{1}:\mathcal {E}\rightarrow \mathcal {M}_{1,1}$ denote the universal elliptic curve and $\mathcal {E}^{n}$ denote the n-fold fiber product of $\mathcal {E}$ with itself over $\mathcal {M}_{1,1}$ . Note that $\mathcal {M}_{1,n}$ is an open substack of $\mathcal {E}^{n-1}$ . By the long exact sequences for the pairs $(\overline {\mathcal {M}}_{1,n},\mathcal {M}_{1,n})$ and $(\mathcal {E}^{n-1},\mathcal {M}_{1,n})$ , we see that there are natural isomorphisms

$$\begin{align*}W_{11}H^{11}(\mathcal{E}^{n-1})\cong W_{11}H^{11}(\mathcal{M}_{1,n})\cong H^{11}(\overline{\mathcal{M}}_{1,n}). \end{align*}$$

One can then study the Leray spectral sequence for the smooth morphism $f^{n-1}:\mathcal {E}^{n-1}\rightarrow \mathcal {M}_{1,1}$ . Let $\mathbb {V}$ denote the local system $R^1f^1_*\mathbb {Q}$ . Then by the Künneth formula,

$$\begin{align*}Rf^{n-1}_*\mathbb{Q}\cong (Rf^{1}_*\mathbb{Q})^{\otimes n-1}\cong (\mathbb{Q} \oplus \mathbb{V}[-1]\oplus \mathbb{Q}(-1)[-2])^{\otimes n-1}. \end{align*}$$

The pure cohomology $W_{11}H^{11}(\mathcal {E}^{n-1})$ arises from the $\binom {n-1}{10}$ summands $\mathbb {V}[-1]^{\otimes 10}$ , each of which gives a copy of $\mathsf {S}_{12}$ . This gives Lemma 2.1. To identify the $\mathbb {S}_n$ representation as in Proposition 2.3, one first notes that as an $\mathbb {S}_{n-1}$ representation, we have

$$\begin{align*}W_{11}H^{11}(\mathcal{E}_{n-1})\cong \mathsf{S}_{12}\otimes \mathrm{Ind}_{\mathbb{S}_{10}\times \mathbb{S}_{n-11}}^{\mathbb{S}_{n-1}}(\mathrm{sgn}\boxtimes \mathrm{triv}). \end{align*}$$

By the Pieri formula and the branching rule for the symmetric group, it follows that as an $\mathbb {S}_n$ representation,

$$\begin{align*}W_{11}H^{11}(\mathcal{E}_{n-1})\cong \mathsf{S}_{12} \otimes V_{n-10,1^{10}}. \end{align*}$$

3.1 Restricting to boundary divisors

Consider the excision long exact sequence associated to the boundary $\partial \mathcal {M}_{g,n}=\overline {\mathcal {M}}_{g,n}\smallsetminus \mathcal {M}_{g,n}$ :

$$ \begin{align*} \cdots\rightarrow H^k_c(\mathcal{M}_{g,n})\rightarrow H^k(\overline{\mathcal{M}}_{g,n})\rightarrow H^k(\partial \mathcal{M}_{g,n})\rightarrow H^{k+1}_c(\mathcal{M}_{g,n})\rightarrow \cdots. \end{align*} $$

Note that this sequence is in fact a long exact sequence of mixed Hodge structures or $\ell $ -adic Galois representations. In particular, when $H^{k}_c(\mathcal{M}_{g,n}) = 0$ , there is an injective morphism

(3.1) $$\begin{align}H^k(\overline{\mathcal{M}}_{g,n})\hookrightarrow H^k(\partial \mathcal{M}_{g,n}). \end{align}$$

Let $\widetilde {\partial \mathcal {M}}_{g,n}$ denote the normalization of $\partial \mathcal {M}_{g,n}$ . Arbarello and Cornalba improve on the injectivity of the map (3.1), as follows.

Lemma 3.1 (Lemma 2.6 of [Reference Arbarello and Cornalba1]).

Suppose $H^k_c(\mathcal {M}_{g,n})=0$ . Then the pullback

$$\begin{align*}H^k(\overline{\mathcal{M}}_{g,n})\rightarrow H^k(\widetilde{\partial \mathcal{M}}_{g,n}) \end{align*}$$

is injective.

For fixed k, the following proposition gives vanishing of compactly supported cohomology in all but an explicit finite collection of cases.

Proposition 3.2 (Proposition 2.1 of [Reference Bergström, Faber and Payne4]).

Assume $g\geq 1$ .

$$ \begin{align*} H^k_c(\mathcal{M}_{g,n})=0 \text{ for } \begin{cases}\kern-10pt & k<2g \text{ and } n=0,1 \\ \kern-10pt & k<2g-2+n \text{ and } n\geq 2. \end{cases} \end{align*} $$

3.2 The case of even degrees $k \leq 12$

Let $RH^*(\overline {\mathcal {M}}_{g,n})\subset H^*(\overline {\mathcal {M}}_{g,n})$ be the tautological cohomology ring. Tautological classes are algebraic and defined over $\mathbb {Z}$ , so if $H^k(\overline {\mathcal {M}}_{g,n}) = RH^k(\overline {\mathcal {M}}_{g,n})$ , then $H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate. The next lemma provides the necessary base cases for the inductive argument.

Proof of Theorem 1.2.

We induct on g and n. By Lemma 3.3, we can assume $2g-2+n>k$ . By Lemma 3.2, we have an injection

(3.2) $$ \begin{align} H^{k}(\overline{\mathcal{M}}_{g,n})\hookrightarrow H^{k}(\widetilde{\partial \mathcal{M}}_{g,n}). \end{align} $$

Each component of $\widetilde {\partial \mathcal {M}}_{g,n}$ is a quotient by a finite group of $\overline {\mathcal {M}}_{g-1,n+2}$ or $\overline {\mathcal {M}}_{g_1,n_1+1}\times \overline {\mathcal {M}}_{g_2,n_2+1}$ , where $g_1+g_2=g$ , $n_1 + n_2 = n$ , and $2g_i-2+n_i>0$ . By induction on g, we know $H^k(\overline {\mathcal {M}}_{g-1,n+2})$ is pure Hodge–Tate. Meanwhile, note that $H^j(\overline {\mathcal {M}}_{g',n'})=0$ for all $(g',n')$ and odd $j \leq k/2 \leq 6$ by [Reference Arbarello and Cornalba1]. Hence, the Künneth formula shows that

(3.3) $$ \begin{align} H^{k}(\overline{\mathcal{M}}_{g_1,n_1+1}\times\overline{\mathcal{M}}_{g_2,n_2+1})=\bigoplus_{i=0}^{k/2} H^{2i}(\overline{\mathcal{M}}_{g_1,n_1+1})\otimes H^{k-2i}(\overline{\mathcal{M}}_{g_2,n_2+1}). \end{align} $$

Inductively, we know that the right-hand side of equation (3.3) is pure Hodge–Tate. Thus, the map (3.2) is an injection of $H^k(\overline {\mathcal {M}}_{g,n})$ into a Hodge structure or Galois representation that is pure Hodge–Tate, and it follows that $H^k(\overline {\mathcal {M}}_{g,n})$ is pure Hodge–Tate as well.

3.3 The case of degree $k=11$

The base cases required to run an analogous induction for $k = 11$ are those where $H^{11}(\overline {\mathcal {M}}_{g,n})$ does not inject into $H^{11}(\widetilde {\partial \mathcal {M}}_{1,11})$ . Recent results of the first two authors rule out any such bases cases with $g \geq 2$ .

Proposition 3.5. For any g, there is an injection

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{g,n})\hookrightarrow \bigoplus H^{11}(\overline{\mathcal{M}}_{1,11}).\end{align*}$$

Proof. The result is known for $g \leq 1$ , so we may assume $g \geq 2$ . By Lemma 3.4, we have an injection

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{g,n})\hookrightarrow H^{11}(\widetilde{\partial \mathcal{M}}_{g,n}). \end{align*}$$

Each component of $\widetilde {\partial \mathcal {M}}_{g,n}$ is a quotient by a finite group of $\overline {\mathcal {M}}_{g-1,n+2}$ or $\overline {\mathcal {M}}_{g_1,n_1+1}\times \overline {\mathcal {M}}_{g_2,n_2+1}$ , where $k_1+k_2=n$ , $2g_i-2+n_i>0$ . Because $H^k(\overline {\mathcal {M}}_{g',n'})=0$ for all $(g',n')$ and $k=1,3,5,7,9$ [Reference Arbarello and Cornalba1, Reference Bergström, Faber and Payne4], the Künneth formula shows that

(3.4) $$ \begin{align} H^{11}(\overline{\mathcal{M}}_{g_1,n_1+1}\times\overline{\mathcal{M}}_{g_2,n_2+1})=H^{11}(\overline{\mathcal{M}}_{g_1,n_1+1})\oplus H^{11}(\overline{\mathcal{M}}_{g_2,n_2+1}). \end{align} $$

Note that either $g_1<g$ or $g_1=g$ and $n_1+1<n$ , and analogously for $(g_2,n_2)$ . Therefore, there is an injective morphism of Hodge structures from $H^{11}(\overline {\mathcal {M}}_{g,n})$ into a direct sum of Hodge structures of the form $H^{11}(\overline {\mathcal {M}}_{\gamma ,\nu })$ , where $\gamma <g$ or $\gamma =g$ and $\nu <n$ . By double induction on g and n and using that $H^{11}(\overline {\mathcal {M}}_{0,n}) = 0$ and $H^{11}(\overline {\mathcal {M}}_{1,n}) = 0$ for $n \leq 10$ , we conclude that there is an injective morphism of Hodge structures or $\ell $ -adic Galois representations

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{g,n})\hookrightarrow \bigoplus H^{11}(\overline{\mathcal{M}}_{1,11}).\\[-42pt] \end{align*}$$

4.1 The first two maps in the weight k complex

Let $\Gamma $ denote a stable n-marked graph of genus g; the underlying graph is connected, each vertex v is labeled by an integer $g_v$ and the valence of the vertex v is denoted $n_v$ . The stability condition is that $2g_v+n_v-2> 0$ . Set $\overline {\mathcal {M}}_{\Gamma }=\prod _v \overline {\mathcal {M}}_{g_v,n_v}$ . There is the natural gluing map

$$\begin{align*}\xi_{\Gamma}\colon\overline{\mathcal{M}}_{\Gamma}\rightarrow \overline{\mathcal{M}}_{g,n}. \end{align*}$$

The normalization of $\xi _{\Gamma }(\overline {\mathcal {M}}_{\Gamma })$ is isomorphic to $\overline {\mathcal {M}}_{\Gamma }/\operatorname {\mathrm {Aut}}(\Gamma )$ . Thus, the first map in the weight k complex, the pullback of $H^k$ to the normalization of the boundary, can be rewritten as:

(4.1) $$ \begin{align} H^{k}(\overline{\mathcal{M}}_{g,n})\xrightarrow{ \ \alpha \ } \bigoplus_{|E(\Gamma)|=1} H^{k}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{\mathrm{Aut}} \Gamma}. \end{align} $$

The target of the jth map in the weight k complex is $\bigoplus _{|E(\Gamma )|=j} (H^{k}(\overline {\mathcal {M}}_{\Gamma }) \otimes \det E(\Gamma ))^{\operatorname {\mathrm {Aut}} \Gamma }.$ Here, $\det E(\Gamma )$ denotes the determinant of the permutation representation of $\operatorname {\mathrm {Aut}}(\Gamma )$ acting on the set of edges. We will only need the first and second maps.

Let us describe the second map. We define

(4.2) $$ \begin{align} \bigoplus_{|E(\Gamma) = 1|} H^{k}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{\mathrm{Aut}}(\Gamma)} \xrightarrow{ \ \beta \ } \bigoplus_{|E(\Gamma) = 2|} (H^{k}(\overline{\mathcal{M}}_{\Gamma}) \otimes \det E(\Gamma))^{\operatorname{\mathrm{Aut}}(\Gamma)} \end{align} $$

as follows. For each graph $\Gamma $ with two edges, choose an ordering of the edges $E(\Gamma )= \{e_1, e_2\}$ and say $e_i$ corresponds to the node obtained by gluing the marked points $p_i$ and $q_i$ . Let $\phi _i(\Gamma )$ denote the graph with one edge obtained by contracting $e_i$ . Gluing $p_i$ and $q_i$ induces a map $\xi _i\colon \overline {\mathcal {M}}_{\Gamma } \to \overline {\mathcal {M}}_{\phi _i(\Gamma )}$ , which in turn gives a map $\xi _i^*\colon H^{k}(\overline {\mathcal {M}}_{\phi _i(\Gamma )}) \to H^{k}(\overline {\mathcal {M}}_{\Gamma })$ . This induces

(4.3) $$ \begin{align} H^{11}(\overline{\mathcal{M}}_{\phi_i(\Gamma)}) \xrightarrow{\xi_i^*} H^{11}(\overline{\mathcal{M}}_{\Gamma}). \end{align} $$

If $E(\Gamma )$ is a nontrivial representation of $\operatorname {\mathrm {Aut}}(\Gamma )$ , then $\xi _2 = \xi _1 \circ \sigma $ , where $\sigma \colon \overline {\mathcal {M}}_{\Gamma } \to \overline {\mathcal {M}}_{\Gamma }$ is the automorphism that simultaneously swaps $p_1$ with $p_2$ and $q_1$ with $q_2$ (corresponding to the automorphism of $\Gamma $ that swaps the two edges). In particular, it follows that the image of $\xi _1^* - \xi _2^*$ lies in the subspace $(H^{k}(\overline {\mathcal {M}}_{\Gamma }) \otimes \det E(\Gamma ))^{\operatorname {\mathrm {Aut}}(\Gamma )} \subset H^{k}(\overline {\mathcal {M}}_{\Gamma })$ . Then $\beta $ is defined by taking the sum over all two-edge graphs. Then $\beta \circ \alpha = 0$ because each component is the difference of the pullbacks under two copies of the same gluing maps.

4.2 Proof of Theorem 1.1, assuming it holds for $g = 2$

We first give a short inductive proof of Theorem 1.1, assuming that $H^{11}(\overline {\mathcal {M}}_{2,n}) = 0$ for all n. Fix $k = 11$ and $g \geq 3$ . By Lemma 3.4, the map $\alpha $ in (4.1) is injective. We claim that the map $\beta $ in (4.2) is also injective. The theorem follows from this claim since $\beta \circ \alpha = 0$ .

Consider the domain of $\beta $ , which is $\bigoplus _{|E(\Gamma ) = 1|} H^{k}(\overline {\mathcal {M}}_{\Gamma })^{\operatorname {\mathrm {Aut}}(\Gamma )}$ . If $\Gamma $ has a single vertex with a loop edge, then $H^{11}(\overline {\mathcal {M}}_\Gamma )^{\operatorname {\mathrm {Aut}}(\Gamma )} = H^{11}(\overline {\mathcal {M}}_{g-1,n+2})^{\mathbb {S}_2}$ , which vanishes by induction on g.

Suppose $\Gamma $ has two vertices joined by an edge. Then $\overline {\mathcal {M}}_\Gamma = \overline {\mathcal {M}}_{a, A \cup p} \times \overline {\mathcal {M}}_{b, A^c \cup q}$ . By the Künneth formula and the vanishing of lower degree odd cohomology,

(4.4) $$ \begin{align} H^{11}(\overline{\mathcal{M}}_{a, A \cup p} \times \overline{\mathcal{M}}_{b,A^c \cup q})=H^{11}(\overline{\mathcal{M}}_{a, A \cup p})\oplus H^{11}(\overline{\mathcal{M}}_{b,A^c \cup q}). \end{align} $$

Assume $a \geq b$ . By induction on g and n, $H^{11}(\overline {\mathcal {M}}_\Gamma )$ vanishes unless $b = 1$ and $|A^c| \geq 10$ , in which case it is $H^{11}(\overline {\mathcal {M}}_{1,A^c \cup q})$ . Note that, in this case, $\operatorname {\mathrm {Aut}}(\Gamma )$ is trivial.

Figure 1

The graph $\Gamma $ on the left and $\Gamma '$ on the right.

Let $\Gamma '$ be the graph obtained by attaching a loop to the vertex of genus $g-1$ (and decreasing its genus accordingly), as shown in Figure 1. As in equation (4.4), we know that $H^{11}(\overline {\mathcal {M}}_{\Gamma '})$ contains $H^{11}(\overline {\mathcal {M}}_{1,A^c \cup q})$ as a summand. Then $\beta $ maps $H^{11}(\overline {\mathcal {M}}_{\Gamma })$ injectively into this summand of $H^{11}(\overline {\mathcal {M}}_{\Gamma '})$ , and it follows that $\beta $ is injective, as claimed.

4.3 Proof of Theorem 1.1 for $g = 2$

The proof that $H^{11}(\overline {\mathcal {M}}_{g,n}) = 0$ for all n follows a similar strategy to the inductive argument for $g\geq 3$ . By Lemma 3.4, we know that $\alpha $ is injective, and we claim that $\beta $ is also injective. The theorem follows from this claim. However, the proof that $\beta $ is injective is more involved in this case.

We begin by describing some of the components of $\beta $ as concretely as possible.

We will show that, for each $1$ -edge graph $\Gamma $ there is a collection of $2$ -edge graphs $\{\Gamma _i\}$ such that $\beta $ maps $H^{11}(\overline {\mathcal {M}}_{\Gamma })^{\operatorname {\mathrm {Aut}}(\Gamma )}$ injectively into $\bigoplus _i (H^{11}(\overline {\mathcal {M}}_{\Gamma _i}) \otimes \det (E_{\Gamma _i}))^{\operatorname {\mathrm {Aut}}(\Gamma _i)}$ . Moreover, we order the one-edge graphs in such a way that, at each step, none of the two-edge graphs $\{\Gamma _i\}$ admit edge contractions to any of the one-edge graphs that came earlier in the order. In this way, we see that $\beta $ can be represented by a block upper diagonal matrix with injective blocks, and hence is injective, as required.

For $n \leq 9$ , the cohomology $H^*(\overline {\mathcal {M}}_{2,n})$ is tautological [Reference Canning and Larson5], and hence $H^{11}(\overline {\mathcal {M}}_{2,n}) = 0$ , as required. The cases $n = 10$ and $11$ can be handled by an argument similar to that used for the cases $n \geq 12$ , below, but the details are more involved. Instead, we note that $\overline {\mathcal {M}}_{2,n}$ is rational in these cases [Reference Casnati6], and hence, $H^{11,0}(\overline {\mathcal {M}}_{2,n}) = 0$ . By Proposition 3.5, it follows that $H^{11}(\overline {\mathcal {M}}_{2,n}) = 0$ . For the remainder of this subsection, we therefore assume $n \geq 12$ .

There are three types of $1$ -edge graphs to consider: those with a vertex of genus $2$ , those with a unique vertex of genus $1$ and a self-edge and those with two vertices of genus $1$ .

4.3.2 Graphs with a single genus 1 vertex

Let $\Gamma $ be the graph with one vertex of genus $1$ , n legs and one self-edge. Consider $\overline {\mathcal {M}}_{\Gamma } = \overline {\mathcal {M}}_{1,n+2}$ labeled $\{1, \ldots , n, p_1, q_1\}$ , where $p_1, q_1$ are glued together to get curves with dual graph $\Gamma $ . As in Section 2.2, given a subset $B \subset \{1, \ldots , n, p_1, q_1\}$ of cardinality $10$ , we let $D_B = \overline {\mathcal {M}}_{1,B \cup p_2} \times \overline {\mathcal {M}}_{0, B^c \cup q_2}$ . (This is nonempty because $n \geq 12$ .) There are four flavors of subsets B: both of $p_1, q_1$ are in B; neither $p_1$ nor $q_1$ is in B; only $p_1$ is in B; or only $q_1$ is in B. These correspond to $4$ types of codimension $1$ strata under the map $D_B \to \overline {\mathcal {M}}_{1,n+2}$ that glues $p_2$ and $q_2$ (see Figure 2).

Figure 2

The four flavors of $D_B$ .

For each subset B, let $\Gamma _B$ be the two-edge graph obtained by further gluing $p_1$ and $q_1$ (see Figure 3). Note that $\overline {\mathcal {M}}_{\Gamma _B} = D_B$ . We want to show that

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{\mathrm{Aut}}(\Gamma)}\rightarrow \bigoplus_{|B|=10} H^{11}(\overline{\mathcal{M}}_{\Gamma_B}\otimes \det E(\Gamma_B))^{\operatorname{\mathrm{Aut}}(\Gamma_B)} \end{align*}$$

is injective.

When we glue $p_1$ and $q_1$ , the last two types of subsets B give the same type of two-edge graph $\Gamma _B$ :

Figure 3

Gluing $p_1$ and $q_1$ .

Swapping $p_1$ and $q_1$ preserves the first two maps $D_B \to \overline {\mathcal {M}}_{1,n+2}$ but exchanges the other two. Correspondingly, the edge representation is trivial in the first two cases and nontrivial in cases three and four.

By Corollary 2.5, we have an injection

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{1,n+2}) \hookrightarrow \bigoplus_{\substack{|B| = 10}} H^{11}(D_B).\end{align*}$$

Now, $\mathbb {S}_2$ acts on both sides by swapping $p_1$ and $q_1$ . Taking $\mathbb {S}_2$ -invariants of both sides, we have an injection from $H^{11}(\overline {\mathcal {M}}_{1,n+2})^{\mathbb {S}_2}$ into

$$\begin{align*}\bigoplus_{\substack{\{p_1, q_1\} \subset B \text{ or } \{p_1,q_1\} \subset B^c \\ |B| = 10} } H^{11}(D_B)^{\mathbb{S}_2} \oplus \bigoplus_{\substack{p_1 \in B \text{ and } q_1 \notin B \\ |B| = 10}}[H^{11}(D_B) \oplus H^{11}(D_{(B\smallsetminus p_1) \cup q_1})]^{\mathbb{S}_2}.\end{align*}$$

The first collection of terms is $(H^{11}(\overline {\mathcal {M}}_{\Gamma _B}) \otimes \det E(\Gamma _B))^{\mathbb {S}_2}$ for the graphs $\Gamma _B$ of the first and second flavor in Figure 3, which have trivial edge representation. For the terms in square brackets, we have $H^{11}(D_B) \cong H^{11}(D_{(B\smallsetminus p_1)\cup q_1})$ and the $\mathbb {S}_2$ -action switches the two factors. Hence, the space of $\mathbb {S}_2$ -invariants is $H^{11}(\overline {\mathcal {M}}_{\Gamma _B})$ . When $p_1 \in B$ and $q_1 \notin B$ , note that $\Gamma _B$ is the type of graph considered in Example 4.2. In particular $\operatorname {\mathrm {Aut}}(\Gamma _B)$ acts by the sign representation, so $(H^{11}(\overline {\mathcal {M}}_{\Gamma _B}) \otimes \det E(\Gamma _B))^{\operatorname {\mathrm {Aut}}(\Gamma _B)} = H^{11}(\overline {\mathcal {M}}_{\Gamma _B})$ . In summary, we have given an injection

(4.5) $$ \begin{align} H^{11}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{\mathrm{Aut}}(\Gamma)} = H^{11}(\overline{\mathcal{M}}_{1,n+2})^{\mathbb{S}_2} \hookrightarrow \bigoplus_{|B| = 10} (H^{11}(\overline{\mathcal{M}}_{\Gamma_B}) \otimes \det E(\Gamma_B))^{\operatorname{\mathrm{Aut}}(\Gamma_B)}. \end{align} $$

4.3.3 Graphs with two genus $1$ vertices

Suppose $\Gamma $ has two genus $1$ vertices, so

$$\begin{align*}\overline{\mathcal{M}}_{\Gamma} = \overline{\mathcal{M}}_{1, A \cup p} \times \overline{\mathcal{M}}_{1, A^c \cup q}.\end{align*}$$

We may assume $|A^c| \leq |A|$ . Figure 4 shows three types of graphs that will be used in the three cases of the argument. Note that none of them have a loop, so none admit edge contractions to the one-edge graphs of Section 4.3.2.

Figure 4

The graphs in Cases 1, 2 and 3.

Case 1: $|A^c| = 0$ . By Corollary 2.5, there is an injection

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{\Gamma}) = H^{11}(\overline{\mathcal{M}}_{1,A \cup p}) \hookrightarrow \bigoplus_{\substack{B \subset A \\ |B| = 10}} H^{11}(\overline{\mathcal{M}}_{1, B \cup p'}) = H^{11}(\overline{\mathcal{M}}_{\Psi_B}),\end{align*}$$

where $\Psi _B$ is the first graph in Figure 4, which has

$$\begin{align*}\overline{\mathcal{M}}_{\Psi_B} = \overline{\mathcal{M}}_{1, B \cup p'} \times \overline{\mathcal{M}}_{0, (A \smallsetminus B) \cup \{q',p\}} \times \overline{\mathcal{M}}_{1, \{q\}}.\end{align*}$$

Case 2: $|A^c| = 1$ . Again, by Corollary 2.5, there is an injection

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{\Gamma}) = H^{11}(\overline{\mathcal{M}}_{1,A \cup p}) \hookrightarrow \bigoplus_{\substack{B \subset A \\ |B| = 10}} H^{11}(\overline{\mathcal{M}}_{1, B \cup p'}) = H^{11}(\overline{\mathcal{M}}_{\Psi_B'}),\end{align*}$$

where this time $\Psi _B'$ is the second graph in Figure 4, which has

$$\begin{align*}\overline{\mathcal{M}}_{\Psi_B'} = \overline{\mathcal{M}}_{1, B \cup p'} \times \overline{\mathcal{M}}_{0, (A \smallsetminus B) \cup \{q',p\}} \times \overline{\mathcal{M}}_{1, A^c \cup q}.\end{align*}$$

Note that our assumptions $n \geq 12$ and $|A^c| = 1$ ensure $|A \smallsetminus B| \geq 1$ , so the middle genus $0$ vertex has at least $3$ markings. Note that no edge contraction of $\Psi _B'$ gives a one-edge graph $\Gamma $ of the type in Case 1 or a different $\Gamma $ of the type in Case 2.

Case 3: $|A^c| \geq 2$ . Choose any $i, j \in A^c$ and $k, \ell \in A$ and define

$$\begin{align*}X_{A^c \cup q} = \overline{\mathcal{M}}_{1,(A^c \smallsetminus \{i,j\}) \cup \{q, p'\}} \times \overline{\mathcal{M}}_{0, \{q',i,j\}}\end{align*}$$

and

$$\begin{align*}X_{A \cup p} = \overline{\mathcal{M}}_{1,(A \smallsetminus \{k,\ell\}) \cup \{p, p'\}} \times \overline{\mathcal{M}}_{0, \{q',k,\ell\}}.\end{align*}$$

Associated to these are two-edge graphs $\Phi _A$ and $\Phi _{A^c}$ so that

$$\begin{align*}\overline{\mathcal{M}}_{\Phi_A} = \overline{\mathcal{M}}_{1, A \cup p} \times X_{A^c \cup q} \qquad \text{and} \qquad \overline{\mathcal{M}}_{\Phi_{A^c}} = X_{A \cup p} \times \overline{\mathcal{M}}_{1, A^c \cup q}\end{align*}$$

and contracting the edge connecting $p'$ and $q'$ gives $\Gamma $ . See the last graph in Figure 4 for a picture of $\Phi _A$ . Notice that no edge contraction of $\Phi _A$ or $\Phi _{A^c}$ gives a one-edge graph appearing in Cases 1 or 2 above or a different one-edge graph of the type in Case 3.

By the Künneth formula,

$$ \begin{align*} H^{11}(\overline{\mathcal{M}}_{\Gamma}) &=H^{11}(\overline{\mathcal{M}}_{1,A\cup p})\oplus H^{11}(\overline{\mathcal{M}}_{1,A^c\cup q}), \\ H^{11}(\overline{\mathcal{M}}_{\Phi_A})&=H^{11}(\overline{\mathcal{M}}_{1,A\cup p})\oplus H^{11}(\overline{\mathcal{M}}_{1,(A^c\smallsetminus \{i,j\})\cup \{q,p'\}}), \end{align*} $$

and

$$ \begin{align*} H^{11}(\overline{\mathcal{M}}_{\Phi_{A^c}})&=H^{11}(\overline{\mathcal{M}}_{1,A^c\cup q})\oplus H^{11}(\overline{\mathcal{M}}_{1,(A\smallsetminus \{k,\ell\})\cup \{p,p'\}}). \end{align*} $$

It follows that $H^{11}(\overline {\mathcal {M}}_{\Gamma })$ injects into the sum

$$ \begin{align*} H^{11}(\overline{\mathcal{M}}_{\Phi_A})\oplus H^{11}(\overline{\mathcal{M}}_{\Phi_{A^c}}). \end{align*} $$

Remark 4.3. It should also be possible to deduce the vanishing of $H^{11}(\overline {\mathcal {M}}_{2,n})$ from the work of Dan Petersen as follows. There are exact sequences

$$\begin{align*}H^{k-2}(\overline{\mathcal{M}}_{1,n+2})(-1)\rightarrow H^k(\overline{\mathcal{M}}_{2,n})\rightarrow W_k H^k(\mathcal{M}^{\mathrm{ct}}_{2,n})\rightarrow 0. \end{align*}$$

Specializing to $k=11$ , we see that

$$\begin{align*}H^{11}(\overline{\mathcal{M}}_{2,n})\cong W_{11} H^{11}(\mathcal{M}^{\mathrm{ct}}_{2,n}). \end{align*}$$

By [Reference Petersen23, Theorem 2.1 and Remark 2.2], there is an isomorphism

$$\begin{align*}H^{11}(\mathcal{M}^{\mathrm{ct}}_{2,n})\cong \bigoplus_{p+q=11} H^{p}(\mathcal{M}_2^{\mathrm{ct}},A^q)\oplus H^{p}(\operatorname{\mathrm{Sym}} ^2 \mathcal{M}_{1,1}, B^q), \end{align*}$$

where $A^q$ and $B^q$ are certain direct sums of Tate twists of symplectic local systems. The cohomology of these local systems has been determined by Petersen in [Reference Petersen22]. Petersen’s work makes significant use of high-powered machinery, including mixed Hodge modules, perverse sheaves, the decomposition theorem for the map $\mathcal {M}_{2,n}^{\mathrm {ct}}\rightarrow \mathcal {M}_{2}^{\mathrm {ct}}$ and the Eichler–Shimura isomorphism concerning modular forms. The proof we present here is relatively elementary and highlights a combinatorial perspective on the vanishing of $H^{11}(\overline {\mathcal {M}}_{2,n})$ .