can be identified with the following three summands:

  * •

$(g,\lambda)=(1,0)$; the summand $\mathbf{Q}\otimes
H_{0}(H_{\infty}^{0,1},\mathbf{Q})$.

  * •

$(g,\lambda)=(-1,0)$; the summand $\mathbf{Q}\otimes
H_{0}(H_{\infty}^{0,1},\mathbf{Q})$.

  * •

$(g,\lambda)=(0,[2])$; the summand $\mathrm{Sym}^{2}(A)\otimes
H_{0}(H_{\infty}^{0,1},\wedge^{2}(V))[-2]$.

######  9.2.5.

The arithmetic 9.2.2 is a direct consequence of the purely topological 9.2.4.
Indeed, after 9.2.4 it suffices to compute the Galois action on the stable
homology in degree $0$. But this is obvious: the action is trivial with
trivial coefficients, and pure Tate of weight $2$ on the homology with
coefficients in $\wedge^{2}\mathbb{V}$, where the nonzero homology arises
since the symplectic form777The word “symplectic form” is strictly speaking
incorrect here: $\mathbb{V}$ is a symplectic vector space only when
$(n,m)=(1,0)$, and we are considering the case $(n,m)=(0,1)$, which puts us in
the situation of Proctor’s odd symplectic groups [Pro88]. defines a direct
summand $\mathbf{Q}_{\ell}(-1)\subset\wedge^{2}\mathbb{V}$. Then the homology
in any other degree and for any other $\lambda$ is determined by the
compatibility with Galois action, and that the Gerstenhaber operad has the
Galois action which in homological degree $k$ is pure Tate of weight $-2k$.

######  9.2.6.

The proof of 9.2.4 uses Koszul duality theory for Gerstenhaber algebras.
Recall the following theorem of Berglund [Ber14]:

######  9.2.7 Theorem (Berglund).

Let $X$ be an $n$-connected Koszul space of finite type. The commutative
algebra $H^{\bullet}(X,\mathbf{Q})$, considered as a Gerstenhaber $n$-algebra
with trivial bracket, is Koszul as a Gerstenhaber $n$-algebra. Its Koszul dual
Gerstenhaber $n$-algebra is $H_{\bullet}(\Omega^{n}X,\mathbf{Q})$.

######  9.2.8 Remark.

In our earlier discussions on Koszul duality we formulated it as a duality
between algebras and coalgebras, but here we have combined with the linear
dual to obtain a duality between algebras and algebras. It is only for this
reason we need $X$ to be of finite type. We work with this formulation to be
consistent with Berglund’s paper, and with Sullivan’s work on rational
homotopy theory.

######  9.2.9.

In formulating this theorem we use that the Gerstenhaber operad is self-dual
up to a degree shift; after making this shift by $n$, any Gerstenhaber
$n$-algebra has a Koszul dual Gerstenhaber $n$-algebra. We will need a modest
generalization of Berglund’s result to the case that $X$ is merely
$(n-1)$-connected.

######  9.2.10 Definition.

Let $A_{\bullet}$ be a nonnegatively graded algebra with $A_{0}$ a polynomial
ring. A _group-completion_ of $A$ is a localization of the form $S^{-1}A$,
where $S$ is a minimal generating set of $A_{0}$.

######  9.2.11 Theorem.

Let $X$ be an $(n-1)$-connected Koszul space of finite type. The commutative
algebra $H^{\bullet}(X,\mathbf{Q})$, considered as a Gerstenhaber $n$-algebra
with trivial bracket, is Koszul. The homology
$H_{\bullet}(\Omega^{n}X,\mathbf{Q})$ is a group-completion of the Koszul dual
Gerstenhaber $n$-algebra.

###### Proof.

This follows from the arguments of Berglund [Ber14, Section 4]. Berglund shows
in general for a simply connected Koszul space $X$ that the Koszul dual
Gerstenhaber $n$-algebra of $H^{\bullet}(X,\mathbf{Q})$ is the free algebra on
the rational homotopy groups of $X$, shifted down $n$ degrees. When $X$ is
$n$-connected, this free algebra is precisely
$H_{\bullet}(\Omega^{n}X,\mathbf{Q})$. When $X$ is merely $(n-1)$-connected,
this fails due to nontriviality of $\pi_{0}(\Omega^{n}X)$:
$H_{0}(\Omega^{n}X,\mathbf{Q})$ is the Laurent polynomial algebra on
$\pi_{n}^{\mathbf{Q}}(X)$, not the polynomial algebra. But this is the only
difference, and forming the group-completion in the Koszul dual Gerstenhaber
$n$-algebra produces $H_{\bullet}(\Omega^{n}X,\mathbf{Q})$. ∎

######  9.2.12 Remark.

It is instructive to consider the case $n=2$ and $X=S^{2}$. One may compute
$H_{\bullet}(\Omega^{2}_{0}S^{2},\mathbf{Q})$ in two different ways. One way
is to use that the $2$-connected cover of $S^{2}$ is $S^{3}$, so
$H_{\bullet}(\Omega^{2}_{0}S^{2},\mathbf{Q})\cong
H_{\bullet}(\Omega^{2}S^{3},\mathbf{Q})$, which by 9.2.7 is a free
Gerstenhaber algebra on a single generator $y$ in degree $1$. Alternatively,
9.2.11 identifies $H_{\bullet}(\Omega^{2}S^{2},\mathbf{Q})$ with the group-
completion of the free Gerstenhaber algebra on a single generator $x$ in
degree $0$. We may identify $H_{\bullet}(\Omega_{0}^{2}S^{2},\mathbf{Q})$ with
the Gerstenhaber subalgebra generated by the bracket $[x,x]=y$.

######  9.2.13.

The crucial fact is now that Koszul dual Gerstenhaber $n$-algebras have dual
quadratic presentations [Ber14, Theorem 2.11]. If $D$ is the space of
indecomposables in a Koszul Gerstenhaber $n$-algebra, then the indecomposables
in the Koszul dual Gerstenhaber $n$-algebra are given by $D^{\vee}[n]$.

###### Proof of 9.2.4.

We compute the Gerstenhaber indecomposables in
$H_{\bullet}(\Omega^{2}W(A),\mathbf{Q})$. By 9.2.11 and §9.2.13 this reduces
to computing the indecomposables in the algebra

$H^{\bullet}(W(A),\mathbf{Q})\cong H^{\bullet}(S^{2}\vee
KA/\langle\iota\rangle,\mathbf{Q})$

which are evidently $\mathbf{Q}[-2]$ (the fundamental class of the $2$-sphere)
and $\mathrm{Sym}^{2}(A^{\vee})$ (which generates the cohomology ring of
$KA/\langle\iota\rangle)$. In the homology of $\Omega^{2}W(A)$ these two
summands correspond to the first and third summand enumerated in 9.2.4. The
second summand of 9.2.4 arises by inverting the first summand in the process
of group-completing. ∎

##  10\. Closed surfaces

######  10.0.1.

In this section we will extend the homological stability results to moduli
spaces of closed surfaces. It will be convenient at this point to switch from
homology to cohomology. Let $\mathcal{H}_{g}$ be the moduli stack of
hyperelliptic curves of genus $g\geq 2$, and let $\mathbb{H}$ be the standard
rank $2g$ local system on $\mathcal{H}_{g}$, i.e. $R^{1}p_{\ast}\mathbf{Q}$,
where $p:\mathcal{C}_{g}\to\mathcal{H}_{g}$ is the universal curve. We opt to
denote this local system $\mathbb{H}$ rather than $\mathbb{V}$ for two
reasons:

  1. (1)

In the arguments we will consider the pullbacks of $\mathbb{H}$ to
$\mathcal{H}_{g}^{1,0}$ and $\mathcal{H}_{g}^{0,1}$. On
$\mathcal{H}_{g}^{0,1}$ there is a short exact sequence

$0\to\mathbf{Q}\to\mathbb{V}\to\mathbb{H}\to 0$

relating the local system that we have so far denoted $\mathbb{V}$ with the
pullback of $\mathbb{H}$, and it will be important to distinguish the two
notationally.

  2. (2)

In §3.5.11 and §8.1.4 we gave a definition of a local system888More precisely,
a cohomologically locally constant complex in $D^{b}(\mathcal{M}_{g})$.
$\mathbb{V}$ on $\mathcal{M}_{g}$, whose restriction to $\mathcal{H}_{g}$ is
not the same as $\mathbb{H}$.

In this section we calculate the cohomology groups
$H^{\bullet}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$ in a stable range, with
their Galois action. In particular, the cohomology stabilizes in the sense
that there are isomorphisms

$H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))\cong
H^{k}(\mathcal{H}_{g+1},S^{\lambda}(\mathbb{H}))$

for $g\gg k$, in spite of the fact that there are no natural stabilization
maps when we work with surfaces without boundary. This fact was previously
proven by an indirect argument in [MPP19].

######  10.0.2.

Let first $\mathcal{H}_{g,1,0}$ resp. $\mathcal{H}_{g,0,1}$ denote the moduli
spaces of hyperelliptic curves with a marked Weierstrass point, resp. a marked
non-Weierstrass point. By keeping track of Galois actions in 4.1.13, one sees
that for any $\ell$-adic sheaf $\mathbb{F}$ on $\mathcal{H}_{g,1,0}$ resp.
$\mathcal{H}_{g,0,1}$, there are isomorphisms

$H^{k}(\mathcal{H}_{g}^{1,0},\mathbb{F})=H^{k}(\mathcal{H}_{g,1,0},\mathbb{F})\oplus
H^{k-1}(\mathcal{H}_{g,1,0},\mathbb{F})(-1)$

and

$H^{k}(\mathcal{H}_{g}^{0,1},\mathbb{F})=H^{k}(\mathcal{H}_{g,0,1},\mathbb{F})\oplus
H^{k-1}(\mathcal{H}_{g,0,1},\mathbb{F})(-1)$

for any $k$.

######  10.0.3 Proposition.

If $k\leq\frac{2g-1-|\lambda|}{2}$, then
$H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$ is pure Tate of weight
$2k+|\lambda|$.

###### Proof.

Since the forgetful map $\mathcal{H}_{g,1,0}\to\mathcal{H}_{g}$ is finite,
$H^{\bullet}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))\to
H^{\bullet}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))$ is injective. (The
trace map gives a left inverse.) But
$H^{\bullet}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))$ injects into
$H^{\bullet}(\mathcal{H}_{g}^{1,0},S^{\lambda}(\mathbb{H}))$ by 4.1.13, and we
proved in 9.2.2 that the latter is pure Tate of this weight in a stable range.
(Note that $\mathbb{V}=\mathbb{H}$ on $\mathcal{H}_{g}^{1,0}$, as opposed to
on $\mathcal{H}_{g}^{0,1}$.) ∎

######  10.0.4.

Once we are armed with this purity result, it suffices to find an expression
for the weight-graded Euler characteristic of the homology in a stable range.
We set

$\Phi_{g}=\sum_{k}\sum_{i}\sum_{\lambda}(-1)^{k}z^{i/2}s_{\lambda}\dim\mathrm{Gr}^{W}_{i}H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$

and define similarly generating series

$\Phi_{g}^{1,0}=\sum_{k}\sum_{i}\sum_{\lambda}(-1)^{k}z^{i/2}s_{\lambda}\dim\mathrm{Gr}^{W}_{i}H^{k}(\mathcal{H}_{g}^{1,0},S^{\lambda}(\mathbb{V}))$

and

$\Phi_{g}^{0,1}=\sum_{k}\sum_{i}\sum_{\lambda}(-1)^{k}z^{i/2}s_{\lambda}\dim\mathrm{Gr}^{W}_{i}H^{k}(\mathcal{H}_{g}^{0,1},S^{\lambda}(\mathbb{V})).$

We may also define evident analogues $\Phi_{g,1,0}$ and $\Phi_{g,0,1}$ by
replacing $\mathcal{H}_{g}^{1,0}$ by $\mathcal{H}_{g,1,0}$, etc. but from
§10.0.2 we see that

$\Phi_{g,1,0}=\frac{1}{1-z}\Phi_{g}^{1,0},\qquad\Phi_{g,0,1}=\frac{1}{1-z}\Phi_{g}^{0,1}.$

We have similarly the stable versions of these generating series when
$g=\infty$. We have not yet proven that $\Phi_{\infty}$ is well-defined, but
this will be deduced shortly.

######  10.0.5 Proposition.

For every $g$, there is an identity

$(1-\frac{\partial}{\partial p_{1}}+z)\Phi_{g}=z\Phi_{g,1,0}+\sum_{j\geq
0}(-1)^{j}e_{j}\Phi_{g,0,1},$

where $\frac{\partial}{\partial p_{1}}$ is the partial derivative with respect
to $p_{1}=h_{1}=e_{1}$, and $e_{j}$ denotes the $j$th elementary symmetric
polynomial.

###### Proof.

We argue that both sides of the identity are given by

$\sum_{k}\sum_{i}\sum_{\lambda}(-1)^{k}z^{i/2}s_{\lambda}\dim\mathrm{Gr}^{W}_{i}H^{k}(\mathcal{C}_{g},S^{\lambda}(\mathbb{H})),$

where $\mathcal{C}_{g}\to\mathcal{H}_{g}$ is the universal curve.

For the left-hand side, we consider the Leray spectral sequence for
$p:\mathcal{C}_{g}\to\mathcal{H}_{g}$. It degenerates e.g. by [Del68], and
$Rp_{\ast}\mathbf{Q}\cong\mathbf{Q}\oplus\mathbb{H}[-1]\oplus\mathbf{Q}(-1)[-2].$
Combining this with the projection formula, we see that

$H^{k}(\mathcal{C}_{g},S^{\lambda}(\mathbb{H}))\cong
H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))\oplus
H^{k-1}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H})\otimes\mathbb{H})\oplus
H^{k-2}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))(-1).$

We should explain why the middle term here leads to $-\frac{\partial}{\partial
p_{1}}$. Indeed, by Schur–Weyl duality the $z^{i/2}$-coefficient of $\Phi_{g}$
is the class in $\widehat{\Lambda}$ of the symmetric sequence
$n\mapsto\mathrm{Gr}_{i}^{W}H^{\bullet}(\mathcal{H}_{g},\mathbb{H}^{\otimes
n})$. But then [GK98, Eq. (7.5)] the $z^{i/2}$-coefficient of
$\frac{\partial\Phi_{g}}{\partial p_{1}}$ is the class in $\widehat{\Lambda}$
of the symmetric sequence
$n\mapsto\mathrm{Gr}_{i}^{W}H^{\bullet}(\mathcal{H}_{g},\mathbb{H}^{\otimes(n+1)})$.

For the right-hand side, we note that $\mathcal{H}_{g,1,0}$ is a closed
substack of $\mathcal{C}_{g}$, with open complement $\mathcal{H}_{g,0,1}$. The
associated Gysin long exact sequence reads

$\ldots\to H^{k-2}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))(-1)\to
H^{k}(\mathcal{C}_{g},S^{\lambda}(\mathbb{H}))\to
H^{k}(\mathcal{H}_{g,0,1},S^{\lambda}(\mathbb{H}))$ $\hskip 142.26378pt\to
H^{k-1}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))(-1)\to\ldots$

Taking Euler characteristics in this long exact sequence gives the result.
Indeed, the terms of the form
$H^{\bullet}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))(-1)$ contribute
exactly $z\Phi_{g,1,0}$ to the right-hand side. (Recall that
$\mathbb{H}=\mathbb{V}$ on $\mathcal{H}_{g,1,0}$.) We claim that the terms of
the form $H^{\bullet}(\mathcal{H}_{g,1,0},S^{\lambda}(\mathbb{H}))$ contribute
exactly $\sum_{j\geq 0}(-1)^{j}e_{j}\Phi_{g,0,1}$. Indeed, once we are taking
Euler characteristics we may as well replace $\mathbb{H}$ with the formal
difference $\mathbb{V}-\mathbf{Q}$ (considered in a suitable Grothendieck
group), cf. §10.0.1, and formally one has

$S^{\lambda}(\mathbb{V}-\mathbf{Q})=\bigoplus
c_{\mu,\nu}^{\lambda}S^{\mu}(\mathbb{V})\otimes S^{\nu}(-\mathbf{Q}),$

where $c_{\mu,\nu}^{\lambda}$ is a Littlewood–Richardson coefficient, cf.
property (4) of §2.2.13. The term $S^{\nu}(-\mathbf{Q})$ vanishes unless
$\nu=(1^{j})$, i.e. $S^{\nu}$ is an exterior power, in which case we have
$\wedge^{j}(-\mathbf{Q})=(-1)^{j}\mathbf{Q}$. Since $e_{j}$ is the symmetric
function corresponding to $\wedge^{j}$ the total contribution from these terms
is $\sum_{j\geq 0}(-1)^{j}e_{j}\Phi_{g,0,1}$. ∎

######  10.0.6.

Note that every symmetric function $f\in\Lambda$ can be uniquely written as
$f_{e}+f_{o}$ with $f_{e}\in\bigoplus_{n}\Lambda_{2n}$ and
$f_{o}\in\bigoplus_{n}\Lambda_{2n+1}$. We call $f_{e}$ and $f_{o}$ the _even_
and the _odd_ part of $f$.

######  10.0.7 Theorem.

The stable generating series for closed and open surfaces are related by

$\Phi_{\infty}=\frac{z+\sum_{j\geq 0}e_{2j}}{1-z^{2}}\Phi_{\infty}^{0,1}.$

###### Proof.

This identity is obtained from the preceding proposition by letting
$g\to\infty$, and then taking the even part of the resulting expression.

Considering first the left-hand side of the preceding proposition, we note
that the generating series $\Phi_{g}$ is even for any $g$: the hyperelliptic
involution implies that $S^{\lambda}(\mathbb{H})$ has no cohomology on
$\mathcal{H}_{g}$ when $|\lambda|$ is odd. Hence on the left-hand side the
even part of the expression is precisely $(1+z)\Phi_{g}$, and
$\frac{\partial\Phi_{g}}{\partial p_{1}}$ is purely odd.

On the right-hand side, we see similarly that $\Phi_{g,1,0}$ is purely even
for all $g$ because of the hyperelliptic involution. The same argument does
not apply to $\Phi_{g,0,1}$, but by homological stability we have
$\Phi_{\infty,1,0}=\Phi_{\infty,0,1}$ and a fortiori it is even as a symmetric
function in the stable range. Hence taking the even part just amounts to
discarding the terms $e_{j}$ with odd $j$. Finally
$\Phi_{\infty,0,1}=\frac{1}{1-z}\Phi_{\infty}^{0,1}$. ∎

######  10.0.8.

We combine this formula with the plethystic formula for the stable Poincaré
series of 4.4.12, to obtain an analogue of 4.4.12 for closed surfaces.

######  10.0.9 Corollary.

There is an identity

$\sum_{k}\sum_{\lambda}\dim
H_{k}(H_{\infty},V_{\lambda})(-z)^{k}s_{\lambda^{\prime}}=\frac{z+\sum_{j\geq
0}z^{j}h_{2j}}{1-z^{2}}\mathbf{Exp}(z^{-1}\mathbf{Log}(z+\sum_{r\geq
0}z^{r}h_{2r})-1-h_{2}).$

###### Proof.

Indeed: 4.4.12 gives a formula for $\Phi_{\infty}^{0,1}$, and we merely need
to plug it into 10.0.7. What needs remarking is firstly that in 10.0.7 the
term $S^{\lambda}(\mathbb{H})$ corresponds to the Schur polynomial
$s_{\lambda}$, and in 4.4.12 we find the transpose $s_{\lambda^{\prime}}$. To
relate the formulas, we have to apply the involution $\omega$ of the ring of
symmetric functions to 10.0.7, replacing $e_{j}$ with $h_{j}$. Secondly,
10.0.7 is about generating series for weight-graded Euler characteristics, and
4.4.12 is a formula for Poincaré series. Since
$H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$ is pure of weight
$2k+|\lambda|$, the two are related by a shift of $|\lambda|$, which results
in replacing $h_{2j}$ with $z^{j}h_{2j}$. ∎

######  10.0.10 Remark.

It is quite curious that the same expression $z+\sum_{j\geq 0}z^{j}h_{2j}$
appears twice in the formula for completely different reasons. We do not have
a good conceptual explanation for why this should be the case.

######  10.0.11 Corollary.

If $k\leq\tfrac{2g-1-3|\lambda|}{4}$ then there are Galois-equivariant
isomorphisms $H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))\cong
H^{k}(\mathcal{H}_{g+1},S^{\lambda}(\mathbb{H}))\cong
H^{k}(\mathcal{H}_{g+2},S^{\lambda}(\mathbb{H}))\ldots$

###### Proof.

Note first that since
$H^{j}(\mathcal{H}_{g}^{1,0},S^{\lambda}(\mathbb{V}))\cong
H^{j}(\mathcal{H}_{g+1}^{1,0},S^{\lambda}(\mathbb{V}))$ for
$j\leq\tfrac{2g-1-|\lambda|}{2}$ by the result of Randal-Williams–Wahl, it
follows from the Deligne bounds on the weights of the unstable cohomology
[Del80] that $\Phi_{g}^{1,0}$ and $\Phi_{g+1}^{1,0}$ have the same coefficient
of $z^{i}s_{\lambda}$ for all $i\leq\tfrac{2g-1+|\lambda|}{4}$. The arguments
of this section show that the same holds for $\Phi_{g}$ and $\Phi_{g+1}$.

We claim now that $\dim H^{k}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$ is the
coefficient of $z^{k+|\lambda|/2}$ in $\Phi_{g}$. Indeed, 10.0.3 shows in fact
that $H^{i}(\mathcal{H}_{g},S^{\lambda}(\mathbb{H}))$ is pure Tate of weight
$2i+|\lambda|$ for all $i\leq 2k$, and by the Deligne bounds the weight
$2k+|\lambda|$ cannot occur above this degree. ∎

######  10.0.12 Remark.

Armed with 10.0.9 instead of 4.4.12 one can give analogous proofs to show that
7.0.2, 7.0.4 and 7.0.12 hold with $H_{\infty}^{0,1}$ replaced by $H_{\infty}$.
Similarly, a version 7.0.15 holds with $H_{\infty,1}$ replaced by
$H_{\infty}$, except one finds a polynomial of degree at most $|\lambda|$
rather than exactly $|\lambda|-1$.

##  11\. Point counting

######  11.0.1.

In this section we study the traces of Frobenius on the cohomology groups of
the moduli space $\mathcal{H}_{g}^{1,0}$ with coefficients in the irreducible
symplectic representations $V_{\lambda}.$ In particular, we show that for any
fixed finite field of odd characteristic and any $\lambda$ of even weight, the
limit as $g\to\infty$ of the traces of Frobenius on these cohomology groups
exists and is a rational function in the cardinality of the finite field. A
similar result for the moduli space $\mathcal{H}_{g}$ of closed hyperelliptic
curves together with an algorithmic procedure to compute the rational
functions occurring in the limit was obtained by the first named author
[Ber09, Section 13]. However, rather than trying to compute these rational
functions directly, it is more natural to study a certain generating function
of the _stable_ traces, to be introduced in Subsection 11.2. As we shall see,
this function is closely related to the Poincaré series of the cohomologies
$H_{\bullet}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda}),$ and is
precisely the so-called _arithmetic_ factor occurring in the (conjectural)
asymptotic formula for the moments of quadratic Dirichlet $L$-functions over
rational function fields.

###  11.1. Zeta functions and quadratic $L$-series

######  11.1.1.

We begin by recalling some basic facts about zeta functions of algebraic
curves over finite fields and quadratic $L$-series in the rational function
field setting.

######  11.1.2.

Let $\mathbf{F}_{q}$ be a finite field with $q$ elements, and let
$\overline{\mathbf{F}}_{q}$ be a fixed algebraic closure of $\mathbf{F}_{q}.$
Let $C$ be a smooth, projective, and geometrically connected curve of genus
$g$ over $\mathbf{F}_{q}.$ The zeta function of $C$ is defined by

$Z_{C}(t)=\exp\\!\Bigg{(}\sum_{n\geq
1}\\#C(\mathbf{F}_{q^{n}})\frac{t^{n}}{n}\Bigg{)}\qquad(\text{for
$|t|<1/q$}).$

######  11.1.3.

For a prime $\ell$ different from the characteristic of $\mathbf{F}_{q}$ and
$n\geq 1,$ one can express the number of points as
$\\#C(\mathbf{F}_{q^{n}})=q^{n}+1-a_{n}(C),$ where

$a_{n}(C)=\operatorname{Tr}(\mathrm{Frob}_{q}^{n}\,|\,H_{\text{\'{e}t}}^{1}(\bar{C},\mathbf{Q}_{\ell}))=\sum_{j=1}^{2g}\omega_{j}(C)^{n}.$

Here $\bar{C}:=C\otimes_{\mathbf{F}_{q}}\overline{\mathbf{F}}_{q}$ and
$\mathrm{Frob}_{q}$ is the endomorphism of the $\ell$-adic étale cohomology
induced by the Frobenius morphism $\bar{C}\rightarrow\bar{C}.$ Thus the zeta
function is a rational function

$Z_{C}(t)=\frac{P_{C}(t)}{(1-t)(1-qt)}$

with numerator $P_{C}(t)\in\mathbf{Z}[t]$ of degree $2g,$ and constant term
$P_{C}(0)=1.$ Moreover, it satisfies the functional equation

$Z_{C}(t)=\big{(}qt^{2}\big{)}^{g-1}Z_{C}(1/qt).$

######  11.1.4.

The zeta function satisfies the Riemann hypothesis [Wei48], that is, the
_normalized_ eigenvalues of Frobenius $q^{-1/2}\omega_{j}(C)$
($j=1,\ldots,2g$) lie on the unit circle. Moreover, from the functional
equation, these eigenvalues have the property that, after an eventual
reordering, $\omega_{j}(C)\omega_{j+g}(C)=q$ for all $1\leq j\leq g.$ In other
words, there exists a unique conjugacy class $\Theta(C/\mathbf{F}_{q})$ in the
similitude group $\mathrm{GSp}(2g)$ such that

$P_{C}(t)=\det(1-t\Theta(C/\mathbf{F}_{q})).$

This conjugacy class is called the Frobenius conjugacy class attached to
$C/\mathbf{F}_{q}.$

######  11.1.5.

Now assume that $\mathbf{F}_{q}$ has odd characteristic.

######  11.1.6 Definition.

For $d,p\in\mathbf{F}_{q}[x],$ with $p$ irreducible, define the quadratic
residue symbol $(\frac{d}{p})$ to be $0$ if $p\mid d,$ $1$ if $d$ is congruent
to a square modulo $p$ and $-1$ otherwise. If $m$ is non-constant, and
$m=p_{1}\cdots\,p_{k}$ with each $p_{i}\in\mathbf{F}_{q}[x]$ irreducible, one
defines $(\frac{d}{m})=(\frac{d}{p_{1}})\cdots(\frac{d}{p_{k}})$.

######  11.1.7.

It is clear that the quadratic residue symbol is completely multiplicative in
both variables, that is,

${\displaystyle\left({\frac{d_{1}d_{2}}{m}}\right)}={\displaystyle\left({\frac{d_{1}}{m}}\right)}{\displaystyle\left({\frac{d_{2}}{m}}\right)}\;\;\;\text{and}\;\;\;{\displaystyle\left({\frac{d}{m_{1}m_{2}}}\right)}={\displaystyle\left({\frac{d}{m_{1}}}\right)}{\displaystyle\left({\frac{d}{m_{2}}}\right)}$

for all $d,d_{1},d_{2},m,m_{1},m_{2}\in\mathbf{F}_{q}[x],$ with
$m,m_{1},m_{2}$ non-constant. It is also easy to check that (see [Ros02,
Proposition 3.2])

${\displaystyle\left({\frac{c}{p}}\right)}=c^{\frac{q-1}{2}\deg p}$

for $c\in\mathbf{F}_{q}$ and $p\in\mathbf{F}_{q}[x]$ irreducible. Thus
$(\frac{c}{p})=1$ if $c\in(\mathbf{F}_{q}^{\times})^{2},$ and
$(\frac{c}{p})=(-1)^{\deg p}$ if
$c\in\mathbf{F}_{q}^{\times}\setminus(\mathbf{F}_{q}^{\times})^{2}.$ By
induction on the irreducible factors, $(\frac{c}{m})=1$ or $(-1)^{\deg m},$
for all $m$ of positive degree, according as $c$ is a square in
$\mathbf{F}_{q}^{\times}$ or not.

######  11.1.8.

The quadratic residue symbol also satisfies the reciprocity law [Ros02,
Theorem 3.5]: for $d,m\in\mathbf{F}_{q}[x]$ monic and non-constant, we have

$\begin{split}{\displaystyle\left({\frac{d}{m}}\right)}&=(-1)^{\frac{|d|-1}{2}\frac{|m|-1}{2}}{\displaystyle\left({\frac{m}{d}}\right)}\\\
&=(-1)^{\frac{q-1}{2}\deg d\deg
m}\,{\displaystyle\left({\frac{m}{d}}\right)}\end{split}$

where for a non-zero element $a\in\mathbf{F}_{q}[x],$ $|a|=q^{\deg a}$ denotes
its norm.

######  11.1.9.

For $d\in\mathbf{F}_{q}[x]$ square-free, define $\chi_{d}(m)=(\frac{d}{m}),$
and extend it to all non-zero polynomials by setting $\chi_{d}(c)=1$ for all
$c\in\mathbf{F}_{q}^{\times}.$ To $\chi_{d}$ there is an associated
$L$-function, first considered by E. Artin in his thesis [Art24], defined for
$s\in\mathbf{C}$ with $\Re(s)>1$ by

$L(s,\chi_{d})\;\,=\sum_{\begin{subarray}{c}m\,\in\,\mathbf{F}_{q}[x]\\\
m\text{
monic}\end{subarray}}\chi_{d}(m)|m|^{-s}=\,\prod_{p}(1-\chi_{d}(p)|p|^{-s})^{-1}$

the product in the right-hand side being over all monic irreducibles. It is
easy to see that indeed both the sum and product converge absolutely when
$\Re(s)>1.$

######  11.1.10.

If $d$ is monic and square-free of degree $D\geq 3,$ the $L$-function and the
numerator $P_{C_{d}}(t)$ of the zeta function of $C_{d}$ defined by the affine
model $y^{2}=d(x)$ are connected by

$L(s,\chi_{d})=(1-q^{-s})^{\delta_{D}}P_{C_{d}}(q^{-s}),\;\;\;\text{with
$\delta_{D}=(1+(-1)^{D})/2$.}$

This follows by combining Proposition 14.9 and Proposition 17.7 in [Ros02]
with the well-known fact that the zeta function of a smooth, projective, and
geometrically connected algebraic curve over a finite field is the same as the
zeta function of its function field. If $d$ is monic of degree $D=1,2,$ then
$L(s,\chi_{d})=(1-q^{-s})^{\delta_{D}},$ and for $d\in\mathbf{F}_{q}^{\times}$
we have:

$L(s,\chi_{d})=\zeta_{\mathbf{F}_{q}[x]}(s)=\frac{1}{1-q^{1-s}}\;\;\text{if
$d\in(\mathbf{F}_{q}^{\times})^{2}$}\;\;\;\text{and}\;\;\;L(s,\chi_{d})=\frac{1}{1+q^{1-s}}\;\;\text{if
$d\notin(\mathbf{F}_{q}^{\times})^{2}$.}$

Thus for $d$ monic and square-free of degree $D>0,$ the $L$-function is a
polynomial in $q^{-s}$ of degree $D-1,$ and it satisfies the functional
equation

$(1-q^{-s})^{-\delta_{D}}L(s,\chi_{d})=q^{(\delta_{D}+1)(s-\frac{1}{2})}|d|^{\frac{1}{2}-s}(1-q^{s-1})^{-\delta_{D}}L(1-s,\chi_{d}).$

######  11.1.11.

In what follows, it will be often convenient to substitute $t:=q^{-s},$ and
denote the $L$-function by $\mathcal{L}(t,\chi_{d}).$

###  11.2. A generating function for stable traces

######  11.2.1.

Let $\mathbf{F}_{q}$ be a finite field of odd characteristic. For $n>0,$ let
$\mathscr{P}_{n}$ denote the set of monic square-free polynomials of degree
$n$ in $\mathbf{F}_{q}[x],$ and for $d\in\mathscr{P}_{n}$, let $\Theta_{d}$
denote the following multiset with $n-1$ elements: if $n$ is odd it is the
eigenvalues occurring in the Frobenius conjugacy class of the hyperelliptic
curve $C_{d}/\mathbf{F}_{q}$ with affine equation $y^{2}=d(x)$; if $n$ is even
then $\Theta_{d}$ contains in addition the number $1$. With this definition,
we have

$\mathcal{L}(t,\chi_{d})=\prod_{\omega\in\Theta_{d}}(1-\omega t)$

for all $d$.

######  11.2.2.

Define for $n>0$ an element
$\mathcal{Z}_{n}\in\widehat{\Lambda\otimes\mathbf{C}}$ by the formula

$\mathcal{Z}_{n}=q^{-n}\sum_{d\in\mathscr{P}_{n}}\sum_{\lambda}s_{\lambda}(\Theta_{d})\cdot
s_{\lambda^{\prime}},$

where $s_{\lambda}(\Theta_{d})$ denotes the Schur polynomial in $n-1$
variables evaluated on the elements of $\Theta_{d}$. The goal of this
subsection is to study the limit $\lim_{n\to\infty}\mathcal{Z}_{n}$. In
particular, the limit exists in the sense that its coefficients (say with
respect to the basis of Schur polynomials) converge pointwise. We call this
limit the generating function of stable traces. It may be written explicitly
as follows:

######  11.2.3 Theorem.

The generating function of the stable traces is given by

$\lim_{n\to\infty}\mathcal{Z}_{n}=\mathbf{Exp}\Big{(}q\,\mathbf{Log}\Big{(}q^{-1}+\sum_{k\geq
0}h_{2k}\Big{)}-1\Big{)}$

where $\mathbf{Exp}$ and $\mathbf{Log}$ are the plethystic exponential and
logarithm, respectively.

######  11.2.4.

The generating series $\mathcal{Z}_{n}$ admits a cohomological interpretation,
given in 11.2.6. Via this interpretation, 11.2.3 follows from 1.1.3 and 1.1.7,
together with the bound on the unstable Betti numbers of the braid groups
given in 11.3.13. However, in this subsection we give a direct proof of 11.2.3
by analytic methods, with an explicit error term. While not logically
necessary, since it follows from our topological results, we believe that the
argument is enlightening and clarifies the difference between 11.2.3 and
1.1.11/1.1.13.

######  11.2.5 Remark.

11.2.3 will be deduced from an asymptotic formula proven in 11.2.11. The
latter formula is in a sense not new, see, for instance, [Rud10, RG12] and
[BJ19], where the authors obtained similar asymptotics, but for averages of
products of power sum polynomials in the eigenvalues of Frobenius rather than
products of elementary symmetric polynomials. However, any of these asymptotic
formulas (especially that in [RG12], which is closely related to ours) is
insufficient for our purposes, mainly due to the restriction to partitions
whose parts grow at least logarithmically in the genus $g.$

######  11.2.6 Lemma.

We have the equality

$\mathcal{Z}_{n}=\begin{cases}\sum_{\lambda}\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}|H_{k}^{\mathrm{\acute{e}t}}(\mathcal{H}_{g}^{1,0}\otimes\overline{\mathbf{F}}_{q},S^{\lambda}(\mathbb{V}))\cdot
s_{\lambda^{\prime}}&\text{if }n=2g+1,\\\
\sum_{\lambda}\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}|H_{k}^{\mathrm{\acute{e}t}}(\mathcal{H}_{g}^{0,1}\otimes\overline{\mathbf{F}}_{q},S^{\lambda}(\mathbb{V}))\cdot
s_{\lambda^{\prime}}&\text{if }n=2g+2.\end{cases}$

###### Proof.

Let us say that $n=2g+1$ is odd (the argument in the even case is identical).
By the Grothendieck-Lefschetz trace formula for stacks [Beh03, Corollary
6.4.10] one has

$\sum_{k}\,(-1)^{k}\operatorname{Tr}(\mathrm{Frob}_{q}\,|\,H_{c,\text{\'{e}t}}^{k}(\mathcal{H}_{g}^{1,0}\otimes_{\mathbf{F}_{q}}\overline{\mathbf{F}}_{q},S^{\lambda}(\mathbb{V}))\,=\,\sum_{[C_{d}]\in\mathcal{H}_{g}^{1,0}(\mathbf{F}_{q})}s_{\lambda}(\Theta_{d}).$

The left-hand side equals $q^{\dim\mathcal{H}_{g}^{1,0}}=q^{2g}$ times the
trace of Frobenius on étale homology, by Poincaré duality. As for the right-
hand side, observe that
$\mathcal{H}_{g}^{1,0}\cong\mathrm{Conf}_{2g+1}(\mathbb{A}^{1})/\mathbb{G}_{a}$,
meaning that $\mathcal{H}_{g}^{1,0}(\mathbf{F}_{q})$ equals
$\mathscr{P}_{2g+1}$ up to the action by translation, which accounts for the
additional factor of $q$. Finally, the multiset $\Theta_{d}$ consists
precisely of the eigenvalues of $\mathrm{Frob}_{q}$ on the stalk of
$\mathbb{V}$ at a geometric point above
$[C_{d}]\in\mathcal{H}_{g}^{1,0}(\mathbf{F}_{q})$. ∎

######  11.2.7.

From the definition of $\mathcal{Z}_{n}$ we see that if we specialize
$\mathcal{Z}_{n}$ to a symmetric function of finitely many variables
$\mathrm{t}=t_{1},t_{2},\ldots,t_{r}$, then

$\mathcal{Z}_{n}(\mathrm{t})=q^{-n}\sum_{d\in\mathscr{P}_{n}}\prod_{i=1}^{r}\,\mathcal{L}(-t_{i},\chi_{d}).$

Indeed, by the dual Cauchy identity (§2.2.8) we have for any
$d\in\mathscr{P}_{n}$ that

$s_{\lambda}(\Theta_{d})s_{\lambda^{\prime}}(t_{1},\ldots,t_{r})=\prod_{\omega\in\Theta_{d}}\prod_{i=1}^{r}(1+\omega
t_{i})=\prod_{i=1}^{r}\mathcal{L}(-t_{i},\chi_{d}).$

######  11.2.8.

Now define

$\mathcal{Z}(\mathrm{t},\xi)\,=\,\sum_{n}\mathcal{Z}_{n}(\mathrm{t})\,\xi^{n}\,=\sum_{d\text{
monic \& sq.
free}}\,\,\prod_{i=1}^{r}\,\mathcal{L}(-t_{i},\chi_{d})\,(\xi/q)^{\deg d}.$

Write
$\mathcal{Z}(\mathrm{t},\xi)=\sum_{\lambda}\mathcal{Z}_{\lambda}(\xi)\mathrm{t}^{\lambda}$,
where $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ is a tuple of nonnegative
integers.

######  11.2.9 Remark.

If $n$ is odd and $|\lambda|$ is odd, then the coefficient of $\xi^{n}$ in
$\mathcal{Z}_{\lambda}(\mathrm{t})$ vanishes. Indeed, this follows by a point-
counting argument: if
$c\in\mathbf{F}_{q}^{\times}\setminus(\mathbf{F}_{q}^{\times})^{2}$, then
$\mathcal{L}(t,\chi_{cd})=\mathcal{L}(-t,\chi_{d})$, and by pairing up the
terms corresponding to $d(x)$ and $c^{-n}d(cx)$ we see that the sum vanishes.
It can also be seen via 4.3.5 and the cohomological interpretation of
$\mathcal{Z}_{n}$.

######  11.2.10 Lemma.

Suppose $\lambda=(\lambda_{1},\ldots,\lambda_{r})$. Then

$\mathcal{Z}_{\lambda}(q\xi)=(-1)^{|\lambda|}(1-q\xi^{2})\sum\frac{\mathcal{L}(\xi,\chi_{u})}{\prod_{p\mid
u}(1-\xi^{2\deg p})\prod_{p\mid v}(1+\chi_{u}(p)\xi^{\deg p})}$

where the sum is over all monic polynomials $m_{1},\ldots,m_{r}$ with $\deg
m_{i}=\lambda_{i},$ and for each choice of $m_{i}$, $1\leq i\leq r$, we write
$m_{1}\cdots\,m_{r}=uv^{2}$ with $u,v$ monic and $u$ square-free.

###### Proof.

This is merely an application of the quadratic reciprocity law

${\displaystyle\left({\frac{d}{m}}\right)}=(-1)^{\frac{q-1}{2}\deg d\deg
m}\,{\displaystyle\left({\frac{m}{d}}\right)}.$

To see this, assume $|t_{i}|<1/q$ for $i=1,\ldots,r,$ and $|\xi|<1/q.$ Then,
by expressing each $\mathcal{L}(-t_{i},\chi_{d})$ by its defining series,
interchanging the order of summation and applying quadratic reciprocity, we
can write

$\mathcal{Z}(\mathrm{t},q\xi)\;\;=\sum_{m_{1},\ldots,\,m_{r}\text{
monic}}\;\prod_{i=1}^{r}(-t_{i})^{\deg m_{i}}\sum_{d\text{ monic \& sq.\
free}}\,{\displaystyle\left({\frac{\prod_{i=1}^{r}m_{i}}{d}}\right)}\xi^{\deg
d}.$

Here we used that the sign in the quadratic reciprocity can always be taken to
be $1$: if not, then $\deg d$ and $\deg m=|\lambda|$ are both odd, but in this
case the corresponding coefficient of $\mathcal{Z}(t,\xi)$ vanishes. For fixed
$m_{1},\ldots,m_{r}$, write $m_{1}\cdots\,m_{r}=uv^{2}$ with $u,v$ monic and
$u$ square-free. Then we can express the inner sum as

$\sum_{d\text{ monic \& sq.\
free}}\,{\displaystyle\left({\frac{\prod_{i=1}^{r}m_{i}}{d}}\right)}\xi^{\deg
d}=\sum_{d\text{ monic \& sq.\ free}}\chi_{uv^{2}}(d)\,\xi^{\deg
d}=\prod_{p\,\nmid\,uv}\big{(}1+\chi_{u}(p)\,\xi^{\deg p}\big{)}$

the product being over all monic irreducibles not dividing $uv$. For each such
$p$, we write

$1+\chi_{u}(p)\,\xi^{\deg p}=\frac{1-\xi^{2\deg p}}{1-\chi_{u}(p)\xi^{\deg
p}},$

and thus

$\sum_{d\text{ monic \& sq.\
free}}\,{\displaystyle\left({\frac{\prod_{i=1}^{r}m_{i}}{d}}\right)}\xi^{\deg
d}=\big{(}1-q\xi^{2}\big{)}\frac{\mathcal{L}(\xi,\chi_{u})}{\prod_{p\mid
u}\big{(}1-\xi^{2\deg p}\big{)}\prod_{p\mid v}\big{(}1+\chi_{u}(p)\xi^{\deg
p}\big{)}}.$

Here we used that $\prod_{p}(1-\xi^{2\deg p})=(1-q\xi^{2})$. The result
follows. ∎

######  11.2.11 Proposition.

For $n>0$ and $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ the coefficient of
$\xi^{n}$ in $\mathcal{Z}_{\lambda}(\xi)$ is given by

$\big{(}1-q^{-1}\big{)}\sum_{\begin{subarray}{c}m_{1}\cdots\,m_{r}=v^{2}\\\
\deg m_{i}=\lambda_{i}\end{subarray}}\;\prod_{p\mid
v}\bigg{(}1+\frac{1}{|p|}\bigg{)}^{-1}\;+\;\;O\Big{(}5^{|\lambda|}q^{|\lambda|-n/2}\Big{)}.$

Note in particular that the sum is empty if $|\lambda|$ is odd.

###### Proof.

First notice that the coefficient $\mathcal{Z}_{\lambda}(\xi)$ of
$\mathrm{t}^{\lambda}$ in $\mathcal{Z}(\mathrm{t},\xi)$ (see 11.2.10) is a
holomorphic function in the open disc $|\xi|<q,$ except for the factor
$\mathcal{L}(\xi/q,\chi_{u})$ which picks up a simple pole at $\xi=1$ when
$u=1$. Thus, the coefficient of $\xi^{n}$ in $\mathcal{Z}_{\lambda}(\xi)$ is
given by

$\frac{1}{2\pi\sqrt{-1}}\;\oint\limits_{|\xi|\,=\,\varepsilon}\frac{\mathcal{Z}_{\lambda}(\xi)}{\xi^{n+1}}\,d\xi$

where $0<\varepsilon<1$. Consider the integral

$\frac{1}{2\pi\sqrt{-1}}\oint_{\partial\mathscr{A}}\frac{\mathcal{Z}_{\lambda}(\xi)}{\xi^{n+1}}\,d\xi$

where $\mathscr{A}=\\{\xi\in\mathbf{C}:\varepsilon\leq|\xi|\leq q^{1/2}\\}.$
Note that

(3)
$\frac{1}{2\pi}\left|\;\,\oint\limits_{|\xi|\,=\,q^{1/2}}\frac{\mathcal{Z}_{\lambda}(\xi)}{\xi^{n+1}}\,d\xi\,\right|\leq
q^{-n/2}\cdot\sup_{|\xi|\,=\,q^{1/2}}|\mathcal{Z}_{\lambda}(\xi)|.$

We now bound
$\sup_{|\xi|\,=\,q^{1/2}}|\mathcal{Z}_{\lambda}(\xi)|=\sup_{|\xi|\,=\,q^{-1/2}}|\mathcal{Z}_{\lambda}(q\xi)|$
using 11.2.10. When $|\xi|=q^{\,-1/2}$ we have the following simple estimates:

$\displaystyle|\mathcal{L}(\xi,\chi_{d})|$
$\displaystyle=\prod_{\omega\in\Theta_{d}}\big{|}1-\omega\xi\big{|}\leq
2^{\deg d-1},$ $\displaystyle\prod_{p\mid u}|1-\xi^{2\deg p}|\prod_{p\mid
v}|1+\chi_{u}(p)\xi^{\deg p}|$ $\displaystyle\geq(1-q^{-1/2})^{\deg u+\deg
v}\geq(1-q^{-1/2})^{|\lambda|}.$

Thus, for $|\xi|=q^{\,-1/2},$ we have

$\begin{split}|\mathcal{Z}_{\lambda}(q\xi)|&\leq
2\sum\sup_{|\xi|\,=\,q^{\,-1/2}}\bigg{(}\frac{|\mathcal{L}(\xi,\chi_{u})|}{\prod_{p\mid
u}\big{|}1-\xi^{2\deg p}\big{|}\prod_{p\mid v}\big{|}1+\chi_{u}(p)\xi^{\deg
p}\big{|}}\bigg{)}\\\
&\ll\bigg{(}\frac{2}{1-q^{-1/2}}\bigg{)}^{|\lambda|}\cdot\,\\#\\{(m_{1},\ldots,m_{r}):\text{$\deg
m_{i}=\lambda_{i}$ for $i=1,\ldots,r$}\\}\\\ &\ll
5^{|\lambda|}q^{|\lambda|}.\end{split}$

Accordingly the integral (3) is $\ll 5^{|\lambda|}q^{|\lambda|-n/2}.$ On the
other hand, by the residue theorem we have

$\begin{split}\frac{1}{2\pi\sqrt{-1}}\oint_{\partial\mathscr{A}}\frac{\mathcal{Z}_{\lambda}(\xi)}{\xi^{n+1}}\,d\xi&=\operatorname{Res}_{\xi=1}\mathcal{Z}_{\lambda}(\xi)\\\
&=-\,\bigg{(}1-\frac{1}{q}\bigg{)}\sum_{\begin{subarray}{c}m_{1}\cdots\,m_{r}=v^{2}\\\
\deg m_{i}=\lambda_{i}\end{subarray}}\;\prod_{p\mid
v}\bigg{(}1+\frac{1}{|p|}\bigg{)}^{-1}\end{split}$

and the asymptotic formula follows. This completes the proof. ∎

######  11.2.12.

We may now prove 11.2.3.

###### Proof of 11.2.3.

Let $\mu(n)$ be the usual Möbius function, and let $i_{n}(q)=n^{-1}\sum_{l\mid
n}\mu(n/l)q^{l}$ be the number of monic irreducibles of degree $n$ in
$\mathbf{F}_{q}[x].$ Then we have by 11.2.11

$\begin{split}\lim_{n\to\infty}\mathcal{Z}_{n}(\mathrm{t})\;\,&=(1-q^{-1})\sum_{m_{1}\cdots\,m_{r}=v^{2}}\;\prod_{p\mid
v}\big{(}1+|p|^{-1}\big{)}^{-1}\cdot\,t_{1}^{\deg m_{1}}\cdots\,t_{r}^{\deg
m_{r}}\\\
&=(1-q^{-1})\prod_{p}\Bigg{(}1\,+\,\big{(}1+|p|^{-1}\big{)}^{-1}\cdot\sum_{k>0}\sum_{\begin{subarray}{c}k_{1},\ldots,\,k_{r}\geq
0\\\ k_{1}+\cdots+k_{r}=2k\end{subarray}}\prod_{j=1}^{r}t_{j}^{k_{j}\deg
p}\Bigg{)}\\\
&=(1-q^{-1})\prod_{n>0}\Big{(}1\,+\,\big{(}1+q^{-n}\big{)}^{-1}\sum_{k>0}h_{2k}(\mathrm{t}^{n})\Big{)}^{i_{n}(q)}\end{split}$

where $\mathrm{t}^{n}:=(t_{1}^{n},\ldots,t_{r}^{n}).$ The result follows by
comparing this expression with 7.0.14. ∎

###  11.3. Asymptotics for moments from homological stability

######  11.3.1.

We will prefer to think about the representation theory of the usual
symplectic groups, rather than the odd symplectic groups. For this reason, we
work consistently in this section with the moduli spaces
$\mathcal{H}_{g}^{1,0}$ and $\mathcal{H}_{g,1,0}$ rather than
$\mathcal{H}_{g}^{0,1}$ and $\mathcal{H}_{g,0,1}$. We will also find it
convenient to consider the _unitarized_ conjugacy class of Frobenius. For
$d\in\mathscr{P}_{2g+1}$ we write $\overline{\Theta}_{d}$ for
$q^{-1/2}\Theta_{d}$, which we may think of as a conjugacy class in
$\operatorname{USp}(2g)$. Then $\overline{\Theta}_{d}$ consists of the
eigenvalues of Frobenius on the stalk above the point $[C_{d}]$ of
$\mathbb{V}(\tfrac{1}{2})$, the half-integral Tate twist of the standard local
system.

######  11.3.2 Definition.

Introduce the notation

$\mathrm{tr}_{\lambda}(g)=\frac{1}{q^{2g+1}}\sum_{d\in\mathscr{P}_{2g+1}}s_{\langle\lambda\rangle}(\overline{\Theta}_{d}),$

where $s_{\langle\lambda\rangle}$ denotes a symplectic Schur function. We also
set

$T_{\lambda}=\lim_{g\to\infty}\mathrm{tr}_{\lambda}(g).$

######  11.3.3.

Applying the Grothendieck–Lefschetz trace formula as in 11.2.6 we have

$\mathrm{tr}_{\lambda}(g)=\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{g}^{1,0}\otimes\overline{\mathbf{F}}_{q},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2})))$

and

$T_{\lambda}=\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{\infty}^{1,0}\otimes\overline{\mathbf{F}}_{q},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2}))).$

######  11.3.4.

In the previous subsection we used crucially that

$\mathcal{Z}_{n}(t_{1},\ldots,t_{r})=q^{-n}\sum_{d\in\mathscr{P}_{n}}\prod_{i=1}^{r}\mathcal{L}(-t_{i},\chi_{d});$

this was a consequence of the dual Cauchy identity. As explained in §2.6.8
there is a perfect symplectic analogue of the dual Cauchy identity when one
considers symmetric functions in _infinitely_ many variables, but it does not
specialize in a useful way to finitely many variables. As an adequate
substitute we will use an identity due to Jimbo–Miwa, see [JM85], [How95,
Theorem 3.8.9.3], and [BG06, Lemma 4].

######  11.3.5 Definition.

Let $\lambda\subseteq(r^{g})$ be a partition, i.e. $\lambda_{1}\leq r$ and
$\ell(\lambda)\leq g$. We set
$\lambda^{{\dagger}}:=(g-\lambda_{r}^{\prime}\geq\cdots\geq
g-\lambda_{1}^{\prime}),$ where the non-zero integers among
$\lambda_{j}^{\prime},$ $1\leq j\leq r,$ are the parts of the conjugate
partition $\lambda^{\prime}$ of $\lambda.$ If we want to make the dependence
of $\lambda^{\dagger}$ on $g$ explicit, we write $\lambda^{\dagger}(g)$.

######  11.3.6 Theorem (Jimbo–Miwa).

$\prod_{i=1}^{g}\prod_{j=1}^{r}(x_{i}+x_{i}^{-1}+t_{j}+t_{j}^{-1})=\sum_{\lambda\subseteq(r^{g})}s_{\langle\lambda\rangle}(x_{1}^{\pm},\ldots,x_{g}^{\pm})s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm}).$

######  11.3.7 Corollary.

For complex variables $t_{1},\ldots,t_{r},$ we have

$\frac{1}{q^{2g+1}}\sum_{d\in\mathscr{P}_{2g+1}}\prod_{j=1}^{r}\mathcal{L}(q^{-1/2}t_{j},\chi_{d})\,=\,(t_{1}\cdots\,t_{r})^{g}\\!\\!\sum_{\lambda\subseteq(r^{g})}\mathrm{tr}_{\lambda}(g)s_{\langle\lambda^{{\dagger}}\rangle}\big{(}t_{1}^{\pm
1}\\!,\ldots,t_{r}^{\pm 1}\big{)}.$

###### Proof.

First rewrite 11.3.6 as

$\prod_{i=1}^{g}\prod_{j=1}^{r}(1+t_{j}x_{i})(1+t_{j}x_{i}^{-1})=(t_{1}\cdots
t_{r})^{g}\sum_{\lambda\subseteq(r^{g})}s_{\langle\lambda\rangle}(x_{1}^{\pm},\ldots,x_{g}^{\pm})s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm}).$

Then set $\\{x_{1}^{\pm},\ldots,x_{g}^{\pm}\\}$ equal to
$\overline{\Theta}_{d}$, and observe that
$\mathcal{L}(-q^{-1/2}t_{j},\chi_{d})=\prod_{\omega\in\overline{\Theta}_{d}}(1+t_{j}\omega)$.
Finally, take the product over $r$ and sum over $d$, and observe that the
result is invariant under $t_{j}\mapsto-t_{j}$ since
$\mathrm{tr}_{\lambda}(g)$ vanishes for $|\lambda|$ odd. ∎

######  11.3.8 Corollary.

$\frac{1}{q^{2g+1}}\sum_{d\in\mathscr{P}_{2g+1}}{L}(\tfrac{1}{2},\chi_{d})^{r}\,=\,\sum_{\lambda\subseteq(r^{g})}\mathrm{tr}_{\lambda}(g)\,\dim
V_{\lambda^{\dagger}},$

where $V_{\lambda^{\dagger}}$ denotes an irreducible representation of
$\mathrm{Sp}(2r)$.

###### Proof.

Set $t_{1}=t_{2}=\ldots=1$ in the preceding corollary. ∎

######  11.3.9.

We will in particular make use of the identity

$\sum_{\lambda\subseteq(r^{g})}\dim V_{\lambda}^{\mathrm{Sp}(2g)}\dim
V_{\lambda^{\dagger}}^{\mathrm{Sp}(2r)}=4^{gr}$

obtained by setting all $x_{i}$ and $t_{i}$ equal to $1$ in 11.3.6. Here the
superscripts $\mathrm{Sp}(2g)$, resp. $\mathrm{Sp}(2r)$, indicate which groups
$V_{\lambda}$, resp. $V_{\lambda^{\dagger}}$, are representations of.

######  11.3.10.

If $\ell(\lambda)>g$ then we can still define $\lambda^{\dagger}$ as in
11.3.5, but $\lambda^{\dagger}$ is no longer dominant. We extend the
definition of $s_{\langle\lambda^{\dagger}\rangle}$ to this case by plugging
$\lambda^{\dagger}$ into the Weyl character formula. Let

$\alpha:=\\{\lambda^{\dagger}+\rho\\}-\rho,$

where $\\{\cdot\\}$ denotes the unique dominant weight conjugate to a given
weight and $\rho$ is the sum of the fundamental weights. Then
$s_{\langle\lambda^{\dagger}\rangle}$ vanishes if $\alpha$ is not dominant,
and $s_{\langle\lambda^{\dagger}\rangle}$ is $\pm$ the character of
$V_{\alpha}$ if $\alpha$ is dominant.

######  11.3.11 Definition.

Let

$\mathcal{Q}_{1}(t_{1},\ldots,t_{r};g,q)=(t_{1}\cdots
t_{r})^{g}\sum_{\lambda_{1}\leq
r}T_{\lambda}\,s_{\langle\lambda^{\dagger}(g)\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm}).$

We will see shortly that the sum converges.

######  11.3.12.

One can show using the Weyl dimension formula that

$\dim V_{\lambda}^{\mathrm{Sp}(2r)}=\frac{\prod_{1\leq i<j\leq
r}(\mu_{i}^{2}-\mu_{j}^{2})\prod_{i=1}^{r}\mu_{i}}{(2r-1)!!}$

where $\mu_{i}=\lambda_{i}+r-i+1$. It is straightforward to see from this
expression that for each $\lambda$ with $\lambda_{1}\leq r$, the function

$g\mapsto\dim V_{\lambda^{\dagger}(g)}$

is a polynomial in $g$ of degree $r(r+1)/2$. In particular,
$\mathcal{Q}_{1}(1,\ldots,1;g,q)$ is a polynomial in $g$ of degree $r(r+1)/2$,
for any fixed $q$.

######  11.3.13 Lemma.

Let $V$ be a representation of the braid group $\beta_{n}$. Then $\dim
H_{k}(\beta_{n},V)\leq\binom{n-1}{k}\dim(V)$.

###### Proof.

By Poincaré duality, it suffices to bound the dimension of
$H^{2n-k}_{c}(\mathrm{Conf}_{n}(\mathbf{C}),V)$. Use the cellular chain
complex associated to the Fuks stratification of
$\mathrm{Conf}_{n}(\mathbf{C})$ (see [FN62, Fuk70] and [ETW17, Theorem 4.3]).
It has exactly $\binom{n-1}{k}$ cells of dimension $2n-k$. ∎

######  11.3.14 Lemma.

Let $t_{1},\ldots,t_{r}$ be on the unit circle, and consider a partition
$\lambda$ with $\lambda_{1}\leq r$. Then
$|s_{\langle\lambda^{\dagger}(g)\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})|\leq
4^{rg}\dim S^{\lambda^{\prime}}(\mathbf{C}^{2r})$.

###### Proof.

We have
$\lambda^{\dagger}=(g-\lambda^{\prime}_{r},\ldots,g-\lambda_{1}^{\prime})$.
Let $\alpha$ be as in §11.3.10. Then we can write $\alpha=\beta+\gamma$ with
$\beta\subseteq(g^{r})$ and $\gamma\subseteq\lambda^{\prime}$. Thus,

$|s_{\langle\lambda^{\dagger}(g)\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})|\leq\dim
V_{\alpha}\leq\dim V_{\beta}\dim V_{\gamma}\leq\dim V_{(g^{r})}\dim
V_{\lambda^{\prime}}\leq 4^{gr}\dim S^{\lambda^{\prime}}(\mathbf{C}^{2r})$

using in the last step §11.3.9 for the first factor, and that
$V_{\lambda^{\prime}}$ is a summand of $S^{\lambda^{\prime}}(\mathbf{C}^{2r})$
for the second factor. ∎

######  11.3.15 Lemma.

For all $\varepsilon>0$ and $N\in\mathbf{N}$ there is a bound

$\sum_{|\lambda|=w}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})\dim
S^{\lambda^{\prime}}(\mathbf{C}^{N})\ll_{N,\varepsilon}2^{\varepsilon k}.$

###### Proof.

Consider the generating function of stable Poincaré series

$\mathbf{Exp}\Big{(}\,z^{-1}\,\mathbf{Log}\Big{(}z+\sum_{k\geq
0}z^{k}h_{2k}(\mathrm{t})\Big{)}-h_{2}(\mathrm{t})-1\Big{)}.$

According to 4.4.12, the left-hand side of 11.3.15 is the coefficient of
$z^{k}t^{w}$ of the function obtained by setting $t_{1}=\ldots=t_{N}=t$,
$t_{i}=0$ for $i>N$, in this generating series. By 12.2.4 and the comparison
explained in §12.2.2 the series converges to a holomorphic function on the
region999For our purposes it would be enough to know convergence in the region
$|z|<1$, $|tz|<1$, which is simpler to show than 12.2.4. $|z|<1$,
$|t^{4}z|<1$. Now apply Cauchy’s inequality, i.e. extract the coefficient of
$z^{k}t^{w}$ using Cauchy’s integral formula applied to the circles
$|z|=2^{-\varepsilon}$, $|t|=1$. ∎

######  11.3.16 Definition.

Let $\theta:\mathbf{R}\to\mathbf{R}$ be a monotone increasing function. We say
that $\theta$ is a _uniform stability bound_ if for each $n\in\mathbf{N}$,

$H_{k}(\beta_{n},V_{\lambda})\to H_{k}(\beta_{n+1},V_{\lambda})$

is an isomorphism for all $k\leq\theta(n)$, for all $\lambda$.

######  11.3.17.

Let $\theta$ be a uniform stability bound. Then
$H_{k}(\beta_{\infty},V_{\lambda})\cong H_{k}(\beta_{n},V_{\lambda})$ for all
$k\leq\theta(n)$. But observe that $V_{\lambda}$ is zero as a representation
of $\beta_{n}$ if $\tfrac{n}{2}<\ell(\lambda)$. Thus,
$H_{k}(\beta_{\infty},V_{\lambda})=0$ for $k<\theta(2\ell(\lambda))$. In
7.0.12 we proved that $H_{k}(\beta_{\infty},V_{\lambda})=0$ for
$k<\ell(\lambda)/2$, and this bound is sharp. Thus, without loss of generality
a uniform stability bound must satisfy
$\theta(n)\leq\lfloor\tfrac{n}{4}\rfloor$. Our conjecture is that this bound
is close to sharp.

######  11.3.18 Conjecture.

There exists a uniform stability bound $\theta(n)=\tfrac{1}{4}n-c$, where
$c\geq 0$ is an absolute constant.

######  11.3.19 Theorem.

Suppose that $\theta$ is a uniform stability bound. Then for any
$t_{1},\ldots,t_{r}$ on the unit circle

$\frac{1}{q^{2g+1}}\sum_{d\in\mathscr{P}_{2g+1}}\prod_{i=1}^{r}\mathcal{L}(q^{-1/2}t_{i},\chi_{d})=\mathcal{Q}_{1}(t;g,q)+O_{\theta}\Big{(}4^{g(r+1)}q^{-\theta(2g+1)/2}\Big{)},$

the implied constant depending upon $\theta$ in the $O$-symbol being
explicitly computable.

###### Proof.

The left-hand side of 11.3.19 equals

$(t_{1}\cdots
t_{r})^{g}\sum_{\lambda\subseteq(r^{g})}\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{g}^{1,0},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2})))s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})$

and $\mathcal{Q}_{1}(t,g)$ equals

$(t_{1}\cdots t_{r})^{g}\sum_{\lambda_{1}\leq
r}\sum_{k}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2})))s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm}).$

Subtracting the two, all terms with $k\leq\theta(2g+1)$ cancel by homological
stability. To estimate the difference, we first consider the contribution from
the unstable homology. Using the Deligne bounds on Frobenius eigenvalues from
[Del80], 11.3.13, and the identity of §11.3.9, we get

$\displaystyle\left|(t_{1}\cdots
t_{r})^{g}\sum_{\lambda\subseteq(r^{g})}\sum_{k>\theta(2g+1)}(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{g}^{1,0},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2})))s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})\right|$
$\displaystyle\leq$
$\displaystyle\sum_{\lambda\subseteq(r^{g})}\sum_{k>\theta(2g+1)}q^{-k/2}\dim
H_{k}(\mathcal{H}_{g}^{1,0},\mathbb{V}_{\lambda})\dim
V_{\lambda^{\dagger}}^{\mathrm{Sp}(2r)}$ $\displaystyle\leq$
$\displaystyle\underbrace{\sum_{k>\theta(2g+1)}q^{-k/2}\binom{2g}{k}}_{\ll\,2^{2g}\,q^{-\theta(2g+1)/2}}\underbrace{\sum_{\lambda\subseteq(r^{g})}\dim
V_{\lambda}^{\mathrm{Sp}(2g)}\dim
V_{\lambda^{\dagger}}^{\mathrm{Sp}(2r)}}_{=\,\,4^{gr}}\ll
4^{g(r+1)}q^{-\theta(2g+1)/2}.$

We now have to estimate the contribution from the stable homology, i.e. to
bound the quantity

$(t_{1}\cdots t_{r})^{g}\sum(-1)^{k}\mathrm{Tr}(\mathrm{Frob}_{q}\mid
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda}(\tfrac{|\lambda|}{2})))s_{\langle\lambda^{\dagger}\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})$

where we sum over $k$ and $\lambda$ such that $\lambda_{1}\leq r$ and _either_
$\ell(\lambda)>g$ or $k>\theta(2g+1)$ (or both).

We split this sum into two: first when $|\lambda|\leq 2g$, then when
$|\lambda|>2g$. Recall that the stable homology is pure of weight $-2k$ in
degree $k$. We claim that the first sum can be bounded by

$\sum_{\begin{subarray}{c}|\lambda|\leq 2g\\\ \lambda_{1}\leq
r\end{subarray}}\sum_{k>\theta(2g+1)}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})s_{\langle\lambda^{\dagger}\rangle}(1^{\pm},\ldots,1^{\pm}).$

Indeed, if $\lambda\subseteq(r^{g})$ it is clear that we can start the
summation from $k>\theta(2g+1)$. When $\ell(\lambda)>g$ we instead observe
that the stable homology vanishes below degree $g/2$ by 7.0.12, which is at
least $\theta(2g+1)$ by §11.3.17. Now using 11.3.14 and 11.3.15 we find that

$\displaystyle\sum_{\begin{subarray}{c}|\lambda|\leq 2g\\\ \lambda_{1}\leq
r\end{subarray}}\sum_{k>\theta(2g+1)}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})s_{\langle\lambda^{\dagger}\rangle}(1^{\pm},\ldots,1^{\pm})$
$\displaystyle\leq 4^{gr}$
$\displaystyle\sum_{w=0}^{2g}\sum_{|\lambda|=w}\sum_{k>\theta(2g+1)}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})\dim
S^{\lambda^{\prime}}(\mathbf{C}^{2r})$ $\displaystyle\ll 4^{gr}$
$\displaystyle\sum_{w=0}^{2g}\sum_{k>\theta(2g+1)}q^{-k}2^{\varepsilon k}\ll
4^{gr}(2g+1)2^{\varepsilon\theta(2g+1)}q^{-\theta(2g+1)}$

for all $\varepsilon>0$. The second sum is treated in the exact same way, but
we use instead that $H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})=0$
for $k<|\lambda|/4$ (7.0.2). We obtain the estimate

$\displaystyle\sum_{\begin{subarray}{c}|\lambda|>2g\\\ \lambda_{1}\leq
r\end{subarray}}\sum_{k\geq|\lambda|/4}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})s_{\langle\lambda^{\dagger}\rangle}(1^{\pm},\ldots,1^{\pm})$
$\displaystyle\leq 4^{gr}$
$\displaystyle\sum_{w>2g}\sum_{|\lambda|=w}\sum_{k\geq w/4}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})\dim
S^{\lambda^{\prime}}(\mathbf{C}^{2r})$ $\displaystyle\ll 4^{gr}$
$\displaystyle\sum_{w>2g}\sum_{k\geq w/4}q^{-k}2^{\varepsilon k}\ll
4^{gr}2^{\varepsilon g/2}q^{-g/2}.\qed$

######  11.3.20 Remark.

As stated in 11.3.18, we expect $\theta(n)\approx n/4$, which leads to an
error term in 11.3.19 which is roughly $O(q^{-g/4})$. On the other hand, as we
explain in the next section of the paper, one expects an asymptotic formula
with successive terms $\mathcal{T}_{n}$, in which the first term
$\mathcal{T}_{1}$ accounts for the contribution from stable homology. One also
expects the error term in the asymptotic formula of 11.3.19 to be comparable
to the second term $\mathcal{T}_{2}$, which is approximately $O(q^{-g/2})$.
Looking through the proof of 11.3.19, the discrepancy comes from the
contributions from the homology of $\beta_{n}$ which is _just outside_ the
stable range. In the stable range we know that Frobenius eigenvalues on
$H_{k}$ are $q^{-k}$, but outside the stable range we only have the Deligne
bounds, giving the upper bound $q^{-k/2}$. Our expectation is that in the
regime just outside the stable range we have _secondary stability_ [GKRW19],
and that just as the stable homology is pure of lowest possible weight, also
the homology in the range where secondary stability holds has weights (close
to) the lowest possible. This would account for the mismatch of error terms in
11.3.19 and the next section.

##  12\. Moments and stability predictions

######  12.0.1.

We now turn our attention to the important problem of determining the
asymptotics of the traces $\operatorname{tr}_{\lambda}(g)$ as the genus $g$
grows, for all partitions $\lambda\subseteq(r^{g})$ (i.e., all partitions
whose Young diagram fit inside a rectangle of $g$ rows and $r$ columns), where
$r$ is a fixed positive integer. Since $\operatorname{tr}_{\lambda}(g)=0$ when
$|\lambda|$ is odd, we can consider only the partitions of even weight. When
$|\lambda|\leq 2g-1,$ an asymptotic formula for
$\operatorname{tr}_{\lambda}(g)$ can be obtained by combining Proposition
11.2.11 with the results of Rudnick [Rud10, Theorem 1] and Roditty-Gershon
[RG12, Theorem 1.1]. Obtaining a similar result in a wider range of $\lambda$
seems to be a challenging problem. In fact, to the best of our knowledge, at
present, there is not even a prediction as to what the asymptotic behavior of
these traces of Frobenius should be, when $|\lambda|$ is quite large compared
with $g.$ The only existing result for general partitions comes from the
seminal work of Katz and Sarnak, see [KS99, Theorem 10.8.2]:

######  12.0.2 Theorem (Katz–Sarnak).

There exist positive constants $A(g)$ and $C(g)$ such that

$|\operatorname{tr}_{\lambda}(g)|\leq C(g)(\dim\,V_{\lambda})q^{-1/2}$

for all $\lambda$, as long as $q=\\#\mathbf{F}_{q}\geq A(g).$ Here
$V_{\lambda}$ is the irreducible representation of $\mathrm{USp}(2g)$ with
highest weight $\lambda.$

###  12.1. The general conjecture

######  12.1.1.

To set up some notation, let $A=(A_{ij})_{1\leq i,j\leq r+1}$ be the symmetric
matrix whose entries are

$A_{ij}=\begin{cases}2&\mbox{if $i=j$}\\\ -1&\mbox{if $i\neq j$ and either
$i=r+1$ or $j=r+1$}\\\ 0&\mbox{otherwise.}\end{cases}$

To $A$ one can attach a Kac-Moody Lie algebra $\mathfrak{g}(A)$ with the set
of roots $\Delta,$ and $\\{\alpha_{i}\\}_{1\leq i,j\leq r+1}\subset\Delta$ a
system of simple roots; the reader interested in the general theory of Kac-
Moody algebras can consult [Kac90]. We have that $\det A=-\,2^{r-1}(r-4).$ By
[Kac90, Theorem 4.3], one finds at once that $A$ is of finite, affine, or
indefinite type when $r\leq 3,$ $r=4,$ or $r\geq 5,$ respectively. For
example, the Dynkin diagram of $A$ is $D_{4}$ when $r=3$ and $\tilde{D}_{4}$
when $r=4;$ for $r\geq 5,$ the Dynkin diagram has a star-like shape. Let
$W_{r}$ denote the Weyl group of $\mathfrak{g}(A),$ and let
$\Delta^{\mathrm{re}}_{+}\subset\Delta$ be the set of positive real roots. For
$\alpha\in\Delta^{\mathrm{re}}_{+},\alpha=\sum_{i}k_{i}\alpha_{i}$
($k_{i}\in\mathbf{N}$), let $d(\alpha)=\sum_{i}k_{i}$ denote the height of
$\alpha,$ and for each positive integer $n,$ let $\Phi_{n}$ be the subset of
$\Delta^{\mathrm{re}}_{+}$ with $k_{r+1}=n;$ by [DT20, Lemma 2.2], $\Phi_{n}$
is a finite set. Finally, for each $\alpha\in\Phi_{n},$ we fix a Weyl group
element $w_{\alpha}$ in reduced form of length $l(w_{\alpha})$ such that
$w_{\alpha}(\alpha)=\alpha_{r+1}.$ This choice is possible since all real
roots are in the orbit $W_{r}\alpha_{r+1},$ see [DT20, Lemma 2.3].

######  12.1.2.

Now consider the $r$th moment-sums of quadratic $L$-functions, that is, the
coefficients of the moment generating function $\mathcal{Z}(\mathrm{t},\xi)$.
Initially, we take the variables to be $t_{i}=q^{-s_{i}},$ for $i=1,\ldots,r.$
In [DT20], it has been conjectured that, for $\Re(s_{i})=\frac{1}{2}$ and
$N\geq 1$ an integer, we have

(4)
$\sum_{d\in\mathscr{P}_{2g+1}}\prod_{i=1}^{r}L(s_{i},\chi_{d})\,=\sum_{n\leq
N}Q_{n,r}(\mathrm{s}^{\prime};2g+1,q)\,+\,O_{\Theta,\,q,\,r}\left(q^{(2g+1)(1+\Theta)/2}\right)$

for any $(N+1)^{-1}<\Theta<N^{-1}.$

######  12.1.3.

The terms $Q_{n,r}(\mathrm{s}^{\prime};2g+1,q),$
$\mathrm{s}^{\prime}:=(s_{1},\ldots,s_{r}),$ have a somewhat complicated
expression of the form

$Q_{n,r}(\mathrm{s}^{\prime};2g+1,q)=n^{-1}\cdot\sum_{\alpha\in\Phi_{n}}\bigg{\\{}\sum_{a\in\\{1,\,\theta_{0}\\}}\;\sum_{\zeta_{a}^{n}=\mathrm{sgn}(a)}\frac{\Gamma_{w_{\alpha}}(a;\zeta_{a})}{2^{l(w_{\alpha})}}S_{\alpha}(\mathrm{s}^{\prime},\zeta_{a})\zeta_{a}^{2g+1}\bigg{\\}}\,q^{\frac{(2g+1)(d(\alpha)+1-2\alpha(\mathrm{s}^{\prime}))}{2n}}$

where $\theta_{0}$ is any fixed non-square in $\mathbf{F}_{q}^{\times},$ and
$\mathrm{sgn}(a)=1$ or $-1$ according as $a=1$ or $\theta_{0}.$ In addition,
for $\alpha=\\!\sum k_{i}\alpha_{i}\in\Delta^{\mathrm{re}}_{+},$ we set
$\alpha(\mathrm{s}^{\prime}):=\sum_{i\leq r}k_{i}s_{i}.$ The function
$S_{\alpha}(\mathrm{s}^{\prime},\zeta_{a})$ decomposes as

$S_{\alpha}(\mathrm{s}^{\prime},\zeta_{a})=\prod_{p}S_{p}^{w_{\alpha}}(\mathrm{s}^{\prime},\zeta_{a})$

over all monic irreducibles, with
$S_{p}^{w_{\alpha}}(\mathrm{s}^{\prime},\zeta_{a})$ given in [DT20, Sec.
5.2]. An explicit formula for
$2^{-l(w_{\alpha})}\Gamma_{\\!w_{\alpha}}\\!(a;\zeta_{a})$ can be found in
[DT20, Sec. 5.1]. When $r=1,2,$ the sets $\Phi_{n}$ ($n\geq 2$) are all
empty, and when $r=3$ the same is true, but for $n\geq 3.$ If $r\geq 4,$ then
$\Phi_{n}$ is non-empty for all $n\geq 1.$ Consequently, $Q_{n,1}=Q_{n,2}=0$
for $n\geq 2,$ and $Q_{n,3}=0$ for $n\geq 3.$ For our purposes here, we shall
only need the first two terms of the asymptotic formula of (4), when $r\geq
3$.

######  12.1.4.

Prior to [DT20], Andrade and Keating [AK14, Conjecture 5] made the weaker
conjecture that

$\sum_{d\in\mathscr{P}_{2g+1}}L(\tfrac{1}{2},\chi_{d})^{r}=\sum_{d\in\mathscr{P}_{2g+1}}Q_{r}(\log_{q}|d|)(1+o(1))$

where $Q_{r}$ is an explicit polynomial of degree $r(r+1)/2.$ Note that
$Q_{r}(\log_{q}|d|)=Q_{r}(2g+1)$ for all $d\in\mathscr{P}_{2g+1},$ and thus
the right-hand side is $\sim\\#\mathscr{P}_{2g+1}\cdot Q_{r}(2g+1)$ as
$g\to\infty.$ One checks the compatibility between this term and the main term
$Q_{1,r}$ in (4). This conjecture has been established only for $r\leq 3$
(with explicit power-saving error terms) in [HR92] for $r=1,$ and [Flo17c,
Flo17b] for $r\leq 3;$ in [Flo17a], Florea also obtained a weaker version of
the asymptotic formula for the fourth moment. The only results towards (4) are
the asymptotics established in [Dia19], for the third moment, and in [DPP21],
for a weighted fourth moment.

######  12.1.5.

The connection between the moment-sums of quadratic $L$-functions and the
traces $\operatorname{tr}_{\lambda}(g)$ is given by 11.3.7. See also [DP18,
Theorem 6.4]. To obtain information about the traces
$\operatorname{tr}_{\lambda}(g),$ we shall apply to Equation (4) the Weyl
integration formula, which, for convenience, we now recall, see [Bum04]:

######  12.1.6 Theorem (Weyl integration formula).

Let $\mathbf{G}$ be a compact connected Lie group and $\mathbf{T}$ a maximal
torus. Let $du$ and $dt$ be Haar measures on these groups, normalized so that
their volumes are $1.$ Then, for any $\mathbf{C}$-valued continuous central
function $f$ on $\mathbf{G},$ we have

$\int_{\mathbf{G}}f(u)\,du=\frac{1}{\\#W}\\!\int_{\mathbf{T}}f(t)|\delta(t)|^{2}\,dt$

where $W=N(\mathbf{T})/\mathbf{T}$ is the Weyl group of $\mathbf{G},$ and

$\delta(t)=\prod_{\alpha>0}\left(e^{\alpha(t)/2}-e^{-\alpha(t)/2}\right)$

the product being over the positive roots of $\mathbf{G}.$

######  12.1.7.

We apply Weyl’s formula when
$\mathbf{G}=\mathrm{USp}(2r),\mathbf{T}=\left\\{\operatorname{diag}\left(e^{\pm\sqrt{-1}\theta_{1}},\ldots,e^{\pm\sqrt{-1}\theta_{r}}\right):\theta_{j}\in\mathbf{R}\right\\},$
and

$f(u)=\sum_{\lambda\subseteq(r^{g})}\operatorname{tr}_{\lambda}(g)s_{\langle\lambda^{{\dagger}}\rangle}(u)$

where $s_{\langle\lambda^{{\dagger}}\rangle}$ is the character of the
irreducible representation of $\mathrm{USp}(2r)$ with highest weight
$\lambda^{{\dagger}}.$ Since each $Q_{n,r}$ in (4) is a function of $q^{\pm
s_{i}},$ $i=1,\ldots,r,$ it is convenient to substitute
$q^{-s_{i}}\\!=q^{-1/2}t_{i},$ hence $|t_{i}|=1$ for all $i;$ we denote the
resulting functions multiplied by the normalizing factor $q^{-2g-1}$ by
$\mathcal{Q}_{n}(t;g,q).$ For $n\geq 1,$ we define

$\mathcal{T}_{n}(\lambda^{{\dagger}})=\frac{1}{\\#W}\\!\int_{\mathbf{T}}\mathcal{Q}_{n}(t;g,q)\psi(t)^{-g}\overline{s_{\langle\lambda^{{\dagger}}\rangle}(t)}\,|\delta(t)|^{2}\,dt$

where $\psi:\mathbf{T}\to\mathbf{C}^{\times}$ is the character defined by
$\psi(t)=t_{1}\cdots\,t_{r},$ and $t=\operatorname{diag}(t_{1}^{\pm
1},\ldots,t_{r}^{\pm 1}).$ Then Proposition 11.3.7 and the conjectural
asymptotic formula (4), combined with the orthonormality of the characters
$s_{\langle\lambda^{{\dagger}}\rangle},$ lead to the following:

######  12.1.8 Conjecture.

For every $\lambda\subseteq(r^{g})$ we have

$\operatorname{tr}_{\lambda}(g)=\sum_{n\leq
N}\mathcal{T}_{n}(\lambda^{{\dagger}})\,+\,O_{\Theta,\,q,\,r}\left(q^{(2g+1)(\Theta-1)/2}\right)\\!.$

######  12.1.9.

While the asymptotic formula (4) has a regular behavior in the sense that
$Q_{n,r}(\mathrm{s}^{\prime};2g+1,q)$ dominates the term
$Q_{n+1,r}(\mathrm{s}^{\prime};2g+1,q)$ for all $n\geq 1,$ as either $g$ or
$q$ gets large, the asymptotics of $\operatorname{tr}_{\lambda}(g)$ behave
quite differently depending upon various regimes of the ratio $|\lambda|/g.$
In what follows, we begin to investigate the asymptotic behavior of
$\operatorname{tr}_{\lambda}(g)$ by analyzing closely the first two terms
$\mathcal{T}_{1}(\lambda^{{\dagger}})$ and
$\mathcal{T}_{2}(\lambda^{{\dagger}})$ in the above conjecture. Our analysis
of the first term, in particular, also suggests that the family
$\\{\mathcal{H}_{g}^{1,0}\\}$ satisfies homological stability for the twisted
coefficients $\mathbb{V}_{\lambda}$ in a range independent of $\lambda,$ as in
11.3.18.

###  12.2. The first term

######  12.2.1.

To study $\mathcal{T}_{1}(\lambda^{{\dagger}}),$ we first note that the set
$\Phi_{1}$ consists of the roots of the form $\alpha=\sum_{i\leq
r}k_{i}\alpha_{i}+\alpha_{r+1},$ with $k_{i}=0$ or $1$ for all $i=1,\ldots,r.$
From [DT20, Section 7], one finds easily that

$\mathcal{Q}_{1}(t;g,q)\psi(t)^{-g}=(1-q^{-1})\sum_{\epsilon}\frac{R_{1}(q^{-1/2}t_{1}^{\epsilon_{1}},\ldots,q^{-1/2}t_{r}^{\epsilon_{r}})}{\prod_{i=1}^{r}t_{i}^{\epsilon_{i}g}\prod_{1\leq
i\leq j\leq r}(1-t_{i}^{\epsilon_{i}}t_{j}^{\epsilon_{j}})}$

the sum being over all tuples
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{-1,1\\}^{r}.$ Here
$R_{1}(\mathrm{t})$ is given by

$R_{1}(\mathrm{t})=\prod_{n\geq 1}B(\mathrm{t}^{n},q^{n})^{i_{n}\\!(q)}$

where

$B(\mathrm{t},q)\;=\prod_{1\leq i\leq j\leq
r}(1-t_{i}t_{j})\,\cdot\,\bigg{(}1+\frac{-\,2+\prod_{i=1}^{r}(1-t_{i})^{-1}+\prod_{i=1}^{r}(1+t_{i})^{-1}}{2(1+q^{-1})}\bigg{)}.$

######  12.2.2.

The function $R_{1}$ is closely related to the generating function of stable
traces. Indeed,

$\displaystyle(1-q^{-1})R_{1}(q^{-1/2}t_{1},\ldots,q^{-1/2}t_{r})$
$\displaystyle=\mathbf{Exp}\Big{(}\,q\,\mathbf{Log}\Big{(}q^{-1}+\sum_{k\geq
0}q^{-k}h_{2k}(\mathrm{t})\Big{)}-h_{2}(\mathrm{t})-1\Big{)}$
$\displaystyle=\sum_{\lambda}T_{\lambda}\,s_{\lambda^{\prime}}(t_{1},\ldots,t_{r}).$

The first equality follows by comparing $R_{1}$ with the expression obtained
by setting $z=1/q$ in 7.0.14, and noting that
$\mathbf{Exp}(-h_{2}(\mathrm{t}))=\prod_{i\leq j}(1-t_{i}t_{j})$, cf. §2.6.8.
The second equality is 4.4.12. Combining this last expression with the
following lemma, we also see that $\mathcal{Q}_{1}$ agrees with the function
defined in the previous section.

######  12.2.3 Lemma.

For any partition $\lambda$ with $\lambda_{1}\leq r$, any $g$, and
indeterminates $t_{1},\ldots,t_{r}$ one has

$s_{\langle\lambda^{{\dagger}}(g)\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})=\sum_{\epsilon}\frac{s_{\lambda^{\prime}}(t^{\epsilon_{1}}_{1},\ldots,t_{r}^{\epsilon_{r}})}{\prod_{i=1}^{r}t_{i}^{\epsilon_{i}g}\prod_{1\leq
i\leq j\leq r}(1-t_{i}^{\epsilon_{i}}t_{j}^{\epsilon_{j}})},$

the sum being over all tuples
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{-1,1\\}^{r}.$

###### Proof.

Recall that

$s_{\langle\lambda^{\dagger}(g)\rangle}(t_{1}^{\pm},\ldots,t_{r}^{\pm})=\frac{\det\left(t_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-t_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq r}}{\mathrm{det}\\!\left(t_{i}^{r+1-j}-t_{i}^{-(r+1-j)}\right)_{1\leq
i,j\leq r}}.$

Write the determinant in the numerator as

$\det\left(t_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-t_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq
r}=\;\sum_{\epsilon}\,\prod_{k=1}^{r}\epsilon_{k}\det\left(t_{i}^{\epsilon_{i}(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq r}$

and note that the denominator can be expressed as

$\det\left(t_{i}^{r+1-j}-t_{i}^{-(r+1-j)}\right)_{1\leq i,j\leq
r}=(-1)^{r(r+1)/2}(t_{1}\cdots\,t_{r})^{-r}\prod_{1\leq i\leq j\leq
r}(1-t_{i}t_{j})\prod_{1\leq k<l\leq r}(t_{k}-t_{l}).$

In the numerator, replace $t_{i}$ by $1/t_{i}$ for all $i,$ and note that

$\det\left(t_{i}^{r+1-j}-t_{i}^{-(r+1-j)}\right)_{1\leq i,j\leq
r}=\prod_{k=1}^{r}\epsilon_{k}\det\left(t_{i}^{\epsilon_{i}(r+1-j)}-t_{i}^{-\epsilon_{i}(r+1-j)}\right)_{1\leq
i,j\leq r}.\qed$

######  12.2.4 Lemma.

For sufficiently small $\varepsilon>0,$ the function $R_{1}(\mathrm{t})$ is
holomorphic on the polydisc $|t_{i}|\leq q^{-\frac{1}{4}-\varepsilon},$ for
$i=1,\ldots,r.$ Moreover, on this polydisc, we have the upper bound

$|R_{1}(\mathrm{t})|<\big{(}1-q^{-4\varepsilon}\big{)}^{-2^{1+r(r+1)/2}}.$

###### Proof.

Using the simple equality

$1+\frac{x-2}{2(1+q^{-1})}=\frac{x}{2}+\frac{2-x}{2(q+1)}$

with $x=\prod_{i=1}^{r}(1-t_{i})^{-1}+\prod_{i=1}^{r}(1+t_{i})^{-1},$ we can
write $B=B_{1}+B_{2},$ where

$B_{1}(\mathrm{t},q)=\frac{1}{2}\prod_{1\leq i<j\leq
r}(1-t_{i}t_{j})\,\cdot\,\bigg{(}\prod_{i=1}^{r}(1+t_{i})\,+\,\prod_{i=1}^{r}(1-t_{i})\bigg{)}$

and $B_{2}(\mathrm{t},q)$ is given by

$\frac{2\prod_{1\leq i\leq j\leq r}(1-t_{i}t_{j})\,-\,\prod_{1\leq i<j\leq
r}(1-t_{i}t_{j})\prod_{i=1}^{r}(1+t_{i})\,-\,\prod_{1\leq i<j\leq
r}(1-t_{i}t_{j})\prod_{i=1}^{r}(1-t_{i})}{2(q+1)}.$

Note that $B_{2}$ has no constant term and that all monomials of degree two in
$B_{1}$ cancel out, that is to say $B_{1}-1$ has no monomial of degree smaller
than four. Thus, if $|t_{i}|\leq q^{\,-\frac{1}{4}-\varepsilon}$ for
$i=1,\ldots,r,$ we have the estimate

$\begin{split}|B(\mathrm{t},q)-1|&\leq|B_{1}(\mathrm{t},q)-1|+|B_{2}(\mathrm{t},q)|\\\
&\leq 2^{-1+r(r+1)/2}q^{-1-4\varepsilon}\,+\,3\cdot
2^{-1+r(r+1)/2}q^{-\frac{3}{2}-2\varepsilon}\\\
&<2^{1+r(r+1)/2}q^{-1-4\varepsilon}.\end{split}$

Applying the inequality $i_{n}(q)\leq q^{n}/n,$ which follows from the well-
known formula $\sum_{d\mid n}di_{d}(q)=q^{n}$ ($n\geq 1$), and the familiar
estimate $\log(1+y)<y$ for $y>0,$ we have

$\log|R_{1}(\mathrm{t})|<2^{1+r(r+1)/2}\sum_{n=1}^{\infty}\frac{q^{n-n(1+4\varepsilon)}}{n}=2^{1+r(r+1)/2}\log\big{(}1-q^{-4\varepsilon}\big{)}^{-1}.$

This implies the absolute and uniform convergence of the product
$R_{1}(\mathrm{t}).$ The upper bound follows by exponentiating the last
inequality. ∎

######  12.2.5 Remark.

By multiplying successively $R_{1}(\mathrm{t})$ by suitable products of zeta
functions, or inverses of such, one can extend $R_{1}(\mathrm{t})$ even
further. For example, the product

$\begin{split}\prod_{1\leq i_{1}<i_{2}\leq
r}\\!\left(1-z_{i_{1}}^{2}z_{i_{2}}^{2}\right)^{-1}&\prod_{1\leq
i_{1}<i_{2}<i_{3}\leq
r}\\!\left(1-z_{i_{1}}^{2}z_{i_{2}}z_{i_{3}}\right)^{-1}\\!\left(1-z_{i_{1}}z_{i_{2}}^{2}z_{i_{3}}\right)^{-1}\\!\left(1-z_{i_{1}}z_{i_{2}}z_{i_{3}}^{2}\right)^{-1}\\\
&\cdot\prod_{1\leq i_{1}<i_{2}<i_{3}<i_{4}\leq
r}\\!\left(1-z_{i_{1}}z_{i_{2}}z_{i_{3}}z_{i_{4}}\right)^{-2}\cdot\,B(\mathrm{t},q)\end{split}$

is $1+O_{r}\left(q^{-1-6\varepsilon}\right)$ for $|t_{i}|\leq
q^{-\frac{1}{6}-\varepsilon},$ from which it follows at once that
$R_{1}(\mathrm{t})$ is holomorphic on any such polydisc, with small positive
$\varepsilon.$ This observation could be used to improve the estimates in
Corollary 12.2.13 and 12.2.16 below. However, to simplify the arguments, we
shall only make use of 12.2.4.

######  12.2.6 Lemma.

Let $a_{1},\ldots,a_{r}$ be distinct non-zero complex numbers such that
$a_{i}a_{j}\neq 1$ for all $1\leq i,j\leq r.$ Suppose $h$ is a symmetric
function of $r$ variables, holomorphic on a domain containing
$\left(a_{1}^{\epsilon_{1}},\ldots,a_{r}^{\epsilon_{r}}\right)$ for all
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{\pm 1\\}^{r}.$ Then

$\sum_{\epsilon}\frac{h(a_{1}^{\epsilon_{1}},\ldots,a_{r}^{\epsilon_{r}})}{\prod_{1\leq
i\leq j\leq
r}(1-a_{i}^{\epsilon_{i}}a_{j}^{\epsilon_{j}})}=\frac{(-1)^{r(r+1)/2}}{r!}\frac{1}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\cdots\oint
h(\mathrm{z})\frac{\prod_{1\leq i<j\leq
r}(z_{j}-z_{i})^{2}(1-z_{i}z_{j})}{\prod_{i,j=1}^{r}(1-z_{i}a_{j})(1-z_{i}a_{j}^{-1})}\,\frac{d\mathrm{z}}{\mathrm{z}^{r}}$

where each path of integration encloses the $a_{j}^{\pm 1},$ but not
$z_{j}\\!=0.$

###### Proof.

This is [DT20, Lemma 7.1]. ∎

######  12.2.7.

We apply Lemma 12.2.6 to the function
$R_{1}(q^{-1/2}\mathrm{z})\prod_{i=1}^{r}z_{i}^{-g}$ with $a_{i}=t_{i}.$ By
Lemma 12.2.4, the paths of integration can be taken to be the boundaries of
$\rho_{1}<|z_{i}|<\rho_{2}$ with the positive orientations; we are assuming
that $\rho_{1}<1<\rho_{2},$ with $\rho_{2}-\rho_{1}$ sufficiently small. It
follows at once that the average

(5)
$\sum_{\epsilon}\frac{R_{1}(q^{-1/2}t_{1}^{\epsilon_{1}},\ldots,q^{-1/2}t_{r}^{\epsilon_{r}})}{\prod_{i=1}^{r}t_{i}^{\epsilon_{i}g}\prod_{1\leq
i\leq j\leq r}(1-t_{i}^{\epsilon_{i}}t_{j}^{\epsilon_{j}})}$

is holomorphic in a neighborhood of $|t_{i}|=1$ $(1\leq i\leq r).$ In
particular, the function $\mathcal{Q}_{1}(t;g,q)\psi(t)^{-g}$ is continuous on
$\mathbf{T}.$

######  12.2.8.

The following proposition provides an explicit formula for the first term
$\mathcal{T}_{1}(\lambda^{{\dagger}})$ in Conjecture 12.1.8.

######  12.2.9 Proposition.

For every partition $\lambda\subseteq(r^{g})$ we have

(6)
$\displaystyle\mathcal{T}_{1}(\lambda^{{\dagger}})=\frac{(1-q^{-1})}{r!}\cdot\frac{(-1)^{r(r-1)/2}}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}R_{1}(q^{-1/2}\mathrm{z})\prod_{1\leq
i<j\leq r}(z_{i}-z_{j})$
$\displaystyle\cdot\,\mathrm{det}\left(z_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-z_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq r}\,\frac{d\mathrm{z}}{\mathrm{z}^{g+r+1}}.$

###### Proof.

By definition

$\mathcal{T}_{1}(\lambda^{{\dagger}})=\frac{1}{\\#W}\\!\int_{\mathbf{T}}\mathcal{Q}_{1}(t;g,q)\psi(t)^{-g}\overline{s_{\langle\lambda^{{\dagger}}\rangle}(t)}\,|\delta(t)|^{2}\,dt.$

The function $\mathcal{Q}_{1}(t;g,q)\psi(t)^{-g}$ is equal to the average (5)
multiplied by the factor $(1-q^{-1})$. Using Lemma 12.2.6, we express (5) as
an $r$-fold integral over the boundaries of $\rho_{1}<|z_{i}|<\rho_{2}$ with
the positive orientations. As above, we take $\rho_{1}<1<\rho_{2},$ with
$\rho_{2}-\rho_{1}$ sufficiently small. This integral can be decomposed as a
sum of $2^{r}$ $r$-fold integrals over circles $|z_{i}|=\rho_{k}$ ($k=1,2$).
Pick one of them, say when the first $n$ integrals are over $|z_{i}|=\rho_{1}$
($i=1,\ldots,n$), while the remaining ones are over $|z_{i}|=\rho_{2}.$ To the
denominator of the expression under the integral, we apply Cauchy’s identity
for $\mathrm{Sp}(2r)$ (see §2.6.8), which states that

$\prod_{i,j=1}^{r}(1-t_{i}x_{\\!j})^{-1}(1-t_{i}x_{\\!j}^{-1})^{-1}\,=\prod_{1\leq
i<j\leq r}(1-t_{i}t_{\\!j})^{-1}\cdot\sum_{\begin{subarray}{c}\mu\\\
l(\mu)\leq r\end{subarray}}s_{\langle\mu\rangle}(x_{1}^{\pm
1}\\!,\ldots,x_{r}^{\pm 1})s_{\mu}(t_{1},\ldots,t_{r}).$

To ensure the absolute convergence of the series involved, we change $z_{i}$
to $z_{i}^{-1}$ for $n+1\leq i\leq r,$ and apply the simple identity

$\frac{1}{(1-z_{i}a_{j})(1-z_{i}a_{j}^{-1})}=\frac{z_{i}^{-2}}{(1-z_{i}^{-1}a_{j})(1-z_{i}^{-1}a_{j}^{-1})}.$

Accordingly, if we set

$\epsilon_{i}=\begin{cases}1&\text{if $1\leq i\leq n$}\\\ -1&\text{if $n+1\leq
i\leq r$}\end{cases}$

then by performing the integral over $\mathbf{T},$ using the orthonormality of
the symplectic Schur functions, we can write the contribution to
$\mathcal{T}_{1}(\lambda^{{\dagger}})$ as

$\begin{split}I_{\epsilon}:=\frac{(1-q^{-1})}{r!}\frac{(-1)^{\frac{r(r+1)}{2}+n}}{\left(2\pi\sqrt{-1}\right)^{r}}\oint\limits_{|z_{r}|=\rho_{2}}\cdots\oint\limits_{|z_{1}|=\rho_{1}}R_{1}(q^{-1/2}\mathrm{z})&\frac{\prod_{1\leq
i<j\leq r}(z_{j}-z_{i})^{2}(1-z_{i}z_{j})}{\prod_{1\leq i<j\leq
r}\big{(}1-z_{i}^{\epsilon_{i}}z_{j}^{\epsilon_{j}}\big{)}}\\\
&\cdot\prod_{k=1}^{r}z_{k}^{-r(1-\epsilon_{k})}\cdot
s_{\lambda^{\dagger}}(\mathrm{z^{\epsilon}})\,\frac{d\mathrm{z}}{\mathrm{z}^{g+r}}\end{split}$

where $\epsilon=(\epsilon_{1},\ldots,\epsilon_{r}).$ Notice that
$(-1)^{n}=(-1)^{-r}\prod_{i=1}^{r}\epsilon_{i}.$ Using the identities

$\frac{(z_{i}-z_{j})(1-z_{i}z_{j})}{1-z_{i}^{\epsilon_{i}}z_{j}^{\epsilon_{j}}}=z_{i}^{1-\epsilon_{i}}z_{j}^{1-\epsilon_{j}}\big{(}z_{i}^{\epsilon_{i}}-z_{j}^{\epsilon_{j}}\big{)}\;\;\;\text{and}\;\;\prod_{1\leq
i<j\leq
r}z_{i}^{1-\epsilon_{i}}z_{j}^{1-\epsilon_{j}}=\prod_{k=1}^{r}z_{k}^{(r-1)(1-\epsilon_{k})}$

we can simplify $I_{\epsilon}$ to:

$\begin{split}I_{\epsilon}:=\frac{(1-q^{-1})}{r!}\frac{(-1)^{\frac{r(r-1)}{2}}\prod_{i=1}^{r}\epsilon_{i}}{\left(2\pi\sqrt{-1}\right)^{r}}\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}R_{1}(q^{-1/2}\mathrm{z})&\prod_{1\leq
i<j\leq
r}(z_{i}-z_{j})\big{(}z_{i}^{\epsilon_{i}}-z_{j}^{\epsilon_{j}}\big{)}\\\
&\cdot\prod_{k=1}^{r}z_{k}^{\epsilon_{k}}\cdot
s_{\lambda^{\dagger}}(\mathrm{z^{\epsilon}})\,\frac{d\mathrm{z}}{\mathrm{z}^{g+r+1}}.\end{split}$

Also, note that the product $\prod_{1\leq i<j\leq
r}\big{(}z_{i}^{\epsilon_{i}}-z_{j}^{\epsilon_{j}}\big{)}$ cancels the
denominator of $s_{\lambda^{\dagger}}(\mathrm{z^{\epsilon}}).$ Summing now
over all tuples
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{-1,1\\}^{r},$ the formula
for $\mathcal{T}_{1}(\lambda^{{\dagger}})$ is an immediate consequence of the
simple identity

$\mathrm{det}\\!\left(z_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-z_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq
r}=\;\sum_{\epsilon}\,\prod_{k=1}^{r}\epsilon_{k}z_{k}^{\epsilon_{k}}\,\cdot\prod_{1\leq
i<j\leq r}\big{(}z_{i}^{\epsilon_{i}}-z_{j}^{\epsilon_{j}}\big{)}\cdot
s_{\lambda^{\dagger}}(\mathrm{z^{\epsilon}})$

which follows from the multilinearity of the determinant. This completes the
proof. ∎

######  12.2.10.

As a first corollary, we obtain the expected connection between the stable
traces $T_{\lambda}$ and $\mathcal{T}_{1}(\lambda^{{\dagger}}).$

######  12.2.11 Corollary.

For any fixed partition $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots)$ with
$\lambda_{1}\leq r,$ we have

(7)
$\begin{split}&\lim_{g\to\infty}\mathcal{T}_{1}(\lambda^{{\dagger}})=T_{\lambda}=\frac{(1-q^{-1})}{r!}\cdot\frac{(-1)^{r(r-1)/2}}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}R_{1}(q^{-1/2}\mathrm{z})\prod_{1\leq
i<j\leq r}(z_{i}-z_{j})\\\ &\hskip
140.0pt\cdot\,\mathrm{det}\Big{(}z_{i}^{-\lambda_{r+1-j}^{\prime}+r+1-j}\Big{)}_{1\leq
i,j\leq r}\,\frac{d\mathrm{z}}{\mathrm{z}^{r+1}}.\end{split}$

###### Proof.

In formula (6), divide the determinant by $\mathrm{z}^{g}.$ Applying the
Riemann-Lebesgue lemma, we see at once that the limit and the $r$-fold
integral in (7) are equal.

To see that the integral is also equal to $T_{\lambda},$ divide the
determinant by $\mathrm{z}^{r},$ and note that

$(-1)^{r(r-1)/2}\mathrm{det}\Big{(}z_{i}^{\lambda_{r+1-j}^{\prime}+j-1}\Big{)}_{1\leq
i,j\leq r}=\;\mathrm{det}\Big{(}z_{i}^{\lambda_{j}^{\prime}+r-j}\Big{)}_{1\leq
i,j\leq r}=\;s_{\lambda^{\prime}}(\mathrm{z})\,\cdot\prod_{1\leq i<j\leq
r}(z_{i}-z_{j}).$

Recalling that $R_{1}(\mathrm{z})$ is the Schur-generating function of the
stable traces $T_{\lambda},$ it follows at once from the Peter-Weyl theorem
[Bum04], applied to the compact group $\mathrm{U}(r),$ that the integral in
(7) can be decomposed as

$\frac{(1-q^{-1})}{r!}\sum_{\begin{subarray}{c}\mu\\\ l(\mu^{\prime})\leq
r\end{subarray}}T_{\mu}\cdot\frac{1}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}s_{\mu^{\prime}}(\mathrm{z})\overline{s_{\lambda^{\prime}}(\mathrm{z})}\prod_{1\leq
i<j\leq r}|z_{i}-z_{j}|^{2}\,\frac{d\mathrm{z}}{\mathrm{z}}.$

Letting $z_{i}=e^{2\pi\sqrt{-1}\theta_{i}},$ $0\leq\theta_{i}\leq 1,$ and
using the orthonormality of the Schur functions, one finds that the last
integral is $1$ or $0$ according as $\mu=\lambda$ or not, which completes the
proof. ∎

######  12.2.12.

From the integral representation (7) of $T_{\lambda},$ we deduce the following
estimate:

######  12.2.13 Corollary.

For any partition $\lambda$ with $\lambda_{1}=l(\lambda^{\prime})\leq r,$ we
have the estimate

$|T_{\lambda}|\ll_{r,\,\varepsilon}q^{\left(-\frac{1}{4}+\varepsilon\right)|\lambda|}.$

###### Proof.

By Lemma 12.2.4, we can integrate in (7) over
$|z_{i}|=q^{\frac{1}{4}-\varepsilon}$ (instead of $|z_{i}|=1$), for
$i=1,\ldots,r,$ and after the simple change of variables $z_{i}\to
q^{\frac{1}{4}-\varepsilon}z_{i},$ we can write

$\begin{split}T_{\lambda}=\frac{(1-q^{-1})q^{\left(-\frac{1}{4}+\varepsilon\right)|\lambda|}}{r!}\cdot\frac{(-1)^{r(r-1)/2}}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}&R_{1}\big{(}q^{-\frac{1}{4}-\varepsilon}\mathrm{z}\big{)}\prod_{1\leq
i<j\leq r}(z_{i}-z_{j})\\\
&\cdot\,\mathrm{det}\Big{(}z_{i}^{-\lambda_{r+1-j}^{\prime}+r+1-j}\Big{)}_{1\leq
i,j\leq r}\,\frac{d\mathrm{z}}{\mathrm{z}^{r+1}}.\end{split}$

Applying the upper bound in Lemma 12.2.4, and the Hadamard inequality for
determinants, we have

$\left|R_{1}\big{(}q^{-\frac{1}{4}-\varepsilon}\mathrm{z}\big{)}\prod_{1\leq
i<j\leq
r}(z_{i}-z_{j})\cdot\,\mathrm{det}\Big{(}z_{i}^{-\lambda_{r+1-j}^{\prime}+r+1-j}\Big{)}_{1\leq
i,j\leq r}\right|<r^{r}\big{(}1-q^{-4\varepsilon}\big{)}^{-2^{1+r(r+1)/2}}$

and our assertion follows. ∎

######  12.2.14 Remark.

This estimate can also be seen from 7.0.2 and 11.3.15, although upon
sufficient unwinding the proofs are not so different. Indeed, if $|\lambda|=w$
and $\lambda_{1}\leq r$ then $T_{\lambda}$ is bounded by

$\displaystyle\sum_{|\lambda|=w}|T_{\lambda}|\dim
S^{\lambda^{\prime}}(\mathbf{C}^{r})$
$\displaystyle\leq\sum_{|\lambda|=w}\sum_{k\geq w/4}q^{-k}\dim
H_{k}(\mathcal{H}_{\infty}^{1,0},\mathbb{V}_{\lambda})\dim
S^{\lambda^{\prime}}(\mathbf{C}^{r})$ $\displaystyle\leq\sum_{k\geq
w/4}q^{-k}2^{\varepsilon k}$

for any $\varepsilon>0$. Similarly one may show, using 7.0.12 instead of
7.0.2, that
$|T_{\lambda}|\ll_{r,\varepsilon}q^{(-\frac{1}{2}+\varepsilon)\ell(\lambda)}$.

######  12.2.15 Remark.

Notice that, by Remark 12.2.5, one can use the holomorphy of
$R_{1}(\mathrm{t})$ in a larger polydisc to improve the estimate in Corollary
12.2.13, and hence the range of $\lambda$ for which
$H^{k}(\mathcal{H}_{g}^{1,0},\mathbb{V}_{\lambda})=0$ for $k<\frac{g}{2}.$ For
instance, using the holomorphy of $R_{1}(q^{-1/2}\mathrm{z})$ on the polydisc
$|z_{i}|\leq q^{\frac{1}{3}-\varepsilon},$ $i=1,\ldots,r,$ one can show as in
the proof of Corollary 12.2.13 that

$|T_{\lambda}|\ll_{r,\,\varepsilon}q^{\left(-\frac{1}{3}+\varepsilon\right)|\lambda|+\frac{r^{2}(r^{2}-1)}{36}}$

and the corresponding bound on $|\lambda|$ is obtained by imposing the
condition

$q^{\left(-\frac{1}{3}+\varepsilon\right)|\lambda|+r^{2}(r^{2}-1)/{36}}<q^{-{g}/{2}}$

which, upon replacing $\varepsilon$ by $\varepsilon/4,$ is implied by
$|\lambda|>(1+\varepsilon)\\!\left(3g/2+r^{2}(r^{2}-1)/12\right)$.

######  12.2.16 Corollary.

For any partition $\lambda$ with $\lambda_{1}=l(\lambda^{\prime})\leq r,$ we
have the estimate

$|\mathcal{T}_{1}(\lambda^{{\dagger}})-T_{\lambda}|\ll_{r,\,\varepsilon}q^{\left(-\frac{1}{4}+\varepsilon\right)|\lambda|-\left(\frac{1}{2}-2\varepsilon\right)\left(g-\lambda_{1}^{\prime}+1\right)}.$

###### Proof.

For a partition $\mu=(\mu_{1}\geq\cdots\geq\mu_{r}\geq 0),$ we set for
notational convenience

$a_{\mu}(\mathrm{z})=\mathrm{det}\Big{(}z_{i}^{\mu_{j}+r+1-j}\Big{)}_{1\leq
i,j\leq r}.$

Then, by the determinant identity in the end of the proof of 12.2.9 we have
that

$\left|\,\det\left(z_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-z_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq
r}-\;a_{\lambda^{{\dagger}}}(\mathrm{z})\,\right|\,\leq\,\sum_{\epsilon}\,|a_{\lambda^{{\dagger}}}(\mathrm{z^{\epsilon}})|$

the sum being over all tuples
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{-1,1\\}^{r}$ with at least
one component $\epsilon_{i}=-1.$ Assuming with no loss of generality that
$\epsilon_{1}=-1,$ we can estimate
$|a_{\lambda^{{\dagger}}}(\mathrm{z^{\epsilon}})|,$ with
$|z_{i}|=q^{\frac{1}{4}-\varepsilon},$ by applying the Laplace expansion along
the first row of $a_{\lambda^{{\dagger}}}(\mathrm{z^{\epsilon}}).$ Notice that
$\big{|}z_{1}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\big{|}\leq
q^{-\left(\frac{1}{4}-\varepsilon\right)(g-\lambda_{1}^{\prime}+1)}$ for all
$j,$ while by Hadamard’s inequality, the corresponding cofactors are all
$\ll_{r}q^{\left(\frac{1}{4}-\varepsilon\right)\left((r-1)g-|\lambda|+\lambda_{1}^{\prime}-1+r(r+1)/2\right)}.$
Thus we get the estimate

$\sum_{\epsilon}\,|a_{\lambda^{{\dagger}}}(\mathrm{z^{\epsilon}})|\,\ll_{r}\,q^{\left(\frac{1}{4}-\varepsilon\right)\left((r-2)g-|\lambda|+2\lambda_{1}^{\prime}-2+r(r+1)/2\right)}.$

Now consider the difference between the integrals (6) and (7). Again, by Lemma
12.2.4, we can integrate over $|z_{i}|=q^{\frac{1}{4}-\varepsilon}\\!,$ and
$|R_{1}(q^{-1/2}\mathrm{z})|<\big{(}1-q^{-4\varepsilon}\big{)}^{-2^{1+r(r+1)/2}}$
on these circles. The remaining piece under the integral is easily seen to be
$\ll_{r}q^{\frac{|\lambda|}{2}-\left(\frac{1}{4}-\varepsilon\right)\left(rg+\frac{r(r+1)}{2}\right)},$
and the estimate in the corollary follows. ∎

######  12.2.17.

The estimates in the last two corollaries imply that

$|\mathcal{T}_{1}(\lambda^{{\dagger}})|\ll_{r,\,\varepsilon}q^{-\left(\frac{1}{4}-\varepsilon\right)|\lambda|}$

for any partition $\lambda$ with $\lambda_{1}=l(\lambda^{\prime})\leq r.$
This, combined with Conjecture 12.1.8, suggests that the traces
$\operatorname{tr}_{\lambda}(g)$ indeed decrease as the weight of $\lambda$
grows with $g$. Again, the estimate in Corollary 12.2.16 (and hence also this
one) can be improved to

$|\mathcal{T}_{1}(\lambda^{{\dagger}})-T_{\lambda}|\ll_{r,\,\varepsilon}q^{\left(-\frac{1}{3}+\varepsilon\right)|\lambda|-\left(\frac{2}{3}-2\varepsilon\right)\left(g-\lambda_{1}^{\prime}+1\right)+\frac{r^{2}(r^{2}-1)}{36}}.$

###  12.3. The second term

######  12.3.1.

To analyze the second term $\mathcal{T}_{2}(\lambda^{{\dagger}})$ in the
asymptotic formula of $\operatorname{tr}_{\lambda}(g),$ one first checks that
the set of roots $\Phi_{2}$ is

$\Phi_{2}=\Bigg{\\{}\\!\sum_{\begin{subarray}{c}j=1\\\ j\neq
j_{1},\,j_{2},\,j_{3}\end{subarray}}^{r}\\!\\!\\!k_{j}\alpha_{j}+\alpha_{j_{1}}\\!+\alpha_{j_{2}}\\!+\alpha_{j_{3}}\\!+2\alpha_{r+1}:\text{$1\leq
j_{1}<j_{2}<j_{3}\leq r$, and $k_{j}\in\\{0,2\\}$ for
$j\neq\\!j_{1},j_{2},j_{3}$}\Bigg{\\}}$

see [DT20]. As in the case of $\mathcal{T}_{1}(\lambda^{{\dagger}}),$ we shall
represent $\mathcal{Q}_{2}(t;g,q)\psi(t)^{-g}$ as an integral. The analogue of
the function $R_{1}(\mathrm{t})$ is defined as follows. For $\zeta$ a $4$th
root of $1,$ define

$\begin{split}&\mathrm{d}_{1}=\frac{\left(1-\frac{\xi_{1}}{\zeta
q^{3/4}\xi_{2}\xi_{3}}\right)\\!\left(1-\frac{\xi_{2}}{\zeta
q^{3/4}\xi_{1}\xi_{3}}\right)\\!\left(1-\frac{\xi_{3}}{\zeta
q^{3/4}\xi_{1}\xi_{2}}\right)}{\left(1-\frac{\zeta\xi_{2}\xi_{3}}{q^{1/4}\xi_{1}}\right)\\!\left(1-\frac{\zeta\xi_{1}\xi_{3}}{q^{1/4}\xi_{2}}\right)\\!\left(1-\frac{\zeta\xi_{1}\xi_{2}}{q^{1/4}\xi_{3}}\right)},\;\;\;\mathrm{d}_{2}=\frac{\zeta}{q^{3/4}\xi_{1}\xi_{2}\xi_{3}}\\\
&\mathrm{d}_{3}=\frac{\left(1+\frac{\xi_{1}}{\zeta
q^{3/4}\xi_{2}\xi_{3}}\right)\\!\left(1+\frac{\xi_{2}}{\zeta
q^{3/4}\xi_{1}\xi_{3}}\right)\\!\left(1+\frac{\xi_{3}}{\zeta
q^{3/4}\xi_{1}\xi_{2}}\right)}{\left(1+\frac{\zeta\xi_{2}\xi_{3}}{q^{1/4}\xi_{1}}\right)\\!\left(1+\frac{\zeta\xi_{1}\xi_{3}}{q^{1/4}\xi_{2}}\right)\\!\left(1+\frac{\zeta\xi_{1}\xi_{2}}{q^{1/4}\xi_{3}}\right)}\end{split}$

and

$\mathrm{e}_{1}=\frac{1-(\zeta q^{3/4}\xi_{1}\xi_{2}\xi_{3})^{-1}}{1-\zeta
q^{\,-1/4}\xi_{1}\xi_{2}\xi_{3}},\;\;\;\mathrm{e}_{2}=(\zeta
q^{1/4}\xi_{1}\xi_{2}\xi_{3})^{-1},\;\;\,\text{and}\;\;\;\,\mathrm{e}_{3}=\frac{1+(\zeta
q^{3/4}\xi_{1}\xi_{2}\xi_{3})^{-1}}{1+\zeta q^{\,-1/4}\xi_{1}\xi_{2}\xi_{3}}.$

We set

$\begin{split}L_{1}&=\,\tfrac{1}{4}\mathrm{e}_{1}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!+2\zeta^{2}\mathrm{d}_{2})\,-\,\tfrac{1}{4}\mathrm{e}_{3}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!-2\zeta^{2}\mathrm{d}_{2})\\\
L_{2}&=\,\tfrac{1}{8}\mathrm{e}_{1}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!+2\zeta^{2}\mathrm{d}_{2})\,+\,\tfrac{1}{8}\mathrm{e}_{3}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!-2\zeta^{2}\mathrm{d}_{2})\,+\,\tfrac{1}{4}\mathrm{e}_{2}(\mathrm{d}_{1}\\!-\mathrm{d}_{3})\\\
L_{3}&=\,\tfrac{1}{8}\mathrm{e}_{1}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!+2\zeta^{2}\mathrm{d}_{2})\,+\,\tfrac{1}{8}\mathrm{e}_{3}(\mathrm{d}_{1}\\!+\mathrm{d}_{3}\\!-2\zeta^{2}\mathrm{d}_{2})\,-\,\tfrac{1}{4}\mathrm{e}_{2}(\mathrm{d}_{1}\\!-\mathrm{d}_{3})\end{split}$

and let $S=S(\xi_{1}^{2},\ldots,\xi_{r}^{2},\zeta^{2},q^{-1/2})$ be defined by

$\begin{split}S=&\,\big{(}1-q^{-1}\big{)}\\!\left\\{\frac{L_{1}}{q^{3/4}\zeta\xi_{1}\xi_{2}\xi_{3}}+\frac{L_{2}}{\prod_{i=1}^{r}(1-q^{-1/2}\xi_{i}^{2})}+\frac{L_{3}}{\prod_{i=1}^{r}(1+q^{-1/2}\xi_{i}^{2})}\right\\}\\\
&\cdot\,\left(1-\frac{\zeta^{2}\xi_{1}^{2}\xi_{2}^{2}\xi_{3}^{2}}{\sqrt{q}}\right)\prod_{1\leq
i\leq
3}\left(1-\frac{\zeta^{2}\xi_{i}^{2}}{\sqrt{q}}\right)\\!\left(1-\frac{\zeta^{2}\xi_{1}^{2}\xi_{2}^{2}\xi_{3}^{2}}{\sqrt{q}\,\xi_{i}^{4}}\right)\\\
&\\!\cdot\;\prod_{i=1}^{3}\prod_{j=4}^{r}\left(1-\frac{\xi_{i}^{2}\xi_{j}^{2}}{q}\right)\\!\left(1-\frac{\xi_{i}^{-2}\xi_{j}^{2}}{q}\right)\prod_{4\leq
k\leq l\leq r}\left(1-\frac{\xi_{k}^{2}\xi_{l}^{2}}{q}\right)\\!.\end{split}$

This function is symmetric in the variables $\xi_{1}^{\pm 1},\xi_{2}^{\pm
1},\xi_{3}^{\pm 1},$ and (separately) in $\xi_{k}$ for $4\leq k\leq r,$ see
[DT20, Section 7]. As the above notation suggests, it is also even in
$\xi_{i}$ ($1\leq i\leq r$) and $\zeta,$ and as a function of $q,$ it is a
polynomial in $q^{-1/2}$ of the form

$S=1+O\Big{(}q^{-\frac{3}{2}}\Big{)}.$

Substitute $t_{i}=\xi_{i}^{2},$ and denote the resulting function by
$S(\mathrm{t},\zeta^{2}\\!,q^{-1/2}).$ We define $R_{2}(\mathrm{t},\zeta)$ by

$R_{2}(\mathrm{t},\zeta)=\prod_{n\geq
1}S(\mathrm{t}^{n}\\!,\zeta^{2n}\\!,q^{-n/2})^{i_{n}\\!(q)}$

the product being absolutely and uniformly convergent for $t_{1},t_{2},t_{3}$
in a small neighborhood of the unit circle, and $|t_{k}|\leq q^{\varepsilon}$
($k=4,\ldots,r$) for sufficiently small positive $\varepsilon.$ Letting

$G(t_{1},t_{2},t_{3},\zeta)=-\,\frac{1+\frac{1}{q}+\frac{\zeta^{2}}{\sqrt{q}}\sum_{j=1}^{3}\left(t_{j}+t_{j}^{-1}\right)}{\left(1-\frac{\zeta^{2}t_{1}t_{2}t_{3}}{\sqrt{q}}\right)\\!\left(1-\frac{\zeta^{2}}{\sqrt{q}\,t_{1}t_{2}t_{3}}\right)\prod_{i=1}^{3}\left(1-\frac{\zeta^{2}t_{1}t_{2}t_{3}}{\sqrt{q}\,t_{i}^{2}}\right)\\!\left(1-\frac{\zeta^{2}t_{i}^{2}}{\sqrt{q}\,t_{1}t_{2}t_{3}}\right)}$

it follows from the computations in [DT20, Section 7] that

$\begin{split}&\mathcal{Q}_{2}(t;g,q)\psi(t)^{-g}=\frac{(1-q^{-1})q^{-g/2-2}}{2^{4}3!(r-3)!}\\\
&\cdot\sum_{\sigma\in\Sigma_{r}}\,\sum_{\begin{subarray}{c}\zeta=1,\sqrt{-1}\\\
\epsilon_{\sigma(i)}=\pm
1\end{subarray}}\,\frac{\zeta^{2g}G\\!\left(t_{\sigma(1)}^{\epsilon_{\sigma(1)}},t_{\sigma(2)}^{\epsilon_{\sigma(2)}},t_{\sigma(3)}^{\epsilon_{\sigma(3)}},\zeta\right)\\!R_{2}\\!\left(t_{\sigma(1)}^{\epsilon_{\sigma(1)}},\ldots,t_{\sigma(r)}^{\epsilon_{\sigma(r)}},\zeta\right)\left(t_{\sigma(4)}^{\epsilon_{\sigma(4)}}\cdots\,t_{\sigma(r)}^{\epsilon_{\sigma(r)}}\right)^{-g}}{\prod_{i=1}^{3}\prod_{j=4}^{r}\left(1-t_{\sigma(i)}^{\epsilon_{\sigma(i)}}t_{\sigma(j)}^{\epsilon_{\sigma(j)}}\right)\left(1-t_{\sigma(i)}^{-\epsilon_{\sigma(i)}}t_{\sigma(j)}^{\epsilon_{\sigma(j)}}\right)\,\cdot\prod_{4\leq
k\leq l\leq
r}\left(1-t_{\sigma(k)}^{\epsilon_{\sigma(k)}}t_{\sigma(l)}^{\epsilon_{\sigma(l)}}\right)}.\end{split}$

As before, using the following slight generalization of Lemma 12.2.6, we can
represent $\mathcal{Q}_{2}(t;g,q)\psi(t)^{-g}$ as an integral.

######  12.3.2 Lemma.

Let $a_{1},\ldots,a_{r}$ be distinct non-zero complex numbers such that
$a_{i}a_{j}\neq 1$ for all $1\leq i,j\leq r.$ Suppose $h$ is a function of $r$
complex variables, holomorphic on a domain containing
$\left(a_{\sigma(1)}^{\epsilon_{1}},\ldots,a_{\sigma(r)}^{\epsilon_{r}}\right)$
for all $\sigma\in\Sigma_{r}$ and
$\epsilon=(\epsilon_{1},\ldots,\epsilon_{r})\in\\{\pm 1\\}^{r}.$ For $0\leq
m<r,$ define $K_{m}(\mathrm{z})$ by

$K_{m}(\mathrm{z})=\frac{h(\mathrm{z})}{\prod_{k=1}^{m}\prod_{l=m+1}^{r}(1-z_{k}z_{l})(1-z_{k}^{-1}z_{l})\prod_{m+1\leq
k\leq l\leq r}(1-z_{k}z_{l})},\quad\mathrm{z}:=(z_{1},\ldots,z_{r}).$

Then we have

$\begin{split}&\sum_{\sigma\in\Sigma_{r}}\,\sum_{\epsilon}K_{m}\left(a_{\sigma(1)}^{\epsilon_{1}},\ldots,a_{\sigma(r)}^{\epsilon_{r}}\right)\\\
&=\frac{(-1)^{r(r+1)/2}}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\cdots\oint
h(\mathrm{z})\frac{\prod_{1\leq i<j\leq
r}(z_{i}-z_{j})^{\mathrm{e}_{ij}}(1-z_{i}z_{j})\cdot\prod_{1\leq k\leq l\leq
m}(1-z_{k}z_{l})\prod_{i=1}^{m}z_{i}^{r-m}}{\prod_{i,j=1}^{r}(1-z_{i}a_{j})(1-z_{i}a_{j}^{-1})}\frac{d\mathrm{z}}{\mathrm{z}^{r}}\end{split}$

where $\mathrm{e}_{ij}=1$ or $2$ according as $i\leq m$ and $j\geq m+1$ or
not, $\mathrm{z}^{r}:=z_{1}^{r}\cdots\,z_{r}^{r},$ and where each path of
integration encloses the $a_{j}^{\pm 1},$ but not $z_{j}\\!=0.$

###### Proof.

The same idea as in the proof of [DT20, Lemma 7.1]. Alternatively, the lemma
can be easily deduced from [DT20, Lemma 7.5]. ∎

######  12.3.3.

We apply this lemma with $m=3$ and

$h(z_{1},\ldots,z_{r})\;=\sum_{\zeta=1,\sqrt{-1}}\,\zeta^{2g}G(z_{1},z_{2},z_{3},\zeta)R_{2}(z_{1},\ldots,z_{r},\zeta)(z_{4}\,\cdots\,z_{r})^{-g}.$

Following the same idea as in the proof of Proposition 12.2.9, one obtains:

######  12.3.4 Proposition.

For every partition $\lambda\subseteq(r^{g})$, we have

$\displaystyle\mathcal{T}_{2}(\lambda^{{\dagger}})=\frac{(1-q^{-1})q^{-g/2-2}}{2^{4}3!(r-3)!}\cdot\frac{(-1)^{r(r-1)/2}}{\left(2\pi\sqrt{-1}\right)^{r}}\\!\oint\limits_{|z_{r}|=1}\cdots\oint\limits_{|z_{1}|=1}h(\mathrm{z})\prod_{1\leq
i<j\leq r}(z_{i}-z_{j})^{\mathrm{e}_{ij}-1}$ $\displaystyle\cdot\prod_{1\leq
k\leq l\leq
3}(1-z_{k}z_{l})\,\prod_{i=1}^{3}z_{i}^{r-3}\cdot\,\det\left(z_{i}^{g-\lambda_{r+1-j}^{\prime}+r+1-j}-z_{i}^{-(g-\lambda_{r+1-j}^{\prime}+r+1-j)}\right)_{1\leq
i,j\leq r}\,\frac{d\mathrm{z}}{\mathrm{z}^{r+1}}.$

######  12.3.5.

For large $g,$ the value of $|\mathcal{T}_{2}(\lambda^{{\dagger}})|$ is as
large as possible for partitions $\lambda$ such that
$g-\lambda_{i}^{\prime}=O(1)$ for $i=1$, $2,3$, and
$\lambda_{i}^{\prime}=O(1)$ for $i\geq 4.$ For instance, take $r=3$ and
$\lambda^{\prime}=(g,g,g)$ with $g$ even, that is, $\lambda=(3,3,3,\ldots)$.
In this case,

$\displaystyle\mathcal{T}_{2}(\lambda^{{\dagger}})=-\,\frac{(1-q^{-1})q^{-g/2-2}}{2^{4}3!}\cdot\frac{1}{\left(2\pi\sqrt{-1}\right)^{3}}\\!\oint\limits_{|z_{3}|=1}\,\oint\limits_{|z_{2}|=1}\,\oint\limits_{|z_{1}|=1}h(z_{1},z_{2},z_{3})$
$\displaystyle\prod_{1\leq i<j\leq 3}(z_{i}-z_{j})\,\cdot\prod_{1\leq k\leq
l\leq
3}(1-z_{k}z_{l})\cdot\,\mathrm{det}\\!\left(z_{i}^{4-j}-z_{i}^{-4+j}\right)_{1\leq
i,j\leq 3}\,\frac{dz_{1}dz_{2}dz_{3}}{z_{1}^{4}z_{2}^{4}z_{3}^{4}}$

with

$h(z_{1},z_{2},z_{3})\;=\sum_{\zeta=1,\sqrt{-1}}G(z_{1},z_{2},z_{3},\zeta)R_{2}(z_{1},z_{2},z_{3},\zeta).$

We have

$\mathcal{T}_{2}(\lambda^{{\dagger}})=-\,(1-q^{-1})q^{-g/2-2}\left(1-q^{-2}+2q^{-3}-9q^{-4}-32q^{-5}+\cdots\right)$

which can be easily checked as follows. The function
$S(z_{1},z_{2},z_{3},\zeta^{2}\\!,x)$ is a polynomial in $x$ of degree $14$ of
the form $S(z_{1},z_{2},z_{3},\zeta^{2}\\!,x)=1+O\\!\left(x^{3}\right)\\!,$
and thus

$R_{2}(z_{1},z_{2},z_{3},\zeta)\,=\prod_{n\leq
10}S(z_{1}^{n},z_{2}^{n},z_{3}^{n},\zeta^{2n}\\!,q^{-n/2})^{i_{n}\\!(q)}\,\cdot\,\left(1+O\\!\left(q^{-\frac{11}{2}}\right)\right)\\!.$

Use this approximation, and truncate the power series expansion of
$G(z_{1},z_{2},z_{3},\zeta)$ to its $q^{-5}$th term. Replacing $G$ and $R_{2}$
under the integral of $\mathcal{T}_{2}(\lambda^{{\dagger}})$ by their
approximations and truncating again at the $q^{-5}$th term, our assertion
follows by direct computation.

######  12.3.6.

On the other hand, §12.2.17 implies that
$\mathcal{T}_{1}(\lambda^{{\dagger}})\ll_{r,\,\varepsilon}q^{\left(-\frac{\rho}{4}+\varepsilon\right)g}$
when $(|\lambda|/g)\geq\rho,$ for a positive $\rho$. In particular, if
$\lambda_{i}=3$ for all $i,$ then

$\operatorname{tr}_{\lambda}(g)\sim\mathcal{T}_{2}(\lambda^{{\dagger}})$

when $g\to\infty,$ which shows that $\mathcal{T}_{2}(\lambda^{{\dagger}})$
dominates $\mathcal{T}_{1}(\lambda^{{\dagger}})$ in this case. In fact
$\operatorname{tr}_{\lambda}(g)\sim\mathcal{T}_{2}(\lambda^{{\dagger}})$ can
be proven unconditionally, since the asymptotic formula (4) when $r=3$ with an
error term on the order of $q^{\left(-\frac{2}{3}+\varepsilon\right)g}$ can be
established by a similar technique to that used in [Dia19] for the cubic
moment of central values of quadratic $L$-functions.

######  12.3.7.

The regular behavior of the second term $\mathcal{T}_{2}(\lambda^{{\dagger}})$
when $g-\lambda_{i}^{\prime}=O(1)$ for $i=1,2,3$ and
$\lambda_{i}^{\prime}=O(1)$ for $i\geq 4$ suggests an explanation in terms of
_secondary stability_ in the sense of Kupers–Galatius–Randal-Williams [GKRW19]
for the coefficients $\mathbb{V}_{\lambda}$. It will be interesting to further
investigate this phenomenon in the different regimes of $\lambda,$ dictated by
significant changes in the behavior of the traces
$\operatorname{tr}_{\lambda}(g).$

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