Traces of high powers of the Frobenius class in the moduli space of hyperelliptic curves

Abstract

The Zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix $\Theta _C$. Following the work of Rudnick (Acta Arith. 143(1), 81–99, 2010), we compute the expected value of ${{\mathrm{tr}}}(\Theta _C^n)$ over the moduli space of hyperelliptic curves of genus g, over a fixed finite field ${\mathbb F}_q$, in the limit of large genus. As an application, we compute the expected value of the number of points on C in ${\mathbb F}_{q^n}$ as the genus tends to infinity. We also look at biases in both expected values for small values of n.

1 Introduction

Let C be a smooth projective curve of genus $g \ge 1$ defined over a fixed finite field ${\mathbb F}_q$ of odd cardinality q. If we let $\#C({\mathbb F}_{q^n})$ denote the number of points on C in finite extensions ${\mathbb F}_{q^n}$ of degree n of ${\mathbb F}_q$, then the Zeta function associated to the curve C is defined by

$$\begin{aligned} {{\mathrm{Z}}}_C(u):=\exp \Big ( \sum _{n=1}^{\infty } \frac{\#C({\mathbb F}_{q^n})}{n} u^n \Big ), \quad |u| < \frac{1}{q}. \end{aligned}$$

(1)

It is known that ${{\mathrm{Z}}}_C(u)$ is a rational function in u of the form

$$\begin{aligned} {{\mathrm{Z}}}_C(u)=\frac{P_C(u)}{(1-u)(1-qu)}, \end{aligned}$$

where $P_C(u) \in {\mathbb Z}[u]$ is a polynomial of degree 2g, with $P_C (0)=1$, satisfying the functional equation

$$\begin{aligned} P_C(u)=(qu^2)^gP_C\left( \frac{1}{qu}\right) . \end{aligned}$$

(2)

It was proven by Weil [9] that the zeros of $P_C(u)$ all lie on the circle $|u|=1/q^\frac{1}{2}$. Hence,

$$\begin{aligned} P_C(u)=\prod _{j=1}^{2g}\left( 1-q^\frac{1}{2} {{\mathrm{e}}}^{i\theta _j(C)}u\right) , \end{aligned}$$

for some angles $\theta _j(C)$, $1\le j \le 2g$, and

$$\begin{aligned} {{\mathrm{Z}}}_C(u) = \exp \Big ( \sum _{n=1}^{\infty } \frac{\#C({\mathbb F}_{q^n})}{n} u^n \Big ) = \frac{\prod _{j=1}^{2g}(1-q^\frac{1}{2} {{\mathrm{e}}}^{i\theta _j(C)}u)}{(1-u)(1-qu)}. \end{aligned}$$

(3)

Now, we may define a unitary symplectic matrix $\Theta _C \in {{\mathrm{USp}}}(2g)$ by

$$\begin{aligned} \Theta _{C_{jk}}:= \left\{ \begin{array}{lll} {{\mathrm{e}}}^{i\theta _j(C)} &{} \quad \text{ if } k=j,\\ 0 &{} \quad \text{ otherwise, }\\ \end{array}\right. \end{aligned}$$

for $1 \le j,k \le 2g$. Then it is clear that the Zeta function associated to C can be expressed in terms of the characteristic polynomial of $\Theta _C$:

$$\begin{aligned} {{\mathrm{Z}}}_C(u)=\frac{\det ({\mathbb I}-u\sqrt{q}\Theta _C)}{(1-u)(1-qu)}, \end{aligned}$$

with $\Theta _C$ unique up to conjugacy. We call the conjugacy class of $\Theta _C$ the unitarized Frobenius class of C.

Let ${\mathscr {H}}_g$ be the moduli space of hyperelliptic curves of genus g over ${\mathbb F}_q$. Each curve in ${\mathscr {H}}_g$ has an affine model

$$\begin{aligned} C_Q:y^2=Q(x), \end{aligned}$$

where $Q\in {\mathbb F}_q[x]$, Q squarefree, with $\deg (Q)=2g+1,2g+2$. In this model, the point at infinity is not smooth, but all the affine points are smooth and we may account for the point at infinity separately. The smooth models are given by the closures of $C_Q$, denoted $\overline{C}_Q$, under the map

$$\begin{aligned}{}[1,x,x^2,\dots ,x^{g+1},y]:C_Q\rightarrow {\mathbb P}^{g+2}. \end{aligned}$$

The ${\mathbb F}_q$-points of ${\mathscr {H}}_g$ are then given by the ${\mathbb F}_q$-points of the smooth projective curves $\overline{C}_Q$, taken up to $\overline{{\mathbb F}}_q$-isomorphism.

In the smooth models, the point at infinity will be replaced by 0, 1, or 2 points, accordingly, and we get that the number of points at infinity is

(4)

where ${{\mathrm{sgn}}}(Q)$ denotes the leading coefficient of Q. Note the relation between Eqs. (4) and (9); namely, the number of points at infinity is $\lambda _Q+1$, with $\lambda _Q$ as in (9).

Remark 1

One can show that $\overline{C}_Q$ consists of two affine components: the first is $C_Q$ itself and the second is the curve given by $y^2=x^{2g+2}Q(\frac{1}{x})$. In fact, $C_Q$ is isomorphic to $\overline{C}_Q\cap \{x_0\ne 0\}$ and taking $x_0=0$ yields the point at infinity; we refer the reader to Silverman [8].

For any function F on ${\mathscr {H}}_g$, we define the expected value of F over ${\mathscr {H}}_g$

$$\begin{aligned} \langle F \rangle _{{\mathscr {H}}_g}:=\frac{1}{\#{\mathscr {H}}_g^\prime } \sum _{C \in {\mathscr {H}}_g} {'\;F(C)}, \end{aligned}$$

(5)

where ${'}$ indicates that the points $C\in {\mathscr {H}}_g$ are counted with weight $\frac{1}{\#{{\mathrm{Aut}}}(C)}.$

In this paper, we study the traces of high powers of the Frobenius class of $C\in {\mathscr {H}}_g$ over a fixed finite field ${\mathbb F}_q$, of odd cardinality q, as g tends to infinity. In particular, we concern ourselves with the expected values $\langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}$ as $g \rightarrow \infty$ and we compare our work with the Random Matrix results of [3].

From the work of Diaconis and Shahshahani [3], the expected value of the traces of powers over the unitarized symplectic group ${{\mathrm{USp}}}(2g)$ is given by

$$\begin{aligned} \int _{{{\mathrm{USp}}}(2g)}{{\mathrm{tr}}}(U^n)dU= \left\{ \begin{array}{ll} -\eta _n &{} \quad \text{ if } 1\le n \le 2g,\\ 0 &{} \quad \text{ if } n > 2g,\\ \end{array}\right. \end{aligned}$$

(6)

where

$$\begin{aligned} \eta _n=\left\{ \begin{array}{lll} 1, &{} \quad n \text{ even },\\ 0, &{} \quad n \text{ odd. }\\ \end{array}\right. \end{aligned}$$

We will prove the following theorem and the accompanying corollary:

Theorem 1.1

For n odd,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^n) \rangle _{{\mathscr {H}}_g}=0, \end{aligned}$$

and for n even,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^n) \rangle _{{\mathscr {H}}_g}= \frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}}\\ {\deg (P)\ne 1} \end{array}} \frac{\deg (P)}{|P|+1} +O(gq^{\frac{-g}{2}})+ \left\{ \begin{array}{lll} -1, &{} \quad 0< n< 2g,\\ -1-\frac{1}{q^2-1}, &{} \quad n = 2g,\\ O(nq^{\frac{n}{2}-2g}) ,&{} \quad 2g < n,\\ \end{array}\right. \end{aligned}$$

where the sum is over all monic irreducible polynomials $P \in {\mathbb F}_q[x]$ and where $|P|:=q^{\deg (P)}$.

Corollary 1.2

If n is odd, then

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^n) \rangle _{{\mathscr {H}}_g} = \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^n) dU. \end{aligned}$$

For n even with $3 \log _q(g)< n < 4g-5 \log _q(g)$ and $n \ne 2g$,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^n) \rangle _{{\mathscr {H}}_g} = \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^n) dU + o\left( \frac{1}{g}\right) . \end{aligned}$$

In [7], Rudnick considers the mean value of ${{\mathrm{tr}}}(\Theta ^n_{C_Q})$ over a family of hyperlliptic curves given by the affine equations ${C_Q}: y^2=Q(x)$, where $Q\in {\mathcal F}_{2g+1}$ and where

$$\begin{aligned} {\mathcal F}_{2g+1}:=\{f\in {\mathbb F}_q[x] : f \text{ monic, } \text{ squarefree }, \deg (f)=2g+1\}. \end{aligned}$$

Rudnick obtains that

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}} = \eta _n\frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}} \end{array}} \frac{\deg (P)}{|P|+1} +O(gq^{-g})+ \left\{ \begin{array}{lll} -\eta _n, &{} \quad 0< n< 2g,\\ -1-\frac{1}{q-1}, &{} \quad n = 2g,\\ O(nq^{\frac{n}{2}-2g}), &{} \quad 2g < n, \end{array}\right. \end{aligned}$$

with

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}} :=\frac{1}{\#{\mathcal F}_{2g+1}}\sum _{Q\in {\mathcal F}_{2g+1}} {{\mathrm{tr}}}(\Theta ^n_{C_Q}). \end{aligned}$$

In particular, if $3 \log _q(g)< n < 4g-5 \log _q(g)$ and $n \ne 2g$, then

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}}=\int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^n) dU + o\Big ( \frac{1}{g} \Big ). \end{aligned}$$

Rudnick then points out that there is a slight deviation in $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}}$ from the Random Matrix Theory results for small values of n and for $n=2g$; namely,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^2) \rangle _{{\mathcal F}_{2g+1}} \sim \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^2) dU + \frac{1}{q+1} \end{aligned}$$

and

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^{2g}) \rangle _{{\mathcal F}_{2g+1}} \sim \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^{2g}) dU - \frac{1}{q-1}. \end{aligned}$$

By considering the average value of ${{\mathrm{tr}}}(\Theta _C^n)$ over ${\mathscr {H}}_g$, we no longer get a deviation from the RMT results for $n=2$ and the deviation at $n=2g$ diminishes:

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^{2}) \rangle _{{\mathscr {H}}_g} \sim \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^{2}) dU \end{aligned}$$

and

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _C^{2g}) \rangle _{{\mathscr {H}}_g} \sim \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^{2g}) dU - \frac{1}{q^2-1}. \end{aligned}$$

Furthermore, our results for odd n are exact and coincide with the RMT results for all values of g. At first glance, this may seem counterintuitive as one expects to have an error term, as in the even case and as in [7]. Using another approach, one can quickly verify the first result of Theorem 1.1 (this is done in Sect 4).

Now, we may apply Theorem 1.1 to compute the average number of points on $C\in {\mathscr {H}}_g$ in finite extensions ${\mathbb F}_{q^n}$ of ${\mathbb F}_q$, denoted $\langle \#C({\mathbb F}_{q^n}) \rangle _{{\mathscr {H}}_g}$. By taking logarithmic derivatives in (3),

$$\begin{aligned} \#C({\mathbb F}_{q^n})= & {} q^n+1-q^\frac{n}{2}\sum _{j=1}^{2g}{{\mathrm{e}}}^{in\theta _j(C)}\\= & {} q^n+1-q^\frac{n}{2}{{\mathrm{tr}}}(\Theta _{C}^n). \end{aligned}$$

Therefore,

Corollary 1.3

• (i) If n is odd, then

  $$\begin{aligned} \langle \#C({\mathbb F}_{q^n}) \rangle _{{\mathscr {H}}_g}=q^n+1. \end{aligned}$$

• (ii) If n is even, then

  $$\begin{aligned} \langle \#C({\mathbb F}_{q^n}) \rangle _{{\mathscr {H}}_g} \sim q^n+q^\frac{n}{2}+1-\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}}\\ {\deg (P)\ne 1} \end{array}}\frac{\deg (P)}{|P|+1}. \end{aligned}$$

Once again, our results for odd n are exact and hold for all values of g. Although we continue to get deviations from the RMT results for even $n\ge 4$, our results hold for $n=2$ and our deviations are different from those obtained in [7].

Another approach to computing $\langle \#C({\mathbb F}_{q^n}) \rangle _{{\mathscr {H}}_g}$ is the work of Alzahrani [1] which uses the distribution of the ${\mathbb F}_q$-points of ${\mathscr {H}}_g$ in ${\mathbb F}_{q^n}$. Using these methods, the results of Alzahrani agree with the Corollary above (albeit with a larger error term).

Finally, we would like to mention that some of the computations done in Sect 2.1 through 3.2 were done independently by Lorenzo et al. in their study of statistics for biquadratic curves; their work is collected in [5].

2 Preliminaries

In this section, we establish some notation, we introduce the main results of [7], and we prove some preliminary results. Since the majority of what follows is based off of the work in [7], we use the same notation and list important results for the convenience of the reader. We use [6] as a general reference.

Throughout this paper, ${\mathbb F}_q$ is a fixed finite field of odd cardinality q, P is solely used to represent monic irreducible polynomials in ${\mathbb F}_q[x]$, and Q will be used to denote squarefree polynomials of degree $2g+1$ or $2g+2$ with $g\ge 1$. Unless stated otherwise, sums/products are over monic elements in ${\mathbb F}_q[x]$ (under the given restrictions). For example, sums/products indexed by P are over all monic irreducible polynomials in ${\mathbb F}_q[x]$, where as sums/products indexed by $\deg (f)=n$ are over all monic polynomials of degree n in ${\mathbb F}_q[x]$. In the case where a sum involves elements $B\in {\mathbb F}_q[x]$ that are not necessarily monic, we write the sum over B n.n.m..

Given any polynomial $D\in {\mathbb F}_q[x]$ that is not a perfect square, we define the quadratic character $\chi _D$ by the quadratic residue symbol for ${\mathbb F}_q[x]$:

$$\begin{aligned} \chi _D(f):=\Big ( \frac{D}{f} \Big ), \end{aligned}$$

where f is any monic polynomial in ${\mathbb F}_q[x]$.

The Zeta function associated to the hyperelliptic curve $C_Q:y^2=Q(x)$ is then given by

$$\begin{aligned} {{\mathrm{Z}}}_{C_Q}(u)=L^*(u,\chi _Q)\zeta _q(u), \end{aligned}$$

where

$$\begin{aligned} \zeta _q(u):=\frac{1}{(1-u)(1-qu)} \end{aligned}$$

is the Zeta function of ${\mathbb F}_q(x)$ and where

$$\begin{aligned} L^*(u,\chi _Q):= & {} (1-\lambda _Q u)^{-1}\prod _{P}(1-\chi _Q(P) u^{\deg (P)})^{-1}\end{aligned}$$

(7)

$$\begin{aligned}= & {} \det ({\mathbb I}-u\sqrt{q} \Theta _{C_Q}), \end{aligned}$$

(8)

with

(9)

which relates to the count in Eq. (4).

Taking logarithmic derivatives in Eqs. (7) and (8), we see that

$$\begin{aligned} \sum _{j=1}^{2g} {{\mathrm{e}}}^{in\theta _j(C_Q)} = {{\mathrm{tr}}}(\Theta _{C_Q}^n)=-\frac{\lambda _Q^n}{q^\frac{n}{2}}-\frac{1}{q^\frac{n}{2}}\sum _{\deg (f)=n} \Lambda (f) \chi _Q(f), \end{aligned}$$

(10)

where

$$\begin{aligned} \Lambda (f):=\left\{ \begin{array}{lll} \deg (P) &{} \quad \text{ if } f=P^k,\\ 0 &{} \quad \text{ otherwise }\\ \end{array}\right. \end{aligned}$$

is the von Mangoldt function.

Now, let ${\mathcal F}$ be any family of squarefree polynomials of degree d in ${\mathbb F}_q[x]$. For any function F on ${\mathcal F}$, we define the expected value of F over ${\mathcal F}$:

$$\begin{aligned} \langle F \rangle _{\mathcal F}:=\frac{1}{\#{\mathcal F}} \sum _{Q\in {\mathcal F}}F(Q). \end{aligned}$$

In particular,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}}=\frac{1}{\#{\mathcal F}} \sum _{Q\in {\mathcal F}}\Big ( -\frac{\lambda _Q^n}{q^\frac{n}{2}}-\frac{1}{q^\frac{n}{2}}\sum _{\deg (f)=n}\Lambda (f) \chi _Q(f)\Big ). \end{aligned}$$

(11)

Let

$$\begin{aligned} {\mathcal F}_d:=\{f\in {\mathbb F}_q[x]:f \text{ monic, } \text{ squarefree }, \deg (f)=d\} \end{aligned}$$

and let

$$\begin{aligned} \widehat{{\mathcal F}}_d:=\{f\in {\mathbb F}_q[x]:f \text{ squarefree }, \deg (f)=d\}. \end{aligned}$$

Then

$$\begin{aligned} \#\widehat{{\mathcal F}}_d=(q-1)\#{\mathcal F}_d \end{aligned}$$

and it is easy to see that (see Lemma 3 of [4], for example)

$$\begin{aligned} \#{\mathcal F}_d=\left\{ \begin{array}{lll} (1-\frac{1}{q})q^d, &{} \quad d\ge 2\\ q, &{} \quad d=1.\\ \end{array}\right. \end{aligned}$$

Using these sets of polynomials, every curve in ${\mathscr {H}}_g$ has an affine model $C_Q:y^2=Q(x)$, where $Q\in \widehat{{\mathcal F}}_{2g+1}\cup \widehat{{\mathcal F}}_{2g+2}$. By averaging ${{\mathrm{tr}}}(\Theta ^n_{C_Q})$ over all $Q\in \widehat{{\mathcal F}}_{2g+1}\cup \widehat{{\mathcal F}}_{2g+2}$, we count each point in the moduli space $q(q^2-1)$ times (this is due to the fact that the projective linear group ${{\mathrm{PGL}}}_2({\mathbb F}_q)$ has order $q(q^2-1)$ and the ${\mathbb F}_q$-points of $C_Q$ are the same as those of $C_Q$ under the action of any element in ${{\mathrm{PGL}}}_2({\mathbb F}_q)$). Moreover,

$$\begin{aligned} \frac{\#\{C\in {\mathscr {H}}_g:{{\mathrm{tr}}}(\Theta ^n_C)=\alpha \}^{'}}{\#{\mathscr {H}}_g^{'}}= \frac{\#\{Q\in \widehat{{\mathcal F}}_{2g+1}\cup \widehat{{\mathcal F}}_{2g+2}:{{\mathrm{tr}}}(\Theta ^n_{C_Q})=\alpha \}}{\#(\widehat{{\mathcal F}}_{2g+1} \cup \widehat{{\mathcal F}}_{2g+2})}, \end{aligned}$$

(12)

where $'$ indicates that the curves $C\in {\mathscr {H}}_g$ are counted with weight $\frac{1}{\#{{\mathrm{Aut}}}(C)}$; we refer the reader to [2]. Ultimately, averaging ${{\mathrm{tr}}}(\Theta ^n_C)$ over $C\in {\mathscr {H}}_g$ is the same as averaging ${{\mathrm{tr}}}(\Theta ^n_{C_Q})$ over $Q \in \widehat{{\mathcal F}}_{2g+1} \cup \widehat{{\mathcal F}}_{2g+2}$. In other words,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}=\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{\widehat{{\mathcal F}}_{2g+1} \cup \widehat{{\mathcal F}}_{2g+2}}. \end{aligned}$$

(13)

In [7], Rudnick averages the trace over ${\mathcal F}={\mathcal F}_{2g+1}$. We begin by considering the average over ${\mathcal F}={\mathcal F}_{2g+2}$ and then obtain the average over ${\mathscr {H}}_g$ by combining our results and by considering the contribution of the point at infinity which differs on each of the families $\widehat{{\mathcal F}}_{2g+1}$, $\widehat{{\mathcal F}}_{2g+2}$.

Let $\mu$ denote the Möbius function and let $D\in {\mathbb F}_q[x]$. Since

$$\begin{aligned} \sum _{A^2|D}\mu (A)=\left\{ \begin{array}{lll} 1 &{} \quad \text {if }D \text { is squarefree},\\ 0 &{} \quad \text{ otherwise, }\\ \end{array}\right. \end{aligned}$$

we may compute the expected value of F by summing over all elements of degree d in ${\mathbb F}_q[x]$ and sieving out the squarefree terms; namely,

$$\begin{aligned} \langle F(Q)\rangle _{\mathcal F}= \frac{1}{\#{\mathcal F}}\sum _{2\alpha +\beta =d}\sum _{\begin{array}{c} {\deg (B)=\beta }\\ B\, \text {n.n.m.} \end{array}}\sum _{\deg (A)=\alpha } \mu (A) F(A^2B). \end{aligned}$$

(14)

For all A,  $B \in {\mathbb F}_q[x]$,

$$\begin{aligned} \chi _{A^2B}(f)=\Big (\frac{B}{f}\Big ) \Big (\frac{A}{f}\Big )^2=\left\{ \begin{array}{lll} \Big (\frac{B}{f}\Big ) &{} \quad \text{ if } (A,f)=1,\\ 0 &{}\quad \text{ otherwise. }\\ \end{array}\right. \end{aligned}$$

With that said, taking $F(Q)=\chi _Q$ in Eq. (14),

$$\begin{aligned} \langle \chi _Q(f) \rangle _{\mathcal F}=\frac{1}{\#{\mathcal F}} \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {\alpha ,\beta \ge 0} \end{array}} \sigma (f;\alpha )\sum _{\begin{array}{c} {\deg (B)=\beta }\\ {B\, \text {n.n.m.}} \end{array}}\Big (\frac{B}{f}\Big ), \end{aligned}$$

where

$$\begin{aligned} \sigma (f;\alpha ):=\sum _{\begin{array}{c} {\deg (A)=\alpha }\\ {(A,f)=1} \end{array}}\mu (A). \end{aligned}$$

We are now in a position to provide the necessary results from [7]. For $P\in {\mathbb F}_q[x]$ with $\deg (P)=n$, we define

$$\begin{aligned} \sigma _n(\alpha ):=\sigma (P^k;\alpha )=\sum _{\begin{array}{c} {\deg (A)=\alpha }\\ {(A,P^k)=1} \end{array}}\mu (A)=\sum _{\begin{array}{c} {\deg (A)=\alpha }\\ {(A,P)=1} \end{array}}\mu (A)=\sigma (P;\alpha ). \end{aligned}$$

Lemma 2.1

([7], Lemma 4)

• (i) For $n=1$,

  $$\begin{aligned} \sigma _1(0)=1, \quad \sigma _1(\alpha )=1-q \quad \,\forall \alpha \ge 1. \end{aligned}$$

  (15)

• (ii) If $n\ge 2$, then

  $$\begin{aligned} \sigma _n(\alpha )=\left\{ \begin{array}{lll} 1, &{} \quad \alpha \equiv 0\mod n,\\ -q, &{} \quad \alpha \equiv 1\mod n,\\ 0, &{} \quad \text{ otherwise. }\\ \end{array}\right. \end{aligned}$$

  (16)

Recall that the Dirichlet L-series associated to $\chi _Q$, denoted $L(u,\chi _Q)$, is a polynomial in u of degree at most $\deg (Q)-1$ (see Proposition 4.3 of [6], for example). In fact,

$$\begin{aligned} L(u,\chi _Q):=\prod _{P}(1-\chi _Q(P) u^{\deg (P)})^{-1}=\sum _{\beta \ge 0}A_Q(\beta )u^\beta , \end{aligned}$$

where

$$\begin{aligned} A_Q(\beta ):=\sum _{\deg (B)=\beta }\chi _Q(B) \end{aligned}$$

and $A_Q(\beta )=0$ for $\beta \ge \deg (Q)$.

Let

$$\begin{aligned} S(\beta ; n):=\sum _{\deg (P)=n}\sum _{\deg (B)=\beta }\left( \frac{B}{P}\right) . \end{aligned}$$

By the Law of Quadratic Reciprocity [6],

$$\begin{aligned} S(\beta ; n)=(-1)^{\frac{q-1}{2}\beta n} \sum _{\deg (P)=n}A_P(\beta ). \end{aligned}$$

The above equality implies that $S(\beta ;n)=0$ $\forall n \le \beta .$

We let $\pi _q(n)$ denote the number of monic irreducible polynomials of degree n in ${\mathbb F}_q[x]$. From the Prime Polynomial Theorem [6],

$$\begin{aligned} \pi _q(n):= & {} \#\{P\in {\mathbb F}_q[x]:\deg (P)=n\}\\= & {} \frac{q^n}{n}+O\Big (\frac{q^{\frac{n}{2}}}{n}\Big ). \end{aligned}$$

Lemma 2.2

([7], Proposition 7)

• (i) n odd, $0\le \beta \le n-1$:

  $$\begin{aligned} S(\beta ;n)=q^{\beta -\frac{n-1}{2}}S(n-1-\beta ;n) \end{aligned}$$

  (17)

  and

  $$\begin{aligned} S(n-1;n)=\pi _q(n)q^{\frac{n-1}{2}}. \end{aligned}$$

  (18)

• (ii) n even, $1\le \beta \le n-2$:

  $$\begin{aligned} S(\beta ;n)=q^{\beta -\frac{n}{2}}\Big (-S(n-1-\beta ;n)+(q-1)\sum _{j=0}^{n-\beta -2}S(j;n)\Big ) \end{aligned}$$

  (19)

  and

  $$\begin{aligned} S(n-1;n)=-\pi _q(n)q^{\frac{n-2}{2}}. \end{aligned}$$

  (20)

Lemma 2.3

[7, Lemma 8] If $\beta <n$, then

$$\begin{aligned} S(\beta ;n)=\eta _\beta \pi _q(n)q^{\frac{\beta }{2}}+O\left( \frac{\beta }{n}q^{\frac{n}{2}+\beta }\right) , \end{aligned}$$

(21)

where $\eta _\beta =1$ for $\beta$ even and $\eta _\beta =0$ for $\beta$ odd.

2.1 Improved estimate for $S(\beta ;n)$ when $\beta$ is even

Initially, we concern ourselves with $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+2}}$; in doing so, we need to estimate $S(\beta ;n)$ for when $\beta$ is even (see Sect. 3.1.1 and 3.1.3). The following theorem makes use of Lemmas 2.2 and 2.3; it is the analogous result to Proposition 9 of [7] (since Rudnick considers the average value over ${\mathcal F}_{2g+1}$, estimates for $S(\beta ;n)$ in [7] involve $\beta$ odd). Furthermore, this result will allow us to compute $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+2}}$ for n near 4g (just as Proposition 9 in [7] allows Rudnick to compute $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}}$ for n near 4g).

Theorem 2.4

If $\beta$ is even, $\beta \ne 0$, and $\beta < n$, then

$$\begin{aligned} S(\beta ;n)=\pi _q(n)\left( q^{\frac{\beta }{2}}-\eta _nq^{\beta -\frac{n}{2}}\right) +O(q^n), \end{aligned}$$

(22)

where

$$\begin{aligned} \eta _n=\left\{ \begin{array}{ll} 1, &{} \quad n\, \,\text{ even },\\ 0, &{} \quad n \,\,\text{ odd. }\\ \end{array}\right. \end{aligned}$$

Remarks 1

• (i) The result above is essentially the same as Proposition 9 in [7] with one additonal term; namely, $\pi _q(n)q^\frac{\beta }{2}$.

• (ii) As Rudnick points out in [7], the main tool in proving Theorem 2.4 is duality; it allows us to improve the error term in estimates of $S(\beta ;n)$ and to get results holding for n near 4g and not only for n near 2g. We would like to mention that the duality present in our character sums $S(\beta ;n)$ is based on the functional equation (2)

  $$\begin{aligned} L^*(u,\chi _P)=(uq^2)^{\lfloor {\frac{\deg (P)-1}{2}}\rfloor } L^*\left( \frac{1}{qu},\chi _P\right) , \end{aligned}$$

  for prime characters $\chi _P$ (see the proof of Proposition 7 in [7]).

Proof

• (i) If n is odd, we apply (17) to $S(\beta ;n)$ and then apply (21) to $S(n-1-\beta ;n)$:

  $$\begin{aligned} S(\beta ;n)= & {} q^{\beta -\frac{n-1}{2}}S(n-1-\beta ;n)\\= & {} q^{\beta -\frac{n-1}{2}}\left( \pi _q(n)q^\frac{n-1-\beta }{2}+O\left( \frac{n-1-\beta }{n}q^{\frac{n}{2}+n-1-\beta }\right) \right) \\= & {} \pi _q(n)q^{\frac{\beta }{2}}+O(q^n). \end{aligned}$$

• (ii) If n is even, we apply (19) to $S(\beta ;n)$ and then apply (21) to $S(n-1-\beta ;n)$:

  $$\begin{aligned} S(\beta ;n)= & {} q^{\beta -\frac{n}{2}}\left( -S(n-1-\beta ;n)+(q-1)\sum _{j=0}^{n-\beta -2}S(j;n)\right) \\= & {} q^{\beta -\frac{n}{2}}\left( O\Big (\frac{n-1-\beta }{n}q^{\frac{n}{2}+n-1-\beta }\right) \\&+(q-1)\sum _{j=0}^{n-\beta -2} \Big (\eta _j\pi _q(n)q^{\frac{j}{2}}+O(\frac{j}{n}q^{\frac{n}{2}+j})\Big )\Biggr ). \end{aligned}$$

The two error terms are $O(q^n)$. Since both n and $\beta$ are even, $n-\beta -2$ is even and we may rewrite the main term as

$$\begin{aligned} \pi _q(n)q^{\beta -\frac{n}{2}}(q-1)\sum _{j=0}^{\frac{n-\beta -2}{2}}q^j. \end{aligned}$$

Hence,

$$\begin{aligned} S(\beta ;n)= & {} \pi _q(n)q^{\beta -\frac{n}{2}}(q-1)\sum _{j=0}^{\frac{n-\beta -2}{2}}q^j+O(q^n)\\= & {} \pi _q(n)q^{\beta -\frac{n}{2}}(q^{\frac{n-\beta }{2}}-1)+O(q^n)\\= & {} \pi _q(n)(q^{\frac{\beta }{2}}-q^{\beta -\frac{n}{2}})+O(q^n). \end{aligned}$$

$\square$

3 Computing $\langle \hbox {tr}(\Theta _{C_Q}^n) \rangle _{\mathcal {F}_{2g+1} \cup \mathcal {F}_{2g+2}}$

In this section, we compute $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}$ by first computing $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+2}}$ and then combining that result with Rudnick’s for $\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}}.$

3.1 Computing $\hbox {tr}(\Theta _{C_Q}^n)$ for $Q\in \mathcal {F}_{2g+2}$

For the time being, we restrict ourselves to ${\mathcal F}_{2g+2}$. Let $Q\in {\mathcal F}_{2g+2}$ and consider the curve $C_Q:y^2=Q(x)$. The trace of the powers of $\Theta _{C_Q}$ is given by Eq. (10):

$$\begin{aligned} {{\mathrm{tr}}}(\Theta _{C_Q}^n)= & {} -\frac{1}{q^{\frac{n}{2}}}-\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (f)=n}\Lambda (f)\chi _Q(f)\end{aligned}$$

(23)

$$\begin{aligned}= & {} -\frac{1}{q^{\frac{n}{2}}}-\frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {P,k}\\ {\deg (P^k)}=n \end{array}}\deg (P)\chi _Q(P^k)\end{aligned}$$

(24)

$$\begin{aligned}= & {} -\frac{1}{q^{\frac{n}{2}}}+{\mathcal P}_n+\square _n + {\mathbb H}_n, \end{aligned}$$

(25)

where ${\mathcal P}_n$ corresponds to $k=1$, $\square _n$ corresponds to the sum over all k even, and ${\mathbb H}_n$ corresponds to the sum over all odd $k\ge 3$.

In the next three sections, we continue to use Rudnick’s methods in order to compute ${\mathcal P}_n$, $\square _n$, and ${\mathbb H}_n$. Not surprinsingly, our results will only slightly differ from Rudnick’s. The addition of $-1/q^\frac{n}{2}$ from (9) will be the main difference. We will also have different cut-off points for n when estimating ${\mathcal P}_n$ (see Sect. 5.3 of [7]).

3.1.1 Contribution of the primes: $\mathcal {P}_n$

The contribution of the primes in (23) is given by

$$\begin{aligned} {\mathcal P}_n=-\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)=n}n\chi _Q (P). \end{aligned}$$

So,

$$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{-n}{(q-1)q^{2g+1+\frac{n}{2}}}\sum _{\deg (P)=n}\sum _{2\alpha +\beta =2g+2}\sigma _n(\alpha )\sum _{\deg (B)=\beta }\Biggr (\frac{B}{P}\Biggr )\\= & {} \frac{-n}{(q-1)q^{2g+1+\frac{n}{2}}}\sum _{2\alpha +\beta =2g+2}\sigma _n(\alpha )S(\beta ;n). \end{aligned}$$

From Lemma 2.1, if $n>g+1$, then

$$\begin{aligned} \sigma _n(\alpha )\ne 0\Rightarrow & {} (\alpha \equiv 0\, \hbox {mod}\, n \,or\, \alpha \equiv 1\, \hbox {mod}\, n)\\\Rightarrow & {} (\alpha =0 \text { or } \alpha =1), \end{aligned}$$

which follows from the fact that $0\le \alpha \le g+1$. Since $\sigma _n(0)=1$ and $\sigma _n(1)=-q$, when $n>g+1$, we have

$$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}=\frac{-n}{(q-1)q^{2g+1 +\frac{n}{2}}} (S(2g+2;n)-qS(2g;n)). \end{aligned}$$

We now compute $\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}$ by considering the case $n\le g+1$ and the case $n>g+1$, which we break into four (non-distinct) ranges:

• (i) $n\le g+1:$ If $S(\beta ;n)\ne 0$, then $\beta <n$; since $\beta$ is even,

  $$\begin{aligned} S(\beta ;n)= & {} \pi _q(n)q^{\frac{\beta }{2}}+O \left( \frac{\beta }{n}q^{\beta +\frac{n}{2}}\right) \\= & {} \frac{q^{n+\frac{\beta }{2}}}{n}+O \Bigg (\frac{q^{\frac{n}{2}+\frac{\beta }{2}}}{n}\Bigg )+O \Bigg (\frac{\beta }{n}q^{\beta +\frac{n}{2}}\Bigg ). \end{aligned}$$

  Then $S(\beta ;n)\ll \frac{\beta }{n}q^{n+\beta },$ which implies that

  $$\begin{aligned} \langle {\mathcal P}_n\rangle _{{\mathcal F}_{2g+2}}\ll & {} \frac{n}{q^{2g+\frac{n}{2}}}\sum _{\beta <n}\frac{\beta }{n}q^{\beta +n}\\\ll & {} \frac{n}{q^{2g+\frac{n}{2}}}q^{2n}=nq^{\frac{3n}{2}-2g}\ll gq^{\frac{-g}{2}}. \end{aligned}$$

• (ii) $g+1<n<2g+1:$ Since $2g+2,2g\ge n$, $S(2g+2;n)=S(2g;n)=0.$ Hence,

  $$\begin{aligned}\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= \frac{-n}{(q-1)q^{2g+1+\frac{n}{2}}}(S(2g+2;n)-qS(2g;n))=0.\end{aligned}$$

• (iii) $n=2g+1:$ Since $2g+2\ge n$, $S(2g+2;n)=0$ and we get that

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{n}{(q-1)q^{2g+1+\frac{n}{2}}} q S(2g;n),\\= & {} \frac{2g+1}{(q-1)q^{3g+\frac{1}{2}}} S(2g;2g+1). \end{aligned}$$

  Using (18),

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{2g+1}{(q-1)q^{3g+\frac{1}{2}}} \pi _q(2g+1)q^{\frac{(2g+1)-1}{2}}. \end{aligned}$$

  By replacing $\pi _q(2g+1)$ and simplifying, we obtain

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{2g+1}{(q-1)q^{3g+\frac{1}{2}}} \left( \frac{q^{2g+1}}{2g+1}+O\left( \frac{q^g}{2g+1}\right) q^g\right) \\= & {} \frac{q^{\frac{1}{2}}}{q-1}+O(q^{-g}). \end{aligned}$$

• (iv) $n=2g+2$: Similarly,

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{n}{(q-1)q^{2g+1+\frac{n}{2}}}q S(2g;n)\\= & {} \frac{2g+2}{(q-1)q^{3g+1}} S(2g;2g+2). \end{aligned}$$

  From Theorem 2.4,

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{2g+2}{(q-1)q^{3g+1}} \left( \left( \frac{q^{2g+2}}{2g+2}+O(\frac{q^g}{2g+2})\right) (q^g-q^{g-1})+O(q^{2g})\right) \\= & {} 1+O(q^{-g})+O(gq^{-g}). \end{aligned}$$

• (v) $n>2g+2:$ We apply Theorem 2.4 to get

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{-n}{(q-1)q^{2g+1+\frac{n}{2}}}\left( S(2g+2;n)-qS(2g;n)\right) \\= & {} \frac{-n}{(q-1)q^{2g+1+\frac{n}{2}}}\left( \pi _q(n)(q^{\frac{2g+2}{2}}-\eta _nq^{2g+2-\frac{n}{2}})\right. \\&\left. -q\pi _q(n)(q^{\frac{2g}{2}}-\eta _nq^{2g-\frac{n}{2}})+O(q^n)\right) . \end{aligned}$$

  Upon further simplification,

  $$\begin{aligned} \langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}= & {} \frac{n}{(q-1)q^{2g+1+\frac{n}{2}}}\Biggr (\eta _n \pi _q(n) (q^{2g+2-\frac{n}{2}}-q^{2g+1-\frac{n}{2}})+O(q^n)\Biggr )\\= & {} \frac{n\eta _n \pi _q(n)}{q^n}+O(nq^{\frac{n}{2}-2g})\\= & {} \eta _n (1+O(q^{\frac{-n}{2}}))+O(nq^{\frac{n}{2}-2g}). \end{aligned}$$

Note 1 When $n=2g+2$, (v) yields (iv).

3.1.2 Contribution of the squares: $\square _n$

For n even, we have the following contribution from the squares of prime powers:

$$\begin{aligned} \square _n= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P^{2k})=n}\Lambda (P^{2k})\chi _Q(P^{2k})\\= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\Lambda (P^{k})\chi _Q(P^{2k}) . \end{aligned}$$

Therefore,

$$\begin{aligned} \langle \square _n \rangle _{{\mathcal F}_{2g+2}}= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\Lambda (P^{k})\langle \chi _Q(P^{2k})\rangle _{{\mathcal F}_{2g+2}}\\ \!= & {} \!-\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\deg (P)\frac{1}{(q\!-\!1)q^{2g+1}}\sum _{0\le \alpha \le g+1}\sum _{\begin{array}{c} {\deg (A)=\alpha }\\ {P\not \mid A} \end{array}}\mu (A)\sum _{\begin{array}{c} {\deg (B)=2g+2-2\alpha },\\ {P\not \mid B} \end{array}}1. \end{aligned}$$

Although Sect. 3.1.1 shows a slight deviation in $\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}$ from $\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+1}}$, we will see that $\langle \square _n \rangle _{{\mathcal F}_{2g+2}}=\langle \square _n \rangle _{{\mathcal F}_{2g+1}}.$

Let $m=\deg (P)$. Since

$$\begin{aligned} \#\{B\in {\mathbb F}_q[x]: B \; \mathrm monic, \deg (B)=\beta , P\not \mid B\}= q^{\beta }\cdot \left\{ \begin{array}{lll} 1 &{}\quad \text{ if } m>\beta , \\ 1 -\frac{1}{|P|} &{}\quad \text{ if } m\le \beta , \end{array}\right. \end{aligned}$$

$$\begin{aligned} \langle \square _n \rangle _{{\mathcal F}_{2g+2}} \!= & {} \!-\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\deg (P)\frac{1}{(q\!-\!1)q^{2g+1}}\sum _{0\le \alpha \le g+1}\sum _{\begin{array}{c} {\deg (A)=\alpha }\\ {P\not \mid A} \end{array}}\mu (A)\sum _{\begin{array}{c} {\deg (B)=2g+2-2\alpha }\\ {P\not \mid B} \end{array}}1\\= & {} -\frac{q}{(q-1)q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\deg (P) \Biggr ( \Big (1-\frac{1}{|P|}\Big ) \sum _{\alpha \ge 0}\frac{\sigma _m(\alpha )}{q^{2\alpha }}+O(q^{-2g})\Biggr ). \end{aligned}$$

Solving the recurrence relation in Lemma 2.1,

$$\begin{aligned} \langle \square _n \rangle _{{\mathcal F}_{2g+2}}= & {} -\frac{q}{(q-1)q^{\frac{n}{2}}}\sum _{\deg (P^{k})=\frac{n}{2}}\deg (P)\Biggr (\Big (1-\frac{1}{|P|}\Big ) \frac{1-\frac{1}{q}}{1-\frac{1}{|P|^2}}+O(q^{-2g})\Biggr )\\= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \deg (P)\Big (\frac{|P|}{|P|+1}+O(q^{-2g})\Big )\\= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \deg (P)\Big (1-\frac{1}{|P|+1}+O(q^{-2g})\Big )\\= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \deg (P) +\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \deg (P) \Big (\frac{1}{|P|+1}+O(q^{-2g})\Big ). \end{aligned}$$

Since

$$\begin{aligned}&q^l=\sum _{\deg (h)=l} \Lambda (h)=\sum _{\deg (P^k)=l}\deg (P)=\sum _{\deg (P)|l}\deg (P),\\&\langle \square _n \rangle _{{\mathcal F}_{2g+2}} = -1+\frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \frac{\deg (P)}{|P|+1}+O(q^{-2g}). \end{aligned}$$

3.1.3 Contribution of the higher prime powers: $\mathbb {H}_n$

Using trivial bounds for ${\mathbb H}_n$, we obtain slightly different results from Rudnick in the case where $n>6g$. This is due to the fact that our estimates of ${\mathbb H}_n$ involve $S(\beta ;n)$ for $\beta$ even, as opposed to $\beta$ odd, and, from Lemma 2.3,

$$\begin{aligned} S(\beta ;n)\ll \left\{ \begin{array}{lll} q^{n+\beta } &{} \quad \text {if }\beta \, \text {is even}, \\ q^{\frac{n}{2}+\beta } &{} \quad \text {if }\beta \, \text { is odd.} \end{array}\right. \end{aligned}$$

We shall see that our bounds for $n>6g$ are absorbed in the error term from $\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}}$ when $n>2g+1$.

The contribution to ${{\mathrm{tr}}}(\Theta _{C_Q}^n)$ from the higher odd prime powers in (23) is:

$$\begin{aligned} {\mathbb H}_n= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {d|n}\\ {3 \le d: \mathrm{odd}} \end{array}}\sum _{\deg (P)=\frac{n}{d}}\frac{n}{d}\chi _Q(P^d)\\= & {} -\frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {d|n}\\ {3 {\le d: \mathrm{odd}}} \end{array}}\sum _{\deg (P)=\frac{n}{d}}\frac{n}{d}\chi _Q(P), \end{aligned}$$

where the last equality follows from the fact that $\chi _Q(P^d)=\chi _Q(P)$ for odd d.

This implies that

$$\begin{aligned} \langle {\mathbb H}_n \rangle _{{\mathcal F}_{2g+2}}= & {} -\frac{1}{(q-1)q^{2g+1+\frac{n}{2}}}\sum _{\begin{array}{c} {d|n}\\ {3 {\le d: \mathrm{odd}}} \end{array}}\frac{n}{d} \sum _{\deg (P)=\frac{n}{d}} \sum _{2\alpha + \beta = 2g+2} \sigma _{\frac{n}{d}}(\alpha ) \sum _{\deg (B)=\beta }\Big (\frac{B}{P}\Big )\\= & {} -\frac{1}{(q-1)q^{2g+1+\frac{n}{2}}}\sum _{\begin{array}{c} {d|n}\\ {3 {\le d: \mathrm{odd}}} \end{array}}\frac{n}{d}\sum _{2\alpha + \beta = 2g+2} \sigma _{\frac{n}{d}}(\alpha )S (\beta ;\frac{n}{d}). \end{aligned}$$

If $S(\beta ;\frac{n}{d})\ne 0$, then $\beta < \frac{n}{d}$; also, $S(\beta ; \frac{n}{d})\ll q^{\frac{n}{d}+\beta }$. Hence,

If $\frac{n}{3}\le 2g$, then $\min (\frac{n}{d},2g)=\frac{n}{d}$ for all $d\ge 3$; and so,

$$\begin{aligned} \langle {\mathbb H}_n \rangle _{{\mathcal F}_{2g+2}}\ll & {} \frac{n}{q^{2g+\frac{n}{2}}}\sum _{\begin{array}{c} {d|n}\\ {3 {\le d: \mathrm{odd}}} \end{array}}q^{\frac{2n}{d}}\ll \frac{n}{q^{2g+\frac{n}{2}}}q^\frac{2n}{3} = \frac{n}{q^{2g}}q^{\frac{n}{6}}\\\ll & {} \frac{g}{q^{2g}} q^g=gq^{-g}. \end{aligned}$$

If $\frac{n}{3}>2g$, then $\min (\frac{n}{d},2g)\le \frac{n}{3}$ for all $d\ge 3$; therefore,

$$\begin{aligned} \langle {\mathbb H}_n \rangle _{{\mathcal F}_{2g+2}} \ll \frac{n}{q^{2g+\frac{n}{2}}} q^\frac{2n}{3} \ll nq^{\frac{n}{6}-2g}. \end{aligned}$$

3.2 Computing $\langle \hbox {tr}(\Theta _{C_Q}^n)\rangle _{\mathcal {F}_{2g+2}}$

Since

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n)\rangle _{{\mathcal F}_{2g+2}} = -\frac{1}{q^{\frac{n}{2}}}+\langle {\mathcal P}_n \rangle _{{\mathcal F}_{2g+2}} + \langle \square _n \rangle _{{\mathcal F}_{2g+2}} + \langle {\mathbb H}_n \rangle _{{\mathcal F}_{2g+2}}, \end{aligned}$$

we obtain:

Theorem 3.1

$$\begin{aligned} \langle {{\mathrm{tr}}}\Theta _{C_Q}^n\rangle _{{\mathcal F}_{2g+2}}= & {} -\frac{1}{q^{\frac{n}{2}}}+\eta _n \frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \frac{\deg (P)}{|P|+1} +O(gq^{\frac{-g}{2}})\\&+ \left\{ \begin{array}{lll} -\eta _n, &{} \quad 0<n< 2g+1,\\ \frac{q^{\frac{1}{2}}}{q-1}, &{} \quad n=2g+1,\\ O(nq^{\frac{n}{2}-2g}), &{} \quad 2g+1<n.\\ \end{array}\right. \end{aligned}$$

3.3 Computing $\langle \hbox {tr}(\Theta _{C_Q}^n)\rangle _{\mathcal {F}_{2g+1}\cup \mathcal {F}_{2g+2}}$

The main result of [7] is given below:

Theorem 3.2

[7, Theorem 1]

$$\begin{aligned} \langle {{\mathrm{tr}}}\Theta _{C_Q}^n\rangle _{{\mathcal F}_{2g+1}} = \eta _n \frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \frac{\deg (P)}{|P|+1} +O(gq^{-g}) + \left\{ \begin{array}{lll} -\eta _n, &{}\quad 0<n<2g,\\ -1-\frac{1}{q-1}, &{}\quad n=2g,\\ O(nq^{\frac{n}{2}-2g}), &{}\quad 2g<n.\\ \end{array}\right. \end{aligned}$$

We would like to find the expected value of ${{\mathrm{tr}}}(\Theta _{C_Q}^n)$ over all curves $C_Q:y^2=Q(x)$ of genus g with Q monic, squarefree. To do this, we use Theorems 3.1 and 3.2. By identifying the family of curves described above with ${\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}$, we see that

$$\begin{aligned}&\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}\\&\quad =\frac{\#{\mathcal F}_{2g+1}}{\#({\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2})}\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1}}+\frac{\#{\mathcal F}_{2g+2}}{\#({\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2})}\langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+2}}, \end{aligned}$$

where

$$\begin{aligned} \#({\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2})= & {} \#{\mathcal F}_{2g+1}+\#{\mathcal F}_{2g+2}\\= & {} (q-1)q^{2g}+(q-1)q^{2g+1}\\= & {} q^{2g}(q-1)(q+1). \end{aligned}$$

Therefore,

Corollary 3.3

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}=- & {} \frac{1}{q^{\frac{n}{2}}}\frac{q}{q+1} +\eta _n \frac{1}{q^{\frac{n}{2}}}\sum _{\deg (P)|\frac{n}{2}} \frac{\deg (P)}{|P|+1} +O(gq^{\frac{-g}{2}})\\+ & {} \left\{ \begin{array}{lll} -\eta _n, &{} \quad 0<n< 2g,\\ -1-\frac{1}{q^2-1}, &{} \quad n=2g,\\ \frac{q^{\frac{3}{2}}}{q^2-1}, &{} \quad n=2g+1,\\ O(nq^{\frac{n}{2}-2g}), &{} \quad 2g+1<n.\\ \end{array}\right. \end{aligned}$$

Note 2 The first main term in Corollary 3.3 does not appear in Theorem 3.2, neither does the term $\frac{q^{\frac{3}{2}}}{q^2-1}$ corresponding to $n=2g+1$. Similarly, for $n=2g$, the constant $\frac{1}{q-1}$ in Theorem 3.2 is scaled down to $\frac{1}{q^2-1}$ in Corollary 3.3. In the next section, we shall see that these differences are diminished when we consider the average of ${{\mathrm{tr}}}(\Theta _{C}^n)$ over ${\mathscr {H}}_g$.

4 Computing $\langle \hbox {tr}(\Theta _{C}^n)\rangle _{\mathscr {H}_g}$

As we mentioned in the introduction, averaging over monic squarefree polynomials of a fixed degree is not the same as averaging over the moduli space of hyperelliptic curves of genus g: in the latter case, we consider polynomials of degree $2g+1$ and $2g+2$. Also, by restricting ourselves to monic polynomials, we introduce a bias in the average value of the trace: the contribution of the point at infinity is related to the leading coefficient of Q, as seen by Eq. (10).

We now turn our attention to finding the average of ${{\mathrm{tr}}}(\Theta _{C}^n)$ over ${\mathscr {H}}_g$. If we let

$$\begin{aligned} \widetilde{{\mathcal F}}_g=\widehat{{\mathcal F}}_{2g+1} \cup \widehat{{\mathcal F}}_{2g+2}, \end{aligned}$$

then, from Eqs. (5), (10), and (13),

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}= & {} \frac{1}{\#\widetilde{{\mathcal F}}_g} \sum _{Q\in \widetilde{{\mathcal F}}_g}\Big ( -\frac{\lambda _Q^n}{q^\frac{n}{2}}-\frac{1}{q^\frac{n}{2}}\sum _{\deg (f)=n}\Lambda (f) \chi _Q(f)\Big ), \end{aligned}$$

(26)

where

Given $D\in {\mathbb F}_q[x]$ with $\deg (D)=d$, we may write $D=A^2 B$, where $A,B\in {\mathbb F}_q[x]$ with A monic, B not necessarily monic, and $\deg (A)=\alpha$, $\deg (B)=\beta$, so that $d=2\alpha + \beta$. From here, we can take the character sum above over all elements of degree $2g+1,2g+2$ by sieving out the squarefree terms (as we did earlier):

$$\begin{aligned} \sum _{Q\in \widetilde{{\mathcal F}}_g}\chi _Q(f)= & {} \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sum _{\begin{array}{c} {\deg (B)=\beta }\\ B\, \text {n.n.m.} \end{array}} \sum _{\deg (A)=\alpha } \mu (A)\Big (\frac{A}{f}\Big )^2\Big (\frac{B}{f}\Big )\\= & {} \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}}\sigma (f;\alpha ) \sum _{\begin{array}{c} {\deg (B)=\beta }\\ {B\, \text {n.n.m.}} \end{array}}\Big (\frac{B}{f}\Big )\\= & {} \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{a\in {\mathbb F}_q^*}\sum _{\deg (B)=\beta }\Big (\frac{aB}{f}\Big )\\= & {} \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{a\in {\mathbb F}_q^*}\sum _{\deg (B)=\beta }\Big (\frac{a}{f}\Big )\Big (\frac{B}{f}\Big )\\= & {} \sum _{a\in {\mathbb F}_q^*}\Big (\frac{a}{f}\Big ) \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{\deg (B)=\beta }\Big (\frac{B}{f}\Big ). \end{aligned}$$

Hence,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}\!=\!-\frac{1}{q^\frac{n}{2} \#\widetilde{{\mathcal F}}_g}\Biggr ( \sum _{Q\in \widetilde{{\mathcal F}}_g}\lambda _Q^n+\! \sum _{\deg (f)=n} \Lambda (f)\sum _{a\in {\mathbb F}_q^*}\Big (\frac{a}{f}\Big ) \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{\deg (B)=\beta }\Big (\frac{B}{f}\Big )\Biggr ). \end{aligned}$$

Since there are exactly $(q-1)/2$ squares and $(q-1)/2$ non-squares in ${\mathbb F}_q^*$, if n is odd,

$$\begin{aligned} \frac{1}{\#\widetilde{{\mathcal F}}_g}\sum _{Q\in \widetilde{{\mathcal F}}_g}\lambda _Q^n=\frac{1}{\#\widetilde{{\mathcal F}}_g}\sum _{Q\in \widetilde{{\mathcal F}}_g}\lambda _Q=0. \end{aligned}$$

On the other hand, if n is even,

$$\begin{aligned} \frac{1}{\#\widetilde{{\mathcal F}}_g}\sum _{Q\in \widetilde{{\mathcal F}}_g}\lambda _Q^n=\frac{1}{\#\widetilde{{\mathcal F}}_g}\sum _{Q\in \widetilde{{\mathcal F}}_g}|\lambda _Q|=\frac{\#\widehat{{\mathcal F}}_{2g+2}}{\#\widetilde{{\mathcal F}}_g}=\frac{q}{q+1}. \end{aligned}$$

Also, if f is a power of some prime in ${\mathbb F}_q[x]$, say $f=P^k$, then for all $a\in {\mathbb F}_q^*$ (see Proposition 3.2 of [6]),

$$\begin{aligned}\Big (\frac{a}{f}\Big )=\Big (\frac{a}{P}\Big )^k=\Big (a^{\frac{q-1}{2}\deg (P)}\Big )^k=a^{\frac{q-1}{2}\deg (f)}.\end{aligned}$$

If $\deg (f)=\deg (P^k)=n$ is even, then $\Big (\frac{a}{f}\Big )=1$ because $|{\mathbb F}_q^*|=q-1$. This tells us that $\sum _{a\in {\mathbb F}_q^*}\Big (\frac{a}{f}\Big )=q-1$. If $\deg (f)=\deg (P^k)=n$ is odd, then we have that $\sum _{a\in {\mathbb F}_q^*}\Big (\frac{a}{f}\Big )=0$ because there are exactly $(q-1)/2$ QR in ${\mathbb F}_q^*$ and exactly $(q-1)/2$ NQR in ${\mathbb F}_q^*$.

So, for n odd,

$$\begin{aligned}\langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}=0.\end{aligned}$$

For n even,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}= & {} -\frac{1}{q^\frac{n}{2}}\frac{q}{q+1}-\frac{1}{\#\widetilde{{\mathcal F}}_g q^{\frac{n}{2}}}\\&\times \sum _{\deg (f)=n} \Lambda (f)\sum _{a\in {\mathbb F}_q^*}\Big (\frac{a}{f}\Big ) \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{\deg (B)=\beta }\Big (\frac{B}{f}\Big )\\= & {} -\frac{1}{q^\frac{n}{2}}\frac{q}{q+1}-\frac{1}{\#({\mathcal F}_{2g+1}\cup {\mathcal F}_{2g+2}) q^{\frac{n}{2}}}\\&\times \sum _{\deg (f)=n} \Lambda (f) \sum _{\begin{array}{c} {2\alpha +\beta =d}\\ {d=2g+1,2g+2} \end{array}} \sigma (f;\alpha )\sum _{\deg (B)=\beta }\Big (\frac{B}{f}\Big )\\= & {} -\frac{1}{q^\frac{n}{2}}\frac{q}{q+1}-\frac{1}{q^\frac{n}{2}}\sum _{\deg (f)=n} \Lambda (f) \langle \chi _Q(f) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}. \end{aligned}$$

In other words,

Theorem 4.1

For n odd,

$$\begin{aligned}\langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}=0,\end{aligned}$$

and for n even,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}= & {} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}\\= & {} \frac{1}{q^{\frac{n}{2}}}\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}}\\ {\deg (P)\ne 1} \end{array}} \frac{\deg (P)}{|P|+1} +O(gq^{\frac{-g}{2}}) + \left\{ \begin{array}{lll} -1, &{} \quad 0<n<2g,\\ -1-\frac{1}{q^2-1}, &{} \quad n=2g,\\ O(nq^{\frac{n}{2}-2g}), &{} \quad 2g<n.\\ \end{array}\right. \end{aligned}$$

In particular,

Corollary 4.2

If n is odd, then

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g} = \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^n) dU. \end{aligned}$$

For n even with $3 \log _q(g)< n < 4g-5 \log _q(g)$ and $n \ne 2g$,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g} = \int _{{{\mathrm{USp}}}(2g)} {{\mathrm{tr}}}(U^n) dU + o\left( \frac{1}{g}\right) . \end{aligned}$$

Proof

The first part is clear. To prove the second part, we treat each non-mainterm in Theorem 4.1 separately and show that each of them contributes an error term of $o(\frac{1}{g})$ in the desired region.

Fix $\epsilon >0$. If $n<4g-(4+\epsilon )\log _q(g)$, then

$$\begin{aligned} \lim _{g\rightarrow \infty }g nq^{\frac{n}{2}-2g} \le \lim _{g\rightarrow \infty } g^2 g^{-2-\frac{\epsilon }{2}} =\lim _{g\rightarrow \infty } g^\frac{-\epsilon }{2}=0; \end{aligned}$$

i.e.,

$$\begin{aligned} O(nq^{\frac{n}{2}-2g})=o\left( \frac{1}{g}\right) . \end{aligned}$$

Note that

$$\begin{aligned} \frac{1}{q^\frac{n}{2}}\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}}\\ {\deg (P)\ne 1} \end{array}}\frac{\deg (P)}{|P|+1}=O\left( \frac{n}{q^\frac{n}{2}}\right) . \end{aligned}$$

If $n=(2+\epsilon )\log _q(g)$, then

$$\begin{aligned} \lim _{g\rightarrow \infty } g \frac{n}{q^\frac{n}{2}} \ll _\epsilon \lim _{g\rightarrow \infty }\frac{g\log _q(g)}{g^{1+\frac{\epsilon }{2}}}=0. \end{aligned}$$

So, for $n>(2+\epsilon )\log _q(g)$,

$$\begin{aligned} \frac{1}{q^\frac{n}{2}}\sum _{\begin{array}{c} {\deg (P)|\frac{n}{2}}\\ {\deg (P)\ne 1} \end{array}}\frac{\deg (P)}{|P|+1}=O\left( \frac{n}{q^\frac{n}{2}}\right) =o\left( \frac{1}{g}\right) . \end{aligned}$$

We have actually shown a stronger version of our statement; namely, for any fixed $\epsilon >0$ and for any even n with $(2+\epsilon ) \log _q(g)< n < 4g-(4+\epsilon ) \log _q(g)$ and $n \ne 2g$,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g} = -1 + o\Biggr (\frac{1}{g}\Biggr ). \end{aligned}$$

$\square$

We now look at another approach which quickly verifies the first result of Theorem 4.1. This argument was provided by Dr. Zeév Rudnick: fix a finite field ${\mathbb F}_q$ of odd cardinality q, let Q be any monic, squarefree polynomial of degree $2g+1$ or $2g+2$ in ${\mathbb F}_q[x]$ , and let $a\in {\mathbb F}_q^*$. Then

$$\begin{aligned} \#C_{aQ}({\mathbb F}_{q^n})= & {} \sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})} \Big (\chi _n(aQ(x_0))+1\Big )\\= & {} q^n+1+\sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})}\chi _n(aQ(x_0)), \end{aligned}$$

where $\chi _n$ is a multiplicative character on ${\mathbb F}_{q^n}$ defined by

$$\begin{aligned} \chi _n(\alpha ):=\left\{ \begin{array}{lll} 1 &{} \quad \text{ if } \alpha \, \text{ is } \text{ a } \text{ square } \text{ in } \, {\mathbb F}_{q^n}^*,\\ 0 &{} \quad \text{ if } \alpha =0,\\ -1 &{} \quad \text{ if } \alpha \, \text{ is } \text{ not } \text{ a } \text{ square } \text{ in } \, {\mathbb F}_{q^n}^*.\\ \end{array}\right. \end{aligned}$$

When $x_0$ is the point at infinity, $Q(x_0)$ is defined by the evaluation of $x^{2g+2}Q(\frac{1}{x})$ at $x=0$; i.e., $\chi _n(Q(\infty ))$ yields $\lambda _Q$ according to the count of (4). Moreover,

$$\begin{aligned} -q^{\frac{n}{2}}{{\mathrm{tr}}}(\Theta _{C_{aQ}}^n)=\sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})} \chi _n(aQ(x_0)). \end{aligned}$$

Therefore,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}= & {} \frac{-1}{q^\frac{n}{2}\#\widetilde{{\mathcal F}}_g}\sum _{a\in {\mathbb F}_q^*}\sum _{Q\in {\mathbb F}_q[x]}{''}\sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})} \chi _n(aQ(x_0))\\= & {} \frac{-1}{q^\frac{n}{2}\#\widetilde{{\mathcal F}}_g}\sum _{a\in {\mathbb F}_q^*}\chi _n(a)\sum _{Q\in {\mathbb F}_q[x]}{''}\sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})} \chi _n(Q(x_0)), \end{aligned}$$

where $\sum _{Q\in {\mathbb F}_q[x]}{''}$ indicates that the sum is over all monic, squarefree polynomials $Q\in {\mathbb F}_q[x]$ such that $\deg (Q)=2g+1$ or $\deg (Q)=2g+2$. When n is odd, there are exactly $\frac{q-1}{2}$ squares and $\frac{q-1}{2}$ non-squares in ${\mathbb F}_q^* \subset {\mathbb F}_{q^n}$, which tells us that

$$\begin{aligned} \sum _{a\in {\mathbb F}_q^*}\chi _n(a)=0. \end{aligned}$$

So, for odd n,

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}=0. \end{aligned}$$

On the other hand, when n is even, computing the average over the entire moduli space reduces to computing the average over the moduli space with the restriction that $a=1$: for even n, every element of ${\mathbb F}_q^*$ is a square in ${\mathbb F}_{q^n}^*$ so that

$$\begin{aligned} \sum _{a\in {\mathbb F}_q}\chi _n(a)=q-1 \end{aligned}$$

and

$$\begin{aligned} \langle {{\mathrm{tr}}}(\Theta _{C}^n) \rangle _{{\mathscr {H}}_g}= & {} -\frac{q-1}{q^\frac{n}{2}\#(\widehat{{\mathcal F}}_{2g+1}\cup \widehat{{\mathcal F}}_{2g+2})}\sum _{Q\in {\mathbb F}_q[x]}{''}\sum _{x_0\in {\mathbb P}^1({\mathbb F}_{q^n})} \chi _n(Q (x_0))\\= & {} \langle {{\mathrm{tr}}}(\Theta _{C_Q}^n) \rangle _{{\mathcal F}_{2g+1} \cup {\mathcal F}_{2g+2}}. \end{aligned}$$

Evidently, $\#(\widehat{{\mathcal F}}_{2g+1}\cup \widehat{{\mathcal F}}_{2g+2})=(q-1)\#({\mathcal F}_{2g+1}\cup {\mathcal F}_{2g+2} )$.