Odd-degree Jacobian mean and its arithmetic correction

Definitions and mathematical statement

John Voight's AIM question asks for the average number of points on the Jacobian of a hyperelliptic curve. The ground field, ensemble, weighting, genus regime, and meaning of average are not specified in the original. In the finite-field interpretation of the report, equation averages, stack-weighted averages, and unweighted curve-isomorphism averages must be distinguished.

Fix an odd prime power $q$. For $g\geq1$, let $\mathcal H_{2g+1,q}$ be the monic squarefree polynomials of degree $2g+1$ in ${\mathbb{F}}_q[T]$, and let $C_D$ be the smooth projective curve with affine equation $y^2=D(T)$. Define the uniform equation average \[ A_{g,q}=\frac1{|\mathcal H_{2g+1,q}|} \sum_{D\in\mathcal H_{2g+1,q}}\#\operatorname{Jac}(C_D)({\mathbb{F}}_q). \] The report attributes to Andrade the fixed-$q$, high-genus asymptotic \[ A_{g,q}\sim C_q q^g\quad(g\to\infty),\qquad C_q=\frac1{1-q^{-1}}\prod_P\left(1-\frac1{|P|^2(|P|+1)}\right), \] where $P$ runs over monic irreducible polynomials in ${\mathbb{F}}_q[T]$ and $|P|=q^{\deg P}$. This is a mean in the specified odd-degree equation ensemble, not automatically an average over all hyperelliptic moduli points.

The candidate correction expansion is, as $q\to\infty$ through odd prime powers, \[ C_q=1+q^{-1}+q^{-3}+2q^{-6}+O(q^{-7}). \] In particular the report asserts that the coefficients of $q^{-2},q^{-4},q^{-5}$ vanish. This expansion concerns the constants arising after the fixed-field large-genus limit; it is not a uniform joint $(q,g)$ asymptotic unless further estimates are given.

The report also claims a square-term local-density divisor-sum identity for $C_q$ over monic polynomials $H$, a stack-groupoid reduction, and an exact genus-one normalization check. The actual summand, stack normalization, and genus-one formula are not supplied in the record, so these cannot serve here as explicit reproducible identities. The report leaves even-degree and characteristic-two ensembles and the unqualified original averaging question distinct.

Short English statement

Specify the averaging ensemble before asking for a Jacobian point-count mean. The report gives an odd-degree equation asymptotic and a large-field expansion of its constant, while retaining separate stack, even-degree, and characteristic-two questions.

Sources

• AIM workshop "Arithmetic statistics over finite fields and function fields," Question 17, attributed to John Voight: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. No PDF was accessed here.
• Local canonical registry aim-algebraic-number-theory-notes.json, notes[46], tag question; permanent UnsolvedMath ID $20000395$, source code catalogue entry. Entire original asks for the average number of points on a hyperelliptic Jacobian. Original remarks and literature field are empty.
• Supplied research attempt 1 attributes the odd-degree asymptotic to Andrade and proposes the correction expansion and divisor-sum audit. No full authoritative bibliographic references or formulas for the auxiliary checks are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.