Moments of Quadratic L-Functions over Function Fields

Definitions and mathematical statement

Let $q=p^r$, where $p$ is an odd prime and $r\geq1$ is an integer, and let ${\mathbb{F}}_q$ be the field with $q$ elements. For each integer $g\geq0$, put \[ \mathcal H_{2g+1}(q)=\{d\in{\mathbb{F}}_q[t]:d\text{ is monic and squarefree},\ \deg d=2g+1\}. \] Here squarefree means that no square of a nonconstant irreducible polynomial divides $d$.

The supplied record does not define $\chi_d$ or normalize its $L$-function. To make the statement precise, adopt the standard hyperelliptic convention as a provisional interpretation: for a monic irreducible polynomial $P$, define $\chi_d(P)=0$ if $P\mid d$, and otherwise define $\chi_d(P)\in\{1,-1\}$ according as the image of $d$ is a square or a nonsquare in $({\mathbb{F}}_q[t]/(P))^\times$. Extend multiplicatively to monic polynomials, with $\chi_d(1)=1$. For $d\in\mathcal H_{2g+1}(q)$, use the associated polynomial \[ \mathcal L_d(u)=\sum_{n=0}^{2g}\left(\sum_{\substack{f\in{\mathbb{F}}_q[t]\text{ monic}\\\deg f=n}}\chi_d(f)\right)u^n, \qquad L(s,\chi_d)=\mathcal L_d(q^{-s}). \] This convention must be checked against the source report; it is not explicitly specified in the supplied record.

The extracted conjecture is that, for every odd prime power $q$ and every integer $k\geq1$, there exists an explicit polynomial $R_{k,q}(X)$, of degree exactly $k(k+1)/2$, such that \[ \lim_{\substack{g\to\infty\\g\in{\mathbb{Z}}_{\geq0}}} \left(q^{-(2g+1)}\sum_{d\in\mathcal H_{2g+1}(q)}L(1/2,\chi_d)^k-R_{k,q}(g)\right)=0. \] Here $q$ and $k$ are fixed in the limit, and $R_{k,q}$ is independent of $g$. No uniformity in $q$ or $k$ is asserted. The normalization is exactly $q^{-(2g+1)}$, rather than division by the cardinality of $\mathcal H_{2g+1}(q)$. The supplied record gives neither a coefficient formula for $R_{k,q}$ nor its coefficient field, and does not specify what form of explicit description is intended.

The source labels this statement Conjecture 3. Its accompanying discussion reports a proof when $q$ is sufficiently large relative to $k$, using homological stability and homotopy theory. It describes an error involving positive constants $A,B$, but the extracted expression $O(4g(k+1)q^{-Ag+B})$ has unresolved typography, particularly the possible exponent on $4$. No precise error estimate or dependence of these constants is asserted here. The discussion says that sufficiently large $q$ gives a power-saving error and permits recovery of every coefficient of the moment polynomials; it presents this as evidence for the Conrey--Farmer--Keating--Rubinstein--Snaith recipe. These are supplied source claims requiring review, not independently verified conclusions.

Short English statement

For each fixed odd prime power $q$ and positive integer $k$, the conjecture predicts that the $k$-th central moment of quadratic $L$-functions over monic squarefree polynomials of degree $2g+1$, normalized by $q^{2g+1}$, equals an explicit polynomial in $g$ of degree exactly $k(k+1)/2$, with an error tending to zero. The supplied curation labels the problem partially solved, reporting results for sufficiently large $q$ relative to $k$. The character convention and the extracted error estimate require source review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006457$, source problem code catalogue entry, title Moments of Quadratic L-Functions over Function Fields, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2025. The registry's corrected_verified statement label is accompanied by a note that only formula delimiters, notation, grammar, and typography were repaired; it does not constitute independent mathematical verification here.
• Supplied report citation: Analytic Number Theory (2026), Oberwolfach Reports, pp. 2701--2764, DOI https://doi.org/10.4171/owr/2025/51. The excerpt labels the target Conjecture 3 and contains the page marker 2723. The supplied literature list calls this Oberwolfach Report 51/2025. The differing year designations are retained without reconciliation.
• Supplied literature reference: Hyperelliptic curves and moments, https://arxiv.org/abs/2302.07664.
• Supplied literature reference: Uniform twisted homological stability, https://arxiv.org/abs/2402.00354.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that the function-field moment formula has been proved with an explicit error term for $q$ sufficiently large relative to $k$, while the formulation for every odd prime power and every $k$ is not fully established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.