4-rank distribution of Picard groups of hyperelliptic curves via $C$-symmetric matrices

TL;DR

This paper analyzes the distribution of the 4-rank of Picard groups of hyperelliptic curves over finite fields, confirming conjectures and heuristics related to class groups and matrix ensembles.

Contribution

It determines the large-genus limiting distribution of the 4-rank of Picard groups, extending previous heuristics and connecting to random matrix models.

Findings

Distribution matches Cohen–Lenstra–Gerth heuristics for certain fields.

Results hold regardless of ramification conditions.

Provides rank distribution for specific random matrix ensembles.

Abstract

We determine the large-genus limiting distribution of the 4-rank of the Picard group of hyperelliptic curves over a fixed finite field $\mathbb F_q$ of odd characteristic. This is a function field analogue of a result of Fouvry and Kl\"uners. Our computation agrees with (the Picard group analogue of) the Cohen--Lenstra--Gerth heuristics in the case $q \equiv 3\pmod{4}$, i.e., in the absence of roots of unity in the base field. When roots of unity are present, the result is of the same form as conjectured distribution for class groups of quadratic extensions of number fields containing roots of unity. The limiting distribution does not change when imposing finitely many conditions on the ramification behavior of the curves. In the process, we determine the rank distribution of a certain class of random matrix ensembles over finite fields determined by symmetry conditions.