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

Elia Gorokhovsky, Mengzhen Liu

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 Klüners. 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.

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

Abstract

We determine the large-genus limiting distribution of the 4-rank of the Picard group of hyperelliptic curves over a fixed finite field of odd characteristic. This is a function field analogue of a result of Fouvry and Klüners. Our computation agrees with (the Picard group analogue of) the Cohen--Lenstra--Gerth heuristics in the case , 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.
Paper Structure (19 sections, 38 theorems, 195 equations)

This paper contains 19 sections, 38 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Outline of the argument
  3. Notation
  4. Rédei Matrix for Function Fields
  5. The Pairing $\left\langle\cdot,\cdot\right\rangle_X$
  6. Divisors, branched covers, and second-order classes
  7. Parametrization of $\operatorname{Pic}^0(X_f)(k)[2]$ and $\operatorname{Hom}(\pi_1^{\operatorname{twist}}(X_f), \mathbb Z/2\mathbb Z)$
  8. The matrix of the pairing for a fixed curve
  9. Parametrization of hyperelliptic curves and their Rédei matrices
  10. Symmetry properties of the Rédei matrix
  11. Theory of Random $C$-symmetric Matrices over Finite Fields
  12. Symmetric matrices
  13. General $C$-symmetric matrices
  14. Equidistribution of Artin Symbols
  15. Equidistribution from high-degree points
  16. ...and 4 more sections

Key Result

Theorem 1.1

Fix an odd prime power $q$. Let $S, S', S" \subset \mathbb P^1_{\mathbb F_q}$ be disjoint finite sets of closed points. For integers $g \geq 1$, let $X_g \to \mathbb P^1_{\mathbb F_q}$ be chosen uniformly at random among hyperelliptic curves of genus $g$ which are ramified at a set of points contain

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['lem:phi-surjection']}.
  • Lemma 2.5
  • proof
  • ...and 62 more