Numerical Evidence for a refinement of Deligne's Period Conjecture for Jacobians of Curves

TL;DR

This paper provides numerical evidence supporting a refined conjecture on the algebraic and integrality properties of special values of L-functions associated with Jacobians of curves, relating them to Tate-Shafarevich groups.

Contribution

It formulates a new conjecture on the integrality of normalized L-values for Jacobians and investigates it numerically via p-adic congruences, extending Deligne's Period Conjecture.

Findings

Numerical evidence supports the conjectured integrality properties.

p-adic congruences reveal relationships between L-values and Tate-Shafarevich groups.

Refined conjecture aligns with observed numerical patterns.

Abstract

Let $A/\mathbb{Q}$ be a Jacobian variety and let $F$ be a totally real, tamely ramified, abelian number field. Given a character $\psi$ of $F/\mathbb{Q}$, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value $\mathcal{L}(A,\psi,1)$ at $z=1$ of the Hasse-Weil-Artin $L$-function of the $\psi$-twist of $A$. We formulate a conjecture regarding the integrality properties of the family of normalised $L$-values $(\mathcal{L}(A,\psi,1))_{\psi}$, and its relation to the Tate-Shafarevich group of $A$ over $F$. We numerically investigate our conjecture through $p$-adic congruence relations between these values.