Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

TL;DR

This paper proves the strong Birch and Swinnerton-Dyer conjecture for certain genus 2 abelian surfaces with absolutely simple Jacobians by showing their Shafarevich-Tate groups are trivial.

Contribution

It provides the first exact verification of the strong BSD conjecture for specific absolutely simple abelian surfaces.

Findings

Shafarevich-Tate group of these Jacobians is trivial

Verifies strong BSD conjecture for 28 specific cases

Confirms conjecture for genus 2 absolutely simple abelian surfaces

Abstract

Let $X$ be one of the $28$ Atkin-Lehner quotients of a curve $X_0(N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich-Tate group of $J/\mathbb{Q}$ is trivial. This verifies the strong BSD conjecture for $J$.