On the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$

TL;DR

This paper investigates the algebraic and analytic ranks of Jacobians of hyperelliptic curves $y^2 = x^5 + m^2$, establishing conditions on $m$ that bound Selmer groups and ensure non-vanishing of $L$-functions, contributing to the Birch--Swinnerton-Dyer conjecture.

Contribution

It provides new conditions on $m$ for bounding Selmer groups and non-vanishing $L$-functions, leading to examples satisfying the BSD rank part.

Findings

Bounded the size of Selmer groups for certain $m$

Identified conditions for non-vanishing of $L$-functions

Constructed Jacobians satisfying BSD rank part

Abstract

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves $y^2 = x^5 + m^2$ for integers $m$. Namely, we first provide a condition on $m$ that gives a bound of the size of Selmer group and then we provide a condition on $m$ that makes $L$-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch--Swinnerton-Dyer conjecture.