Two-descent on some genus two curves

TL;DR

This paper investigates the rank bounds, 2-torsion in the Shafarevich-Tate group, and rational points of a specific family of genus two hyperelliptic curves over Q, providing new insights into their arithmetic properties.

Contribution

It introduces bounds for the Jacobian rank, identifies cases with 2-torsion in the Shafarevich-Tate group, and explores rational points on these curves.

Findings

Bounds for the Jacobian rank over Q

Many cases with 2-torsion in Shafarevich-Tate group

Results on rational points of the curves

Abstract

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C_p.