Diophantine Criterion for Non-trivial Shafarevich-Tate Groups

TL;DR

This paper links the solvability of specific Diophantine quartic equations to the rank and Shafarevich-Tate groups of certain elliptic curves, revealing a dichotomy based on prime conditions.

Contribution

It establishes a criterion connecting Diophantine solutions with the structure of Shafarevich-Tate groups for a family of elliptic curves.

Findings

Elliptic curves with primes p ≡ 1 mod 8 exhibit a rank 2 or 0 dichotomy.

The Shafarevich-Tate group is trivial or isomorphic to Klein four group depending on solutions.

A sharp criterion determines the elliptic curve's rank and Tate-Shafarevich group structure.

Abstract

The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with generalized Fermat's equation. In this work, we are interested in the integer solutions of a similar quartic equation $pu^{2} = v^{2}+w^{2}$. For a particular form of $u,v$, and $w$, we prove that the elliptic curves $E_p: y^2 = x(x-1)(x+p^2)$, for primes $p \equiv 1 \pmod{8}$ where $q = (p^2+1)/2$ is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either $\mathrm{rank}(E_p(\mathbb{Q})) = 2$ with trivial Shafarevich-Tate group or $\mathrm{rank} = 0$ with 2- torsion subgroup of the Shafarevich-Tate group is isomorphic to the Klein four group.