On the density of abelian surfaces with Tate- Shafarevich group of order five times a square

TL;DR

This paper develops an algorithm to determine the order of the Tate-Shafarevich group for certain abelian surfaces formed from elliptic curves with rational five-torsion points, based on extensive computational analysis.

Contribution

It introduces a new algorithm for analyzing the Tate-Shafarevich group of abelian surfaces derived from elliptic curves with rational five-torsion points, supported by large-scale computational data.

Findings

Approximately 49.16% of the analyzed surfaces have a Tate-Shafarevich group of non-square order.

The algorithm effectively distinguishes between square and five-times-square orders of the Tate-Shafarevich group.

Large dataset of 20 million abelian surfaces was examined to study the distribution of Tate-Shafarevich group orders.

Abstract

Let A=E_1xE_2 be be the product of two elliptic curves over QQ, both having a rational five torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the Tate-Shafarevich group of the abelian surface B has square order or order five times a square, assuming that we can find a basis for the Mordell-Weil groups of both E_i, and that the Tate-Shafarevich groups of the E_i are finite. We considered all pairs (E_1,E_2), with prescribed bounds on the conductor and the coefficients on a minimal Weierstrass equation. In total we considered around 20.0 million of abelian surfaces of which 49.16% have a Tate-Shafarevich group of non-square order.