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arXiv · 1303.3500

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

Stefan Keil·Remke Kloosterman
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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.

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Stefan Keil, Remke Kloosterman. 2013-03-14. On the density of abelian surfaces with Tate- Shafarevich group of order five times a square. https://doi.org/10.2140/obs.2013.1.413

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