Tenth Algorithmic Number Theory Symposium ANTS-X University of California, San Diego July 9 – 13, 2012 Tenth Algorithmic Number Theory Symposium (ANTS-X) July 9 – 13, 2012

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

Stefan Keil and Remke Kloosterman

Abstract: Let A=E_1 x E_2 be be the product of two elliptic curves over Q, both having a rational 5-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), such that the E_i have conductor or coefficients smaller than some given bounds. This gives 20.0 million pairs and we could apply the algorithm to 18.6 million of them. It turns out that about 49% of these pairs have a Tate-Shafarevich group of non-square order.

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