Tenth Algorithmic Number Theory Symposium ANTSX

On the density of abelian surfaces with TateShafarevich 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 5torsion point P_i. Set B=A/<(P_1,P_2)>. In this paper we give an algorithm to decide whether the TateShafarevich group of the abelian surface B has square order or order five times a square, assuming that we can find a basis for the MordellWeil groups of both E_i, and that the TateShafarevich 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 TateShafarevich group of nonsquare order.
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