Vol. 1, No. 1, 2013

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

Stefan Keil and Remke Kloosterman

Vol. 1 (2013), No. 1, 413–435
Abstract

Let A = E1 × E2 be the product of two elliptic curves over , each having a rational 5-torsion point Pi. Set B := A(P1,P2). In this paper we give an algorithm to decide whether the order of the Tate-Shafarevich group of the abelian surface B is square or five times a square, under the assumptions that we can find a basis for the Mordell-Weil groups of E1 and E2 and that the Tate-Shafarevich groups of E1 and E2 are finite.

We considered all pairs (E1,E2) with prescribed bounds on the conductor and the coefficients in a minimal Weierstrass equation. In total we considered around 20.0 million abelian surfaces, of which 49.16% have Tate-Shafarevich groups of nonsquare order.

Keywords
Tate-Shafarevich groups, abelian surface, Cassels-Tate equation
Mathematical Subject Classification 2010
Primary: 11G10
Secondary: 11G40, 14G10, 14K15
Milestones
Published: 14 November 2013
Authors
Stefan Keil
Institut für Mathematik
Humboldt-Universität zu Berlin
Unter den Linden 6
D-10099 Berlin
Germany
Remke Kloosterman
Institut für Mathematik
Humboldt-Universität zu Berlin
Unter den Linden 6
D-10099 Berlin
Germany