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

Stefan Keil, Remke Kloosterman

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.

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

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.

Paper Structure

This paper contains 7 sections, 10 theorems, 25 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing a family of abelian surfaces
  4. Determining the global factor
  5. Determining the local factor
  6. Algorithm
  7. Results

Key Result

Lemma 3.1

Suppose $L=\mathbf{Q}$. Then $\ker \varphi_{\mathbf{Q}}\cong \mathbf{Z}/5\mathbf{Z}$ and $\ker \varphi^{\vee}_{\mathbf{Q}}=0$.

Theorems & Definitions (19)

  • Conjecture 2.1: Birch and Swinnteron-Dyer
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 4.1
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • ...and 9 more