The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point

Zev Klagsbrun,Robert J Lemke Oliver

doi:10.1186/s40687-014-0015-4

Abstract

In recent work, Bhargava and Shankar have shown that the average size of the 2-Selmer group of an elliptic curve over is exactly 3, and Bhargava and Ho have shown that the average size of the 2-Selmer group in the family of elliptic curves with a marked point is exactly 6. In contrast to these results, we show that the average size of the 2-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance.

Highlights

In recent work [1], Bhargava and Shankar showed that when all elliptic curves over Q are ordered by height, the average size of the 2-Selmer group is equal to 3
Similar work in a preprint of Bhargava and Ho shows that the average size is 6 when the average is taken over all elliptic curves with a marked point
Unlike the case of the generic marked point considered by Bhargava and Ho, the existence of this point affects the average size of the 2-Selmer group in an essential way - in particular, we show that the average size is no longer bounded

Summary

Introduction

In recent work [1], Bhargava and Shankar showed that when all elliptic curves over Q are ordered by height, the average size of the 2-Selmer group is equal to 3. Results Given an elliptic curve E/Q with a rational isogeny φ : E → E of degree p, one can associate to E a finite p-group called the φ-Selmer group, which we denote by Selφ(E/Q). We consider the distribution of T (E/E ) as E ranges over the set of elliptic curves with a rational two-torsion point.

Results

Conclusion