Distribution of Selmer Ranks of Twists of Elliptic Curves with Partial Two-Torsion

On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion

Zev Klagsbrun
Abstract.

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.

1. Introduction

1.1. Distributions of Selmer Ranks

Let be an elliptic curve defined over and let be its 2-Selmer group (see Section 2 for its definition). We define the 2-Selmer rank of , denoted , by

For a given elliptic curve and non-negative integer , we are able to ask what proportion of the quadratic twists of have 2-Selmer rank equal to .

Let be the set of squarefree natural numbers less than or equal to . Heath-Brown proved that for the congruent number curve , there are explicits constants summing to one such that

for every , where is the quadratic twist of by [HB]. This result was extended by Swinnerton-Dyer and Kane to all elliptic curves over with that do not have a cyclic 4-isogeny defined over [Kane], [SD]. More recently, Klagsbrun, Mazur, and Rubin showed that a version of this result is true for curves with and when a different method of counting is used [KMR]. These results state that if the mod-4 representation of a curve satisfies certain conditions, then there is a discrete distribution on 2-Selmer ranks within the quadratic twist family of . We show that this is not the case when and does not have a cyclic 4-isogeny defined over . Specifically, we prove the following:

Theorem 1.

Let be an elliptic curve defined over with that does not have a cyclic isogeny defined over . Then for any fixed ,

and

In particular, this shows that there is not a distribution function on 2-Selmer ranks within the quadratic twist family of .

Theorem 1 is proved by way of the result.

Theorem 2.

Let be an elliptic curve defined over with that does not have a cyclic isogeny defined over . Then the normalized distribution

converges weakly to the Gaussian distribution

where

for , , and as defined in Section 2.

Theorem 2 will follow from a variant of the Erdös-Kac theorem for squarefree numbers which is proved in Appendix LABEL:apf.

Xiong and Zaharescu proved results similar to Theorems 1 and 2 in the special case when and has a cyclic 4-isogeny defined over [XZ].

1.2. Layout

We begin in Section 2 by recalling the definitions of the 2-Selmer group and the Selmer groups associated with a 2-isogeny and presenting some of the connections between them. Section LABEL:twistplace examines the behavior of the local conditions for the -Selmer groups under quadratic twist. We prove Theorems 1 and 2 in Section LABEL:pfofmain by combining the results of Sections 2 and LABEL:twistplace with a variant of the Erdös-Kac theorem for squarefree numbers which we prove in Appendix LABEL:apf.

1.3. Acknowledgements

I would like to express my thanks to Karl Rubin for his helpful comments and suggestions, to Ken Kramer for a series of valuable discussions, as well as to Michael Rael and Josiah Sugarman for helpful conversations regarding the Erdös-Kac theorem.

2. Selmer Groups

We begin by recalling the definition of the 2-Selmer group. If is an elliptic curve defined over a field , then maps into via the Kummer map. The following diagram commutes for every place of , where is the Kummer map.

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