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

 d2(E/Q)=dimF2Sel2(E/Q)−dimF2E(Q)[2].

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

 limX→∞|{d∈S(X):d2(Ed/Q)=r}||S(X)|=αr

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 ,

 liminfX→∞∣∣{d∈S(X):d2(Ed/Q)≥r}∣∣|S(X)|≥12

and

 liminfX→∞∣∣{±d∈S(X):d2(Ed/Q)≥r}∣∣2|S(X)|≥12.

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

 Pr(T(E/E′),X)√12loglogX

converges weakly to the Gaussian distribution

 G(z)=1√2π∫z−∞ew22dw,

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.

You are adding the first comment!
How to quickly get a good reply:
• Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
• Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
• Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters