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 $E$ be an elliptic curve defined over $Q$ and let ${S e l}_{2} (E / Q)$ be its 2-Selmer group (see Section 2 for its definition). We define the 2-Selmer rank of $E / Q$ , denoted $d_{2} (E / Q)$ , by

d_{2} (E / Q) = {d i m}_{F_{2}} {S e l}_{2} (E / Q) - {d i m}_{F_{2}} E (Q) [2] .

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

Let $S (X)$ be the set of squarefree natural numbers less than or equal to $X$ . Heath-Brown proved that for the congruent number curve $y^{2} = x^{3} - x$ , there are explicits constants $α_{0}, α_{1}, α_{2}, \dots$ summing to one such that

lim X \to \infty \frac{| {d \in S (X) : d_{2} (E^{d} / Q) = r} |}{| S (X) |} = α_{r}

for every $r \in Z^{\geq 0}$ , where $E^{d}$ is the quadratic twist of $E$ by $d$ [HB]. This result was extended by Swinnerton-Dyer and Kane to all elliptic curves $E$ over $Q$ with $E (Q) [2] ≃ Z / 2 Z \times Z / 2 Z$ that do not have a cyclic 4-isogeny defined over $Q$ [Kane], [SD]. More recently, Klagsbrun, Mazur, and Rubin showed that a version of this result is true for curves $E$ with $E (Q) [2] = 0$ and $G a l (Q (E [2]) / Q) ≃ S_{3}$ when a different method of counting is used [KMR]. These results state that if the mod-4 representation of a curve $E$ satisfies certain conditions, then there is a discrete distribution on 2-Selmer ranks within the quadratic twist family of $E$ . We show that this is not the case when $E (Q) [2] ≃ Z / 2 Z$ and $E$ does not have a cyclic 4-isogeny defined over $Q (E [2])$ . Specifically, we prove the following:

Theorem 1.

Let $E$ be an elliptic curve defined over $Q$ with $E (Q) [2] ≃ Z / 2 Z$ that does not have a cyclic isogeny defined over $Q (E [2])$ . Then for any fixed $r$ ,

liminf X \to \infty \frac{∣ ∣ {d \in S (X) : d_{2} (E^{d} / Q) \geq r} ∣ ∣}{| S (X) |} \geq \frac{1}{2}

and

liminf X \to \infty \frac{∣ ∣ {\pm d \in S (X) : d_{2} (E^{d} / Q) \geq r} ∣ ∣}{2 | S (X) |} \geq \frac{1}{2} .

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

Theorem 1 is proved by way of the result.

Theorem 2.

Let $E$ be an elliptic curve defined over $Q$ with $E (Q) [2] ≃ Z / 2 Z$ that does not have a cyclic isogeny defined over $Q (E [2])$ . Then the normalized distribution

\frac{P_{r} (T (E / E^{'}), X)}{\sqrt{\frac{1}{2} log log X}}

converges weakly to the Gaussian distribution

G (z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{z} e^{\frac{w^{2}}{2}} d w,

where

for $X \in R^{+}$ , $r \in Z^{\geq 0}$ , and $T (E^{d} / E^{' d})$ 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 $E (Q) [2] ≃ Z / 2 Z$ and $E$ has a cyclic 4-isogeny defined over $Q$ [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 $ϕ$ and presenting some of the connections between them. Section LABEL:twistplace examines the behavior of the local conditions for the $ϕ$ -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 $E$ is an elliptic curve defined over a field $K$ , then $E (K) / 2 (K)$ maps into $H^{1} (K, E [2])$ via the Kummer map. The following diagram commutes for every place $p$ of $Q$ , where $δ$ is the Kummer map.

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