On the Distribution of 2-Selmer Ranks within Quadratic Twist Families of Elliptic Curves with Partial Rational Two-Torsion
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.
1.1. 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:
Let be an elliptic curve defined over with that does not have a cyclic isogeny defined over . Then for any fixed ,
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.
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
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.
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.
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.