On the Joint Distribution Of and in Quadratic Twist Families
If is an elliptic curve with a point of order two, then work of Klagsbrun and Lemke Oliver shows that the distribution of within the quadratic twist family tends to the discrete normal distribution as .
We consider the distribution of within such a quadratic twist family when has a fixed value . Specifically, we show that for every , the limiting probability that is given by an explicit constant . The constants are closely related to the -probabilities introduced in Cohen and Lenstra’s work on the distribution of class groups, and thus provide a connection between the distribution of Selmer groups of elliptic curves and random abelian groups.
Our analysis of this problem has two steps. The first step uses algebraic and combinatorial methods to directly relate the ranks of the Selmer groups in question to the dimensions of the kernels of random -matrices. This proves that the density of twists with a given -Selmer rank is given by for an unusual notion of density. The second step of the analysis utilizes techniques from analytic number theory to show that this result implies the correct asymptotics in terms of the natural notion of density.
Recently, there has a lot of interest in the arithmetic statistics related to the quadratic twist family of a given elliptic curve . Much progress has been made towards understanding how 2-Selmer ranks are distributed in these families when either or has an Galois action. In both of these cases, there are explicit constants summing to one such that the proportion of twists with 2-Selmer rank is given by [Kane], [KMR2].
Strikingly, this is not true when has a single rational point of order two. In this case has a degree two isogeny and an associated Selmer group . Work of Xiong shows that if does not have a cyclic 4-isogeny defined over , then the distribution of the ranks of as varies among the squarefree integers less than tends to the distribution as , where is the discrete normal distribution with mean and variance [X]. In this case, maps into , showing that for any fixed , at least half of the quadratic twists of have 2-Selmer rank greater than .
This same result can be deduced by studying how varies under quadratic twist, where is the Selmer group associated to the dual isogeny of . In [KLO], Lemke Oliver and the second author shows that as varies among the squarefree integers less than , the distribution of tends to as .
This article studies the joint distribution of and conditional on a fixed value of . In particular, we prove the following:
Suppose is an elliptic curve with that does not have a cyclic 4-isogeny defined over and . Define
Then for any ,
Theorem 1 is similar to the results of Thorne and the first author regarding the distribution of -Selmer groups in the family of elliptic curves [KT].
1.1. Connections With the Cohen-Lenstra Heuristics
In 1984, Cohen and Lenstra conjectured that if is an imaginary quadratic field, then the probability that is isomorphic to a fixed finite abelian -group should be proportional to . This conjecture infers a distribution on the -rank of and Washington observed that this distribution is identical to one appearing in random matrix theory [WashCL]. Assuming the Cohen-Lenstra heuristic, the probability that has -rank is the same as the probability that a random matrix over has nullity as [FulmanGoldstein].
In their original paper, Cohen and Lenstra also defined a notion of the -probability of a group . Let be a random -group chosen with probability proportional to and be elements of chosen uniformly at random. The -probability of is the probability that . There is a similar notion for -ranks and Cohen and Lenstra obtain the following result.
Theorem 1.1 (Theorem 6.3 in [Cl]).
Define the -probability that has rank as the probability that . The -probability that a -group has rank is given by
While the notion of -probability is only sensible for , we may nonetheless extend the definition to include by defining it to be as in (1) if and zero otherwise. As can be seen, the contants in Theorem 1 are given by . That is, if , then for any , the probability that is equal to the probability that a random -group has rank . Other than the related results in [KT], this is the only instance in which these have been provably shown to arise in the context of arithmetic statistics.
1.2. Methods and Organization
The constants in Theorem 1.1 appear in the following well-known theorem from random matrix theory.
Let be a randomly chosen matrix over . Then the probabilty that the left nullspace of has dimension tends to exponentially quickly as .
This limiting behavior was known at least as far back as [KLS]. The fact that this convergence is exponential in follows from Theorem 1.1 in [FulmanGoldstein], for example. ∎
As described in Sections 2-LABEL:sec:selcoker, we equate the dimension of a co-dimension one subgroup of with the dimension of the left-nullspace of an matrix with entries in . If the entries of were independent and random, then we would be done. Unsurprisingly however, there are dependencies between the entries in . Nonetheless, in Section LABEL:sec:probapproach, we show that under some mild assumptions regarding the values of certain characters involving , is equivalent to a block diagonal matrix with high probability, where is an identity matrix and is an matrix with independent random entries. Section LABEL:sec:analytictech then uses techniques from analytic number theory to show that the assumptions we made regarding the characters involving are satisfied with sufficiently high probability. As a result, we obtain Theorem 1.
We would like to thank Benedek Valko for explaing to us how a result similar to Theorem LABEL:thm:SDLimitThm may be obtained via a generalization of the Markov chain approach developed in [KMR2]. We would also like to thank Jordan Ellenberg for pointing out the relationship between the constants and the notion of -probabilities in the work of Cohen and Lenstra.
We begin by defining the Selmer groups and and then giving an explicit description of the Selmer groups and associated to the quadratic twist of an elliptic curve by a squarefree integer .
Let be an elliptic curve with a single point of order two defined by
and set . There is an isogenous curve given by a model
and an isogeny with kernel . There is a Kummer map
where is the discriminant of .
We have similarly defined local Kummer maps
for every completion of which give a commutative diagram for every place of , where the restriction map is the natural map .