Large families of elliptic curves ordered by conductor

Abstract

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

Squarefree values of polynomial discriminants I

“…They have been recently adapted in [8] to determine the density of squarefree discriminants of elliptic curves over Q having two marked rational points. Other applications include determining the density of conductors in some families of elliptic curves [27] and the density of squarefree values taken by a 4 + b 3 ([24]).…”

The mean number of 2-torsion elements in the class groups of $n$-monogenized cubic fields

“…, then the left-hand side of ( 19) could not equal zero as the x 3 -term would dominate the other two terms. As in [36,Lemma 4.7], the number of integers α ∈ C with bounded height can be controlled by the discriminant and skewness of C.…”