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Large families of elliptic curves ordered by conductor

Part of: Algebraic number theory: global fields Arithmetic algebraic geometry Forms and linear algebraic groups

Published online by Cambridge University Press:  22 June 2021

Ananth N. Shankar ,
Arul Shankar  and
Xiaoheng Wang
Ananth N. Shankar
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI53706, USAashankar@math.wisc.edu
Arul Shankar
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ONM5S 2E4, Canadaashankar@math.toronto.edu
Xiaoheng Wang
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ONN2L 3G1, Canadax46wang@uwaterloo.ca
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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.

Keywords

elliptic curvesconductordiscriminant

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11R29: Class numbers, class groups, discriminants 11R45: Density theorems 11E76: Forms of degree higher than two
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 7 , July 2021 , pp. 1538 - 1583
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Copyright
© The Author(s) 2021

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