## Abstract

In this paper we study the family of elliptic curves E/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 Δ(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 β_{E} := log |Δ(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.

Original language | English (US) |
---|---|

Pages (from-to) | 1538-1583 |

Number of pages | 46 |

Journal | Compositio Mathematica |

Volume | 157 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2021 |

### Funding

It is a pleasure to thank Manjul Bhargava, Étienne Fouvry, Benedict Gross, Hector Pasten, Peter Sarnak, and Jacob Tsimerman for many helpful conversations and comments. The second named author is supported an NSERC Discovery Grant and a Sloan Fellowship. The third named author is supported by an NSERC Discovery Grant.

## Keywords

- conductor
- discriminant
- elliptic curves

## ASJC Scopus subject areas

- Algebra and Number Theory