Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields

TL;DR

This paper uses Weil's explicit formulas to derive explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields, with applications to non-existence results for certain varieties.

Contribution

It provides the first explicit bounds on conductors over number fields and applies these to prove non-existence of certain abelian varieties with good reduction.

Findings

Explicit lower bounds for conductors over number fields

Bounds for conductors with specified bad reduction

Non-existence of certain abelian varieties with everywhere good reduction

Abstract

Following the work of Mestre, we use Weil's explicit formulas to compute explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields. Moreover, we obtain bounds for the conductor of elliptic curves and abelian varieties over $\mathbb{Q}$ with specified bad reduction and over number fields. As an application, for specific fields, we prove the non-existence of abelian varieties with everywhere good reduction.