Endomorphism rings of reductions of elliptic curves and abelian varieties

TL;DR

This paper investigates the properties of endomorphism rings of reductions of elliptic curves over number fields, demonstrating the existence of infinitely many reductions with prescribed divisibility properties of their discriminants, and extends some results to abelian varieties.

Contribution

It proves the existence of infinitely many reductions of elliptic curves with endomorphism discriminants satisfying specific divisibility and coprimality conditions, extending to abelian varieties.

Findings

Infinitely many places with discriminant divisible by N

Discriminant ratios relatively prime to NM

Results inspired by Serre's exercise and questions of Papikian and Cojocaru

Abstract

Let $E$ be an elliptic curve without CM that is defined over a number field $K$. For all but finitely many nonarchimedean places $v$ of $K$ there is the reduction $E(v)$ of $E$ at $v$ that is an elliptic curve over the residue field $k(v)$ at $v$. The set of $v$'s with ordinary $E(v)$ has density 1 (Serre). For such $v$ the endomorphism ring $End(E(v))$ of $E(v)$ is an order in an imaginary quadratic field. We prove that for any pair of relatively prime positive integers $N$ and $M$ there are infinitely many nonarchimedean places $v$ of $K$ such that the discriminant $\Delta(v)$ of $End(E(v))$ is divisible by $N$ and the ratio $\Delta(v)/N$ is relatively prime to $NM$. We also discuss similar questions for reductions of abelian varieties. The subject of this paper was inspired by an exercise in Serre's "Abelian $\ell$-adic representations and elliptic curves" and questions of Mihran Papikian and Alina Cojocaru.