Average Cyclicity for Elliptic Curves in Torsion Families

TL;DR

This paper establishes asymptotic formulas for the cyclicity of elliptic curve reductions within torsion families, aligning with existing conditional and average results, and introduces a new estimate for elliptic curves with specific torsion over finite fields.

Contribution

It provides the first asymptotic formulas for cyclicity in torsion families of elliptic curves and develops a novel analogue of Vlut's result for finite fields.

Findings

Asymptotic formulas match conditional and average results

New estimates for elliptic curves with specific torsion points

Results extend understanding of elliptic curve cyclicity in torsion families

Abstract

We prove asymptotic formulas for cyclicity of reductions of elliptic curves over the rationals in a family of curves having specified torsion. These results agree with established conditional results and with average results taken over larger families. As a key tool, we prove an analogue of a result of Vl\u{a}du\c{t} that estimates the number of elliptic curves over a finite field with a specified torsion point and cyclic group structure.