A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves

TL;DR

This paper proves a uniform version of a finiteness conjecture for CM elliptic curves, showing that only finitely many primes can have certain unramified torsion field extensions, with explicit bounds for small degree fields.

Contribution

It establishes a uniform finiteness result for primes related to CM elliptic curves and provides explicit bounds for fields of degree up to 100.

Findings

Finiteness of primes for CM elliptic curves with specific torsion field properties

Explicit bounds for primes when the number field degree ≤ 100

Validation of a uniform version of Rasmussen and Tamagawa's conjecture

Abstract

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we require these fields to be both a pro-$\ell$ extension of $F(\mu_{\ell^{\infty}})$ and unramified away from $\ell$, examples are quite rare. Indeed, it is expected that for a fixed dimension and field of definition, there exists such an abelian variety for only a finite number of primes. We prove a uniform version of the conjecture in the case where the abelian varieties are elliptic curves with complex multiplication. In addition, we provide explicit bounds in cases where the number field has degree less than or equal to 100.