Counting elliptic curves with torsion, and a probabilistic local-global principle

Abstract: Can we detect torsion of a rational elliptic curve $E$ by looking modulo primes? Well, for almost all primes $p$, the torsion subgroup $E(\mathbb{Q})_{\operatorname{tor}}$ maps injectively into $E(\mathbb{F}_p)$; but the converse statement holds only up to isogeny, by a theorem of Katz. In this talk, we consider a probabilistic refinement for the elliptic curves themselves: if $m | \#E(\mathbb{F}_p)$ for almost all primes $p$, what is the probability that $m | \#E(\mathbb{Q})_{\operatorname{tor}}$? We answer this question in a precise way by giving an asymptotic count of rational elliptic curves by height with certain prescribed Galois image.

This is joint work with John Cullinan and Meagan Kenney.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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