Counting elliptic curves with torsion, and a probabilistic local-global principle
Heilbronn Number Theory Seminar
17th March 2021, 4:00 pm – 5:00 pm
Can we detect torsion of a rational elliptic curve E by looking modulo primes? Well, for almost all primes p, the torsion subgroup E(Q)_tor maps injectively into E(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(F_p) for almost all primes p, what is the probability that m | #E(Q)_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.