# Algebra/Geometry Seminar, Speaker Lance Miller, University of Arkansas: Isogeny-Torson graphs for elliptic curves over imaginary quadratic fields

### Event Description:

Title: Isogeny-Torson graphs for elliptic curves over imaginary quadratic fields

Abstract: Given an elliptic curve E/Q and a prime degree isogeny E -> E', can one compute the torsion subgroup of E'(Q) from E(Q)? Generalizing this question, Chiloyan and Lozano-Robledo gave a complete classification of so called isogeny-torsion graphs for elliptic curves E/Q. These are graphs with edges marked by primes representing prime degree isogeny, and nodes labeled by torsion subgroups of the source and target. There are exactly 52 isomorphism types. The critical theorem driving this is a classic precise bound by Mazur and Kenku for the size of isogeny graphs. We discuss work in progress over certain imaginary quadratic fields K, isolated by Bandwait, Najam, Padurariu, for which we can replicate the Mazur-Kenku theorem and discuss progress towards a possible classification of such isogeny-torsion graphs for elliptic curves over these K. This is work in progress with C. Boothe and M. Logal.