## Number Theory Seminar

### Thursday, November 21st, 2019

**Time:** 4:30-5:30 p.m. **Place:** Jeffery Hall 422

**Speaker:** Richard Leyland

**Title:** Isogenies of Elliptic Curves with Complex Multiplication.

**Abstract:** In my thesis work I seek to answer Mazur's Question which asks if there exists any isomorphisms of mod $N$ Galois representations attached to elliptic curves that are not induced by isogenies. The first step in answering this question is determining which isogenies of elliptic curves are defined over a field $F$. In this talk, I will show how to construct isogenies between CM elliptic curves by using ideals of the endomorphism rings. In particular, we will see that if the field of definition $F$ does not contain the CM field, then we can reduce the problem to finding cyclic isogenies.