Title: Serre's Open Image Theorem
Speaker: Aaron Greicius
Speaker Info: Loyola University Chicago
The theorem in question, proved by J.-P. Serre in 1972, is a celebrated result about elliptic curves E/K, where K is a number field, and the families of two-dimensional p-adic representations (one for each prime p) of the absolute Galois group of K they generate. Serre showed that for a non-CM elliptic curve E/K these representations have open image for all p, and are in fact surjective for almost all p.Date: Wednesday, January 17, 2018
Beyond representing a landmark advance in number theory, Serre's theorem offers to the curious an excellent entry point to a wide vista of theory (profinite groups, arithmetic geometry, algebraic groups, p-adic Galois representations), complete with a diverse wealth of potential research directions, ranging from hands-on computational problems to deep conjectures.
I will endeavor both to give a nuts and bolts description of Serre's result, assuming no prior knowledge beyond basic field theory, and to convey a sense of the larger theoretical landscape it opens up to.