**Title:** Serre's Open Image Theorem

**Speaker:** Aaron Greicius

**Speaker Info:** Loyola University Chicago

**Brief Description:**

**Special Note**:

**Abstract:**

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.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.

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