This is a working seminar focusing on l-adic representations associated to elliptic curves. Elliptic curves (smooth curves of genus 1 with a fixed point) are fundamental objects in today's number theory. They posses very rich arithmetic and complex structure, and are subject to major open questions such as the Birch and Swinnerton-Dyer Conjecture and the Lang-Trotter Conjectures. Moreover, they have been used in cryptography starting with the 1980s and (most importantly) they played a crucial role in Wiles' celebrated proof of Fermat's Last Theorem from the mid 1990s.

Associated to an elliptic curve defined over the field of rational numbers (say) there is a family of l-adic Galois representations. In 1972, Serre proved that if the elliptic curve is without complex multiplication (the "generic" case), then each l-adic representation has image as large as possible provided that l is sufficiently large. The focus in this seminar is to understand what "sufficiently large" means in Serre's result, and in generalizations of Serre's result (due to Ribet) to l-adic representations associated to modular forms. We will attempt to survey works of Mazur, Serre, Kraus/Cojocaru, Masser and Wustholz, Duke, Cojocaru and Hall, Imin Chen, Darmon and Merel, Merel and Rebolledo-Hochart (based on Gross-Kudla).