The aim of this talk is to understand $\ell$-adic Galois representations and associate them to normalized Hecke eigenforms of weight $2$. We will also associate these representations to elliptic curves over $\mathbb{Q}$. This will enable us to state the Modularity Theorem. We will also mention its special case which was proved by Andrew Wiles and led to the proof of Fermatâ€™s Last Theorem.

We will develop most of the central objects involved - modular forms, modular curves, elliptic curves, and Hecke operators, in the talk. We will directly use results from algebraic number theory and algebraic geometry.

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- Seminars for 2022

Last updated: 29 Nov 2023