Analytic continuation of p-adic modular forms and applications to modularity

The lecture series will start with a  brief introduction to rigid analytic geometry. I will then introduce modular curves from various viewpoints (complex analytic, algebraic, and p-adic analytic) and use them to give a geometric definition of  p-adic and overconvergent modular forms and Hecke operators. I will next show how to use the p-adic geometry of the modular curves towards p-adic analytic continuation of overconvergent modular forms. Finally, I will demonstrate an application of these results to modularity of certain Galois representations which can itself be used to prove certain cases of the Artin conjecture. If time allows, I would explain briefly how these ideas extend to higher dimensions by illustrating the easier case of Hilbert modular surfaces.

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