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The first goal of this summer school is to understand and prove the following theorem: Let rho:G_Q->GL2(C) be irreducible and odd with projective image A5. Assume that is unramified at 5. Then there exists a cuspidal weight one form f such that L(f,s)=L(rho,s). In particular,  satisfies the Artin conjecture.
This theorem is due to Taylor, Buzzard and Shepherd-Barron and was initiated just after the proof of Fermat Last Theorem by Wiles. We will see indeed that many ingredients of Wiles are used in the proof. Remark also that the case of projective image A5 was the only unknown case in dimension two. This theorem is also due to Khare and Wintenberger as a corollary of their proof of Serre’s modularity conjecture (so latter with a different proof). They even managed to remove the unramifiedness hypothesis at 5. Today one can adapt the proof of Taylor and friends to arbitrary totally real fields, and remove by the same occasion the unramifiedness assumption at 5.
The other goal of the summer school is to introduce p-adic modular forms, and to prove various classicality theorems for such p-adic modular forms. Here a classicity theorem is a statement saying that a particular p-adic modular form with good properties is in fact a usual modular form, perhaps just with a q-expansion in Qp[[q]] and not in C[[q]]. The link between both subjects will be a classicality result for 5-adic modular forms of weight one (we will try to understand why 5 plays such a strange role). Indeed, methods of Wiles will allow us to construct an object f associated to  as in the theorem, but f will be a 5-adic modular form of weight one, and perhaps not a classical form. To know the holomorphy of L(f,s) (and in fact already to define L(f,s) as a function of a complex variable s) one would need to prove the classicality of f.
All talks, including the introduction one, will last 1h30. Except in the introduction talk, every statement should be proven. You should use the notations of the following text, and not those of the references provided, because this will provide some uniformity between the different talks. Speakers can exchange material between adjacent talks. If you have any question on your talks (mathematical questions, references,...) you should contact Benoit Stroh at benoit.stroh AT gmail.com.