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Wiles' work on Fermat's last Theorem highlighted the power of $p$-adicmethods to prove the existence of analytic continuations of $\zeta$ and $L$functions. These methods have become considerably more sophisticated in recentyears, and have produced a wealth of beautiful results: Hasse--Weil conjecturefor genus $2$ curves, holomorphy of $L$-functions of symmetric powers ofmodular forms, etc. We present some of these advances.

Prolongement analytique de fonctions $\zeta$ et de fonctions $L$