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Modular forms and Galois representations. (English) Zbl 1005.11031

Summary: The goal of this survey is to review some results concerning the deep relation between modular forms and “geometric” Galois representations with values in \(\text{GL}(2,K)\) (\(K\) a \(p\)-adic field). Many questions in this classical setting remain open despite the momentous breakthrough by Wiles and Taylor-Wiles. One can even ask similar questions for more general automorphic forms like Siegel modular forms, replacing \(\text{GL}(2)\) by \(\text{GSp}(4)\); however this topic will not be touched in this paper [see the author and E.Urban, Ann. Sci. Ec. Norm. Supér. (4) 32, 499-574 (1999; Zbl 0991.11016)]. This paper is a written version of a talk given in Anogia (Creta) during the third Panhellenic Conference in Number Theory and Algebra, Sept. 1-3, 2000.

MSC:

11F80 Galois representations
11F11 Holomorphic modular forms of integral weight

Citations:

Zbl 0991.11016
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