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On \(p\)-adic families of automorphic forms. (English) Zbl 1166.11322
Cremona, John (ed.) et al., Modular curves and Abelian varieties. Based on lectures of the conference, Bellaterra, Barcelona, July 15–18, 2002. Basel: Birkhäuser (ISBN 3-7643-6586-2/hbk). Prog. Math. 224, 23-44 (2004).
Summary: R. Coleman and B. Mazur [in: Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)] have constructed “eigencurves”, geometric objects parametrising certain overconvergent \(p\)-adic modular forms. We formulate definitions of overconvergent \(p\)-adic automorphic forms for two more classes of reductive groups – firstly for \(\text{GL}_1\) over a number field, and secondly for \(D^{\times}\), \(D\) a definite quaternion algebra over the rationals. We give several reasons why we believe the objects we construct to be the correct analogue of an overconvergent \(p\)-adic modular form in this setting.
For the entire collection see [Zbl 1032.11002].

11F33 Congruences for modular and \(p\)-adic modular forms
11F80 Galois representations
11F85 \(p\)-adic theory, local fields
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