# zbMATH — the first resource for mathematics

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].

##### MSC:
 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations 11F85 $$p$$-adic theory, local fields
Full Text: