By Fernando Q. Gouvea

The significant subject of this study monograph is the relation among p-adic modular varieties and p-adic Galois representations, and particularly the speculation of deformations of Galois representations lately brought via Mazur. The classical concept of modular types is thought recognized to the reader, however the p-adic idea is reviewed intimately, with plentiful intuitive and heuristic dialogue, in order that the ebook will function a handy aspect of access to analyze in that quarter. the implications at the U operator and on Galois representations are new, and may be of curiosity even to the specialists. a listing of additional difficulties within the box is incorporated to lead the newbie in his study. The publication will therefore be of curiosity to quantity theorists who desire to know about p-adic modular kinds, top them swiftly to attention-grabbing examine, and in addition to the experts within the subject.

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**Example text**

P-adic M o d u l a r F u n c t i o n s 29 a classical m o d u l a r form of weight k - i, level Np, and nebentypus w i-k, so that the isomorphism maps the space of modular forms of weight i on FI(N) N F0(p) to the space of m o d u l a r forms of weight k, level Np, and nebentypus w ~-k (which is precisely the space of m o d u l a r forms of level Np which have p-adic weight (i, k)). A similar s t a t e m e n t could be m a d e for forms of level Np" with the a p p r o p r i a t e nebentypus characters, so that we m a y say that the isomorphism we have obtained preserves the classical subspaces (in the case when k E Z).

3. p-adic M o d u l a r F u n c t i o n s 27 i. f~(q) --~ f(q) in the p-adic topology of B[[q]], ii. X(x) - x k~ ( m o d p ~ ) f o r a l l x E Z ; . We have seen above t h a t the continuous characters X E Horn ..... ( Z ~ , Z ; ) can be indexed by pairs (i, k) E ( Z / ( p - 1 ) Z ) x Z p , via the decomposition Z x = ( Z / ( p - 1 ) Z ) x F ; recall t h a t the correspondence is given by the formula = where the second factor makes sense for any k E Zp because x/w(x) E F is a one-unit. Thus, it is clear that, for any X = X(i,k) as above, there exists a sequence k,~ as in the definition above, so t h a t It is useful to note t h a t this condition determines ks modulo p'~-~(p- 1), and that we m a y chose the kn to be increasing with n in the definition above (by multiplying the f,~ by a p p r o p r i a t e Eisenstein series).

For any subgroup H of order ~ in E, we can consider the quotient curve E/ti-I; let 7r denote the canonical projection E ~ E/H , and let /r denote the dual isogeny. Since £ does not divide Np ~, b o t h ~r and ~r induce isomorphisms between the kernels of multiplication b y Np~in E and in the quotient curve, so t h a t we m a y define a level structure ,' on E/~-I by ~' = ~r-10z. E ]~N ~r E/H In the same way, we can define a trivialization T' = T °7r-1 (this makes sense because 7r induces an isomorphism on the formal group over Zp, since ( t , p ) = 1).