By Fernando Q. Gouvea
The critical 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 by means of Mazur. The classical idea of modular kinds is thought identified to the reader, however the p-adic idea is reviewed intimately, with considerable intuitive and heuristic dialogue, in order that the publication will function a handy aspect of access to investigate in that zone. the implications at the U operator and on Galois representations are new, and may be of curiosity even to the specialists. a listing of extra difficulties within the box is incorporated to steer the newbie in his study. The publication will therefore be of curiosity to quantity theorists who desire to find out about p-adic modular kinds, major them quickly to attention-grabbing learn, and likewise to the experts within the subject.
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Extra resources for Arithmetic of p-adic Modular Forms
Since for every n there are maps M ( B / p n B , kn, N;r) -~ M ( B / p " B , k , . N ; 1 ) , taking the inverse limit gives a map M(B, X(i,k), N; r) , M(B, X(,,k), N; 1). It is not clear that this map is an inclusion, because the maps modulo p'~ are not injective. 1 Let the spaces M(B,X(i,~),N;r) and the maps M(B,x(~,k),N;r) , M(B, X(i,k), N; 1) be defined as above. Are the maps c~ inclusions? In other words, can we think of overconvergent forms of weight (i,k) (as defined above) as a certain kind of p-adic modular forms of weight (i, k) ?
P We claim there is an injection D ~ V ( B , N). To see this, let ~" E B be a uniformizer, and let f = ~ f~ E D, where fi E M(K, Np ~',i). T h e n we have f(q) E B[[q]], and, for some n, 7r'~f E ~ M(B,i, Np~), hence ~r"f E V. Then, since (Tr'~f)(q) = 7r'~f(q), f(q) B((q))/V. 1 above), it follows t h a t there exists ] E V such t h a t ](q) = f(q). Hence we m a y define ot D f ,--+ , , V ] . 12) Note t h a t the injectivity follows at once from the equality of the q-expanslons, since B is flat over Zp.
This allows us to define "slope a eigenspaces" for U which generalize (the integral weight case of) Hida's space of "ordinary p-adic modular forms". This will also show that there are few eigenforms for U outside its kernel, in the precise sense that if we fix the weight of f and the valuation of )~, one gets only a finite dimensional space of overconvergent forms of the given weight with eigenvalues of the given valuation. In contrast, it is clear that, even in the overconvergent case, ker(U) is quite large (in fact, infinite-dimensional), because of the Frobenius endomorphism: given any f E M(B,k,N;r), we have f - F r o b V f C ker(U).
Arithmetic of p-adic Modular Forms by Fernando Q. Gouvea