By Fernando Q. Gouvea
The imperative subject of this study monograph is the relation among p-adic modular varieties and p-adic Galois representations, and particularly the idea of deformations of Galois representations lately brought by means of Mazur. The classical concept of modular types is believed identified to the reader, however the p-adic concept is reviewed intimately, with plentiful intuitive and heuristic dialogue, in order that the publication will function a handy element of access to investigate in that sector. the consequences at the U operator and on Galois representations are new, and may be of curiosity even to the specialists. an inventory 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 find out about p-adic modular types, major them speedily to attention-grabbing study, and in addition to the experts within the subject.
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Extra info for Arithmetic of p-adic Modular Forms
In the last part of this section, we obtain some preliminary results and make some conjectures as to what should be the case. 1 Definition To define the U operator, we start with the Frobenius endomorphism Frob : V ~ V, which, as was shown in the last section, is locally free of rank p. 3. The U Operator 43 a trace h o m o m o r p h i s m TrFrob : V ~V, defined by (TrFrobf)(E/A, T, z) : ~ f(E1, ~1, zl), where the sum is taken over the triples (E1,~'1,*1) which m a p (by quotient by the f u n d a m e n t a l subgroup) to the given triple (E,T,z).
Note, however, that the difficulty disappears if we consider all weights together, since it is easy to see that any m o d u l a r form of weight 1 over k will always have the same q-expansion as the reduction of some m o d u l a r form of weight p = 1 + (p - 1) (just multiply by Ep-1 and note that the reduction m a p for m o d u l a r forms of weight greater t h a n one is onto). C h a p t e r II T h e H e c k e and U O p e r a t o r s In this chapter we define p-adic versions of the classical Hecke operators.
I=2 with ord(a) > 0 (because E is supersingular), ord(ci) _> 1 for i ~ l(modp), and ord(cv) = 0 (because the formal group is of height 2). Note that since a (rood p) for any nonvanishing differential w on E, we have that, if ord(a) < 1, then ord(a) = ord(Ev_l(E,w)). We want to determine the curves (if any) that are mapped to E by quotient by their fundamental group. T h e o r e m I I . 3 . 5 Let 0 < ord(a) < p/(1 + p), so that the canonical subgroup H0 C E is defined, and let H1, H2, . . , Hp be the other finite fiat subgroup schemes of rank p orE.
Arithmetic of p-adic Modular Forms by Fernando Q. Gouvea