Download Arithmetic of p-adic Modular Forms by Fernando Q. Gouvea PDF
By Fernando Q. Gouvea
The primary subject of this examine monograph is the relation among p-adic modular kinds and p-adic Galois representations, and particularly the idea of deformations of Galois representations lately brought via Mazur. The classical concept of modular kinds is thought recognized to the reader, however the p-adic thought is reviewed intimately, with abundant intuitive and heuristic dialogue, in order that the publication will function a handy aspect of access to investigate in that region. 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 booklet will therefore be of curiosity to quantity theorists who desire to find out about p-adic modular kinds, best them swiftly to fascinating learn, and in addition to the experts within the subject.
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Extra resources for Arithmetic of p-adic Modular Forms
38 C h a p t e r II. Hecke and U Operators Proof: For all but the last statement, see [Ka73, Thm. 8. The decrease in overconvergence (from r p to r) is, of course, due to the fact that the quotient curve is more supersingular; the non-integrality is due to the fact that the pull-back of a non-vanishing differential along the quotient map (or along the dual isogeny) is not non-vanlshing (because both isogenies are of degree p).  Thus, the Frobenius endomorphism defined above preserves the space of p-adic modular forms with growth condition 1, but, except in the case of weight zero, maps overconvergent forms to (less) overconvergent forms only up to multiplication by a power of r.
The goal of this section is to try to understand the action of U on the various spaces of overconvergent forms. T h e difficulties are the same as for Frob: first, the question of the existence of the fundamental subgroup, and second (for the case where the weight is not zero), the problem of pulling back a non-vanishing differential via an isogeny of degree p. We want to determine to what extent U preserves overconvergence; since Frob only had good properties with respect to overconvergence up to tensoring with the fraction field K of B, we expect the same sort of behavior in the current case.
We want to determine to what extent U preserves overconvergence; since Frob only had good properties with respect to overconvergence up to tensoring with the fraction field K of B, we expect the same sort of behavior in the current case. The first crucial result is due to Katz. P r o p o s i t i o n I I . 3 . 2 Suppose N > 3 and p X N. Then, for any r 6 B such that ord(r) < 1/(p + 1), the homomorphism F r o b : M(B,0, N ; r p) @ K ) M(B,0, N ; r ) @ K is finite and dtale of rank p. 1]. 9 above.