Download Galois Representations in Arithmetic Algebraic Geometry by Anthony James Scholl, Richard Lawrence Taylor, Editors PDF

By Anthony James Scholl, Richard Lawrence Taylor, Editors

This publication is a convention complaints in line with the 1996 Durham Symposium on "Galois representations in mathematics algebraic geometry". The identify used to be interpreted loosely and the symposium coated fresh advancements at the interface among algebraic quantity thought and mathematics algebraic geometry. The booklet displays this and incorporates a mix of articles. a few are expositions of matters that experience got immense fresh recognition: Erez on geometric tendencies in Galois module idea; Mazur on rational issues on curves and types; Moonen on Shimura types in combined features; Rubin and Scholl at the paintings of Kato at the Birch-Swinnerton-Dyer conjecture; and Schneider on inflexible geometry. a few are learn papers by way of: Coleman and Mazur, Goncharov, Gross, Serre.

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If BK[0,1] and BK(0,1) denote the affinoid and wide open unit disks over K, At(BQp[0,1]BQp(° 1)/Bq (0,1))° may be identified as the ring of power series in the parameter variable T over AM which are of the form E000 A,,Tn Ilan] where Ii is the maxsuch that for some positive real number a, An E imal ideal of AM. ) It is less straightforward to define the notion of R. Coleman & B. Mazur 38 overconvergent modular form or family of overconvergent modular forms. Cf. 4 below for the general definition but, for now, we define overconvergence only for integral weights: Let wk for k E Z denote the k-th tensor power of w, and consider the vector space of rigid analytic sections of wk over the affinoid Z, (N p-) (v).

1 of Hida's book [H-ET] for further proofs and discussion. Let W = WN for N = 1 and W+ denote that part of weight space W consisting of characters i such that K(-1) = 1. We have the p-adic (-function (*(rc) (notation as in section B1 of [C-BMF]) which is a rigid analytic function on W+ away from the point it = 1 (at which it has a simple pole). ) = Lp(x,1 - s), where Lp(x, s) is the Kubota-Leopoldt p-adic L-function. For n > 1 set djn,(d,p)=1 which we view as an Iwasawa function on W+. ) 0 0, set °O vk(n)g EK(q) = 1 + S*pj n=1 and for rc = 1, put Erjq) = 1.

6. 1. If F(q) and G(q) are the q-expansions of overconvergent forms of weight-characters a and ,Q, then F(q)G(q) is the q-expansion of an overconvergent form of weight character a,d. 2. Let F be a convergent family of modular forms over an affinoid subspace X of W, with q-expansion coefficients in A°(X). Then the q-expansion of F is the q-expansion of a Katz modular function F over A°(X). Proof. What we have is that F(q)/E(q) is the q-expansion of a rigid analytic function on X x Z1(q). But A°(X x Z1(q)) = A°(X)®A°(Z1(q)) The Eigencurve 47 is just V00,i(A°(X)), where i = 2 if p = 2 and i = 1 otherwise, so F(q)/E(q) is the q-expansion of a Katz modular function over A° (X ).

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