By Anthony James Scholl, Richard Lawrence Taylor, Editors

This publication is a convention court cases in accordance with the 1996 Durham Symposium on "Galois representations in mathematics algebraic geometry". The identify used to be interpreted loosely and the symposium lined contemporary advancements at the interface among algebraic quantity idea and mathematics algebraic geometry. The publication displays this and incorporates a mix of articles. a few are expositions of matters that experience bought huge fresh recognition: Erez on geometric tendencies in Galois module idea; Mazur on rational issues on curves and kinds; Moonen on Shimura kinds 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 study papers by means of: Coleman and Mazur, Goncharov, Gross, Serre.

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One would eventually wish to treat all primes p and all tame levels N but in the present article we prove our main theorems only for p > 2 and tame level N = 1. For the rest of this section, then, we let p be an odd prime number and N = 1. 1 below we shall define a certain collection of residual GQ,s- representations into finite fields of characteristic p which include, up to equivalence, (the semi-simplification of) all characteristic p residual Galois representations coming from classical modular forms of p-power level.

2. This yields the existence of the factorization. Now suppose Q(T) and H(T) are entire series with constant term 1 in A(X){{T}} such that Q(T) is prime and Q(T)H(T) = F(T). y(T) divides H(T) or Q(T). One of them, say Pk(T), must divide Q(T) since Q(T) is not a unit and A(X){{T}} is an integral domain. 2. From this, we may deduce that the factorization of F(T) is unique. In the following, if U C X is an affinoid subdomain, and Z -> X a rigid space (over X) we let ZU = Z x x U denote the fiber product.

Denoting G,,(T) := En>0an(x)Tn we see that GE(T) E K{{T}}, has a zero in k of absolute value strictly less than 1. The same must be true about FA(T) which is patently false. Hence, G(T) E A{{T}} and we reach the same conclusion for H(T). 7. 11. , Tm]] be a power series ring in m variables over 0, the ring of integers in a field K which is finite over Qp. Denote its associated rigid space over K by XA. Let F(T) be a non-trivial Fredholm series over A. Pk(T)nk ... Proof. 1. In particular, for each rational number, 1 > r > 0, we have an affinoid subdomain of X,.

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