By Haruzo Hida

This publication presents a finished account of a key, might be crucial, idea that types the foundation of Taylor-Wiles evidence of Fermat's final theorem. Hida starts with an outline of the speculation of automorphic types on linear algebraic teams after which covers the fundamental thought and up to date effects on elliptic modular varieties, together with a considerable simplification of the Taylor-Wiles facts via Fujiwara and Diamond. He deals a close exposition of the illustration idea of profinite teams (including deformation theory), in addition to the Euler attribute formulation of Galois cohomology teams. the ultimate bankruptcy offers an evidence of a non-abelian classification quantity formulation.

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**Example text**

1-2, its dimension must be finite. In other words, the structure of V* as a right vector-space over K is defined by the formula (8) (av,v*)v = (v, v* a)v. Conversely, if V and V* are dual groups, and V* has a structure of right vector-space over K, (8) may be used in order to define V as a left vectorspace over K. Thus we may still identify V with the dual of V* when their structures as vector-spaces over K are taken into account. If L is any closed subgroup of V, the subgroup L* of V* associated with L by duality consists of the elements v* of V* such that (v, v*)v = 1 for all VEL; in view of (8), this implies that, if L is a left module for some subring of K, L* is a right module for the same subring, and conversely.

V d } of Kover Qp such that R = I Zp Vi. By (3), we have now, for 1 ~ i~d, v~ 1, aiEZp: and therefore: (4) n (l+p2 vi)pV-l d ai =1+pV+I i= I d I aivi (pv+2R). avivi i with av;EZp for 1 ~ i ~d, and then 1 + Xv + I = (1 + xv) n(1 + p2 Vi) i It is now clear that we have (5) 1+ Xl = n(1 + P2Vi)b i 4· i p v-I avi • Lattices and duality over local fields 34 II where the bi are given, for 1 ~ i ~ d, by +00 b i = L pv-l avi · v= 1 This shows that, as a multiplicative Zp-module, the group 1 + p2 R is generated by the d elements 1 + p2 Vi; as it is an open subgroup of the compact group 1 + P, hence of finite index in 1 + P, and as 1 + P, as a Zp-module, is generated by the elements 1 + p2 Vi and by a full set of representatives of the classes modulo 1 + p2 R in 1 + P, this implies that 1 +P is finitely generated.

The concept of lattice, as developed for p-fields in §§ 1-2, cannot be applied to R-fields. The appropriate concept is here as follows: DEFINITION 3. By an R-lattice in a vector-space V of finite dimension over an R-field, we understand a discrete subgroup L of V such that VIL is compact. We have to recall here some elementary facts about discrete subgroups. Let G be a topological group, r a discrete subgroup of G, and