By Professor André Weil (auth.)

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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