By Professor André Weil (auth.)

ISBN-10: 3662000466

ISBN-13: 9783662000465

ISBN-10: 3662000482

ISBN-13: 9783662000489

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Then, if f is any continuous function with compact support in G, we have (6) Sf(g)dlX(g)= S ( I. f(gy))dlX'(g); G GfT yeT here we have put g= cp(g), and the integrand in the right-hand side, which is written as a function of g but is constant on co sets g r, is to be understood as a function of g. This, in fact, is clear if the support of f is contained in any set g U, and the general case follows from this at once; also, as well known in integration theory, the validity of (6) for continuous functions with compact support implies its validity for all integrable functions, and for all measurable functions with values in R+.

Moreover, the above calculations show also that this expression is unique, which proves our assertions. § 4. Lattices over R. 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

The image of q, in the multiplicative group (Z/nZ) x of the integers prime to n modulo n, must have a finite order N; then qN 1 (n). As zn = 1 for every ZE r, we get now zqNV = z for all v~ 0 and all ZEr, hence w(z)=z, so that z= 1 (P) implies Z= 1. This shows that the morphism of r into k x = (R/pr induced by p is injective, and therefore that r is cyclic, that its order divides q -1, and that, if it is of order q -1, ru {O} is a full set of representatives of R/P in R. In particular, if K is commutative, we see that w induces on R x a morphism of R x onto the group M X of the (q-l)-th roots of 1 in K, that it maps R onto M = M x u {O}, and that it determines a bijection of R/Ponto M; moreover, every subgroup r of K x of order prime to p is then contained in M x; in particular, M x contains all the roots of 1 of order prime to p in K.

### Basic Number Theory by Professor André Weil (auth.)

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