By Andre Weil

This quantity comprises the unique lecture notes awarded by way of A. Weil within which the concept that of adeles used to be first brought, along with a variety of elements of C.L. Siegel’s paintings on quadratic varieties. those notes were supplemented by means of a longer bibliography, and by way of Takashi Ono’s short survey of next examine. Serving as an advent to the topic, those notes can also offer stimulation for extra examine.

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C'¥(O)/log q). '(O) = f D

. (il) Z4>(s) = Z'¥(n-s). B. In the function-field case, q is the number of elements of the field of constants of k). 2. The projective group of a central division algebra. 1. Let G be a locally compact unimodular group, g a closed subgroup of the center of G; put G' = G/g, and let dx, d'x', dgZ be Haar measures matching together topologically on G.

L! G and g are unimodular, there exists on G/g a gauge-form invariant by G. Let dx, do, dP be respectively a left-invariant gauge-form on G, a left-invariant gauge-form on g, and a relatively invariant gauge-form on G/g belonging to the character X of G; let canonical mapping of G onto G/g, and put P = ~(x) ~*(dP) is a differential form on G. Put a(x) = xg = ~*(dP), ~ be the for XE. G. e. such that s(sx) induces on g the form do. It is easily seen that the form a(x) II s(x) on G is a gauge-form which does not depend upon the choice of 13; this will be denoted symbolically by dp·do.

3 Isogenies. We recall that an isogeny is a homomorphism of an algebraic ,roup onto another of the same dimension; two groups G, G' are called - 44 isogenous if G" can be found so that there are isogenies of G" onto G and onto G'. 1, will give for instance the Tamagawa number of the special linear group of a division algebra. 1. If two groups G, G' are isogenous over k, every set of convergence factors for G is a set of convergence factors for G' • Assume that there is an isogeny f of G onto G' by means of representations of G, G' over k; into special linear groups, we can consider them as affine varieties; then, if x' = f(x), the coordinates of x' can be written as polynomials in those of x.