Adeles and Algebraic Groups by Andre Weil

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.

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