By Andre Weil

This quantity comprises the unique lecture notes provided via A. Weil during which the concept that of adeles used to be first brought, together with a number of points of C.L. Siegel’s paintings on quadratic kinds. those notes were supplemented by way of a longer bibliography, and by way of Takashi Ono’s short survey of next study. Serving as an advent to the topic, those notes can also supply stimulation for additional learn.

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Extra resources for Adeles and Algebraic Groups

Example text

3. e. a representation of Zi into Gm, rational over k; moreover, the group of all such characters of Zi is generated by vi' and the group of all characters of Z* = TTiZi, rational over k, is generated by the characters As U is an isogeny of Z* into T, one concludes from this (by well- known elementary arguments in the theory of algebraic toruses) that the group of the characters of T, rational over k, is generated by r characters Xi' and that the characters Xi 0 U of Z* generate a sub- group of finite index of the similar group for Z*.

In the function-field case, q denotes the number of ele- ments of the field of constants of k). Let first F be a continuous function with compact support on the multiplicative group 8+ (resp. 1 shows that F(IN(x)l) has compact support in DA/D k, so that the integral in the left-hand side is convergent. For any to~ 8+ (resp. for any t o =qv) there is XoE DA* such that IN(x)1 =t; repla0 0 c1ng then, in the left-hand side, x by xox, we see that it does not change if F(t) is replaced by F(tot); in other words, as a function Of F.

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.