Download Adeles and Algebraic Groups by Andre Weil PDF

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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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.

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