By Andre Weil
This quantity comprises the unique lecture notes awarded through A. Weil within which the concept that of adeles used to be first brought, together with numerous points of C.L. Siegel’s paintings on quadratic varieties. those notes were supplemented through 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 extra learn.
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Extra info for Adeles and Algebraic Groups
1: XO(xy) determines an isomorphism of 0A onto its dual group, and Ok is selforthogonal for this isomorphism. As a consequence, if we define the Fourier transform ~(y) ~(x) of a function on 0A by the formula we have the inversion formula and the Poisson summation formula L ~(Il) under suitable conditions for are valid if both ~ such that the series and ~ = '1'([3) L f3£ Ok IltO k ~ and ~. For instance, these formulas are continuous, absolutely integrable, and La~(X+Il), L[3~(y+[3) convergent; when that is so, we say that are absolutely and uniformly ~, ~ are "of Poisson type".
E. kv = ~) and Xv >o. Moreover, for almost all p, Ro = Mmn(Qp)' so that the image of f by ¢ is (Gm) = Up (the -p Qp Qp unit-group of 0). Algebraically, our conclusion follows at once from -p these facts; the same holds topologically (so that we may identify ¢(fA)/¢(f k) with Ik/k*) As in view of the final remark of Chapter II. (x ,v) .... x is an isogeny of f onto R*, (1-N(p)-1) set of convergence factors for f; let d' (x,v) is a be the corresponding measure. We shall discuss the number-field case; the function-field case can be treated quite similarly.
Groups, p. 65), proved by an elementary category argument. The same theorem will be used occasionally, sometimes without reference, in the next chapters. 1. The zeta-function of a central division algebra As formerly, whenever V is a variety, defined over a field k, we denote by Vk the set of points of V, rational over k; a vectorspace of dimension dover k can always be denoted by Rk, where R is an affine space of dimension d in the sense of algebraic geometry. In particular, any algebra over k can be so written; the obvious extension to R of the multiplication-law on the algebra Rk makes R into an algebra-variety, defined over k (which means that the multiplication-law on R is defined over k).
Adeles and Algebraic Groups by Andre Weil