By Andre Weil

This quantity includes the unique lecture notes provided through A. Weil within which the idea that of adeles used to be first brought, at the side of quite a few elements of C.L. Siegel’s paintings on quadratic varieties. those notes were supplemented through a longer bibliography, and through Takashi Ono’s short survey of next study. Serving as an advent to the topic, those notes can also offer stimulation for additional learn.

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J. of Math. 78 : see last five lines of p. 561). 1, is a division algebra. 3 shows that (1) is such a set for R*/Z*, hence also for every group isogenous to R*/Z* (Ap) (in particular, for is such a set for every group isogenous to R(1)) and that (R*/Z*) x Gm. In what follows, we shall use these facts freely. , we denote Tamagawa measures derived from the set of convergence factors (1_N(p)-1); for instance, on I k, we use the Tamagawa measw'e (dt/t)'. 1, let Dk n2 be a division algebra of dimension over its center Zk = k; D being the algebra variety defined by the center Z, take R= Mm(D); call center.

1 this is the same as 1 (R~ 1) ) k (R ~ ( 1) )Ak. 2, implies T(R~1)) 1 of the operation = 1. / k, the algebraic group R~1), over the universal 1 - 55 - d. domain, is isomorphic to (SL(min i )) 1 where di = [k i : kJ and n2i is the dimension of Di. Write N for the mapping of R into its center Z = <±l iZi . 3, the norm mapping N: z+zmn of Z* into itself was factored into z+zmn/v+zmn. Similarly, we as- sume now that we have factored N as follows: where T is a commutative group-variety, defined over k, and ~, v Ire isogenies of Z* onto T and of Tonto Z*, also defined over k' then T is a torus, isogenous to Z*.

Ga and of induction assumption XI = 12n - 2 , is normal; gl glgl,. G and of I ; gl, consisting of the matrices = 12n - 2 and u = 0, is normal, and gl gl X for is the semidirect pro- ql • Ig" ,. (G a )2n-2. 3, that (1) is a set of convergence factors, first for gl, and then for g, that T(gl) = 1, and that T(g) = 1. 4, we see that we can identify H with :8/9. 3, (1) is a set of convergence factors for G. 2 to GA, gA' Gk , gk; this Itves - 54 - f f(x)dx f (l: f (XC;) ) dX GA/G k C;C:H k = HA (A k)2n; also, In the left-hand side, we can replace HA by Hk is the k2n - {O}.