By Andre Weil

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

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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*. 3, fo is the connected component of the identity in the center - 56 - of r; it is isomorphic to Z*; the group G= fir 0 R(1) is isogenous to (it is perhaps the most general group isogenous to R(1) over k, but this will not be discussed here), and we investigate ,(G).

1. (i) s and of Z (s) is the sum of an entire function of - 41 - where Pk = [(s-1 )sk(sl] s=l' and '¥ is the Fourier transform of 4>. (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. G' , g. Let 6 be a discrete subgroup of G; put H= G/6, 0 = I:,("\g, 6' =gMg = Mo, H' = G' /6 ' = G/g6, Y= g/o = gM6.

Column-vectors of order m) over D, and let Rm act on ,om by (X,x)"'Xx for X£Rm, xEDm. Put - 48 - e= (we write (1) 1 for the unit-element of D). Call is generic over kin Dm if X der the action of is so in Rm, H is a Zariski-open subset of Dm over k. More preci- sely, if K is any field containing lows. D K, such that the p(x) over D' H the orbit of e un- (mr,r)-matrix = has the rank r. From now on, we assume that m~ 2, and we put G = R~ 1) , G' = R(1). it is easily seen that H is also the orbit of e under G.