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Download Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.) PDF

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

ISBN-10: 0821808958

ISBN-13: 9780821808955

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Example text

And the normalized Haar measure on o A If we put we obviously have vo' G is a eonneeted simply eonneeted simple CL l ; and We may assume that ~ is the produet of the o Tamagawa measure on §l. [gx,gy] = vo(g) [x,y] such that vo [x,y] ). in 40 w = ws' it has poles of order Z(w s ) other words 13, 14. and We observe that if (x - A) 's; cf. Z(ws ) at 1 at has poles of order 0, 2, 9, 11, 17, 19, 26, 28. In 1 1 1 1 1 at 0, 1, 42 , ~, SZ, 82, b(s) = rr(s + A), then they are the Kimura [22], p. 78. s = 13, 14 A's At any rate the residues of 21 are respectively -times We might mention that in general the functional equation for Z(w) is easier to obtain than the Siegel-Weil formula because neither precise information on Tamagawa numbers nor the Poisson formula in §2 is needed; the classical Poisson formula (P) in §l (pU) is enough.

5, Springer-Verlag (1965). -P. Serre, Arbres, amalgames, SL 2 , ast~risque 46 (1977). -P. Serre, Que1ques app1ieations de theoreme de densit~ de Chebotarev, Pub. math. S. 54 (1981), 123-201. 47 [44] T. Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132-188. [45] L. Strauss, Poles of a two-variable p-adic complex power, Trans. Amer. Math. Soc. 278 (1983),481-493. [46] T. Tamagawa, Adeles, Proc. Symp. pure Math. 9 (1966), 113-121.

G defined over If G k, we have This was proved by Haris [10] using Ono's formula for the Tamagawa numbers of isogenous connected semisimple algebraic groups; cf. [33]. Therefore we may assume that G is simply connected 26 if that is convenient. Now for every and di, the quotient of gauge forms, say i x and on gives a gauge form Si(x) = (dx/df(x»i on U(i); and it gives rise to an intrinsic Tamagawa measure on U(i)A. In addition to and I'(~) E'(~) dx Isil A we introduce Z I"(~) ie:k r Then modulo "Hasse principles," one of which will be explained in §3, (SW) will fol10w from, in fact it is almost equivalent to, the combination of (T) "k(HS) "k (G) (pi;) I" (

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Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

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