By Serge Lang (auth.)

The small ebook by means of Shimura-Taniyama with reference to complicated multi is a vintage. It offers the implications received via them (and a few by way of Weil) plication within the larger dimensional case, generalizing in a non-trivial means the strategy of Deuring for elliptic curves, by way of aid mod p. partially in the course of the paintings of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and a few others (Serre, Tate, Kubota, Ribet, Deligne etc.) it really is attainable this present day to make a extra snappy and large presentation of the basic effects than was once attainable in 1961. numerous folks have stumbled on my lecture notes in this topic valuable to them, and so i've got determined to submit this brief publication to lead them to extra broadly to be had. Readers familiar with the traditional thought of abelian kinds, and who desire to get quickly an idea of the elemental evidence of advanced multi plication, are urged to seem first on the major theorems, bankruptcy three, §6 and bankruptcy four, §1, in addition to the remainder of bankruptcy four. The purposes of bankruptcy 6 may be profitably learn early. i'm a lot indebted to N. Schappacher for a cautious analyzing of the manu script leading to a couple of worthwhile feedback. S. LANG Contents bankruptcy 1 Analytic complicated Multiplication four I. optimistic convinced Involutions . . . 6 2. CM varieties and Subfields. . . . . eight three. program to Abelian Manifolds. four. development of Abelian Manifolds with CM 14 21 five. Reflex of a CM style . . . . .

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The dcfinition is phrased in such a way as to make A represent the functor of rational points in smooth extensions of the base. Such rational points correspond to sections of the Neron model. The Neron model commutes with smooth extensions of the base. It is also one of the basic properties that it commutes with the completion of 0 and the ground field k. Let s be the special point of S and As the special fiber. Then As is an algebraic group, not necessarily connected. Now for the Serre-Tate theorem, if A has non-degenerate reduction, then it is elementary and classical that the action of the Galois group is unramified on 1[CA).

Proof. Let wEHr, •. I' Then 1',. • ,1 = w1',. •. 1, we have, up to a permutation, We first remark that if (w-1x) Now we distinguish cases. = x for x = r, s, t then w- 1 = 1 so w = 1. 27 §6. Application to Cyclotomic Fields Suppose (w- 1 x) = x for some value of x, say x = t. Then the other two values r, s are permuted and so w2 = 1. If Hr,s,t # 1, then w # ± 1. Write r = r'D, s = s'D, t = t'D. If r' is not primitive mod M, then s' is also not primitive mod M (because of the permutation by w - 1) and so t' is also not primitive, contradicting the definition of D.

The general case is considerably more complicated, and we refer to Koblitz- Rohrlich for the proof. When N = p then D = 1, M = N. Furthermore, r, s, t are prime to p. 28 1. Analytic Complex Multiplication For a E Z(p)* let St(a) = L ceZ(p)* Bl ( - ca») (Jc- 1 E Q[Gal(Kp/Q)] P be the Stickelberger element, where B1(X) = X -!. Also put St(O) = O. The function St from Z(p) into the group algebra is called the Stickelberger distribution. It is obviously an odd function. Applying an odd character X of the Galois group to the Stickelberger element St(a) yields x(St(a)) = x(a)B1,x where This number is -# 0 because of the classical fact that 0-# L(1, X) = Bl,x times a non-zero factor.