By Serge Lang (auth.)

ISBN-10: 146125485X

ISBN-13: 9781461254850

ISBN-10: 1461254876

ISBN-13: 9781461254874

The small publication through Shimura-Taniyama almost about advanced multi is a vintage. It offers the consequences received via them (and a few via Weil) plication within the greater dimensional case, generalizing in a non-trivial manner the tactic of Deuring for elliptic curves, by means of aid mod p. partially during the paintings of Shimura himself (cf. [Sh 1] [Sh 2], and [Sh 5]), and a few others (Serre, Tate, Kubota, Ribet, Deligne etc.) it's attainable at the present time to make a extra snappy and wide presentation of the basic effects than used to be attainable in 1961. numerous individuals have stumbled on my lecture notes in this topic helpful to them, and so i've got made up our minds to submit this brief ebook to cause them to extra broadly to be had. Readers accustomed to the normal conception of abelian forms, and who desire to get speedily an idea of the basic proof of advanced multi plication, are instructed to appear 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 examining of the manu script leading to a couple of valuable feedback. S. LANG Contents bankruptcy 1 Analytic complicated Multiplication four I. confident convinced Involutions . . . 6 2. CM varieties and Subfields. . . . . eight three. software to Abelian Manifolds. four. development of Abelian Manifolds with CM 14 21 five. Reflex of a CM kind . . . . .

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The model over 0 is uniquely determined, and so is its reduction mod 13. 4 (Serre-Tate). Let A be an abelian variety defined over k. Then A has good reduction mod 13 if and only if13 is unramified in k(A(l) for some prime I ¥- p, and hence for all such primes l. This theorem is substantially deeper than the preceding ones, cf. [Se-T], Theorem 1. Unlike the other theorems which we have stated without proof, the Serre-Tate theorem depends on an extensive theory, that of the Neron model. To develop that theory would require an entire book: Neron's own paper has the length of a book.

I have recalled with proofs some elementary definitions and properties, and without proof some of the more advanced results in this direction. 3 of the next chapter. If the reader is willing to admit these, then he can completely disregard this chapter, except for §3, which recalls the basic facts concerning the l-adic representations and their reductions mod p. Only §3 of this chapter will be used in the sequel, after the congruence relations have been proved. §1. Invariant Differential Forms Let k be a field, called the constant field, which in practice will definitely have characteristic p > 0 in important cases.

Let P be a simple point of V. We say that functions ({Jl' .. ,({In in keY) are a family of local parameters at P if the following conditions are satisfied: PAR 1. They form a separating transcendence basis for k(V) over k. PAR 2. They are defined at P, that is they lie in the local ring at P. PAR 3. Iff E k(V) andfis defined at P, then 8f/8({J; lies in this local ring for all i. Suppose that V is affine, and let x = (Xl"'. ,x m ) be a generic point of V over k. Let P = (x'). Then ({Jl' ... ,({In form a family oflocal parameters at P if and only if the following conditions are satisfied: PAR 1'.

### Complex Multiplication by Serge Lang (auth.)

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