By Ching-Li Chai, Visit Amazon's Brian Conrad Page, search results, Learn about Author Central, Brian Conrad, , Frans Oort

Abelian forms with advanced multiplication lie on the origins of sophistication box concept, and so they play a significant position within the modern conception of Shimura forms. they're distinct in attribute zero and ubiquitous over finite fields. This ebook explores the connection among such abelian kinds over finite fields and over arithmetically attention-grabbing fields of attribute zero through the learn of a number of ordinary CM lifting difficulties which had formerly been solved purely in detailed situations. as well as giving whole options to such questions, the authors offer quite a few examples to demonstrate the final conception and current a close remedy of many primary effects and ideas within the mathematics of abelian kinds, akin to the most Theorem of advanced Multiplication and its generalizations, the finer features of Tate's paintings on abelian types over finite fields, and deformation conception. This publication offers a great representation of ways smooth recommendations in mathematics geometry (such as descent conception, crystalline tools, and crew schemes) should be fruitfully mixed with category box conception to reply to concrete questions on abelian kinds. will probably be an invaluable reference for researchers and complex graduate scholars on the interface of quantity concept and algebraic geometry

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**Extra info for Complex multiplication and lifting problems**

**Example text**

1 isotypicity is preserved as well. 8. CM abelian varieties. 4 as an abstract ring in its own right, and to thereby regard the embedding P → End0 (A) as additional structure on A. This is encoded in the following concept. 1. Deﬁnition. Let A be an abelian variety over a ﬁeld K, and assume that A has suﬃciently many complex multiplications. Let j : P → End0 (A) be an embedding of a CM algebra P with [P : Q] = 2 dim(A). Such a pair (A, j) is called a CM abelian variety (with complex multiplication by P ).

Using the above table, we can prove the following additional facts when the simple A admits suﬃciently many complex multiplications. 4. Proposition. Let A be a simple abelian variety of dimension g > 0 over a ﬁeld K, and assume that A admits suﬃciently many complex multiplications. Let D = End0 (A). 2). (2) If char(K) > 0 then D is of Type III or Type IV. Proof. By simplicity, D is a division algebra. Its center Z is a commutative ﬁeld. First suppose char(K) = 0. Let P ⊂ D be a commutative semisimple Qsubalgebra with [P : Q] = 2g.

1) implies that L acts faithfully on the Q -vector space V (A) of rank 2g. But L = w| Lw , where w runs over all -adic places of L, so each corresponding factor module V (A)w over Lw is non-zero as a vector space over Lw . Hence, 2g = dimQ V (A) = [Lw : Q ] = [L : Q] dimQ V (A)w w| w| with equality if and only if V (A) is free of rank 1 over L . Assume that equality holds, so V (A) is free of rank 1 over L . If A is not isotypic then by passing to an isogenous abelian variety we may arrange that A = B × B with B and B non-zero abelian varieties such that Hom(B, B ) = 0 = Hom(B , B).