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
Read or Download Complex multiplication and lifting problems PDF
Best algebraic geometry books
Introduction to modern number theory : fundamental problems, ideas and theories
This variation has been known as ‘startlingly up-to-date’, and during this corrected moment printing you may be definite that it’s much more contemporaneous. It surveys from a unified viewpoint either the fashionable nation and the developments of continuous improvement in numerous branches of quantity concept. Illuminated by means of simple difficulties, the crucial rules of recent theories are laid naked.
From the stories of the 1st printing of this publication, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My normal influence is of a very great booklet, with a well-balanced bibliography, instructed! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors provide the following an up-to-the-minute consultant to the subject and its major functions, together with a couple of new effects.
An introduction to ergodic theory
This article presents an advent to ergodic concept compatible for readers understanding simple degree idea. The mathematical must haves are summarized in bankruptcy zero. it really is was hoping the reader should be able to take on study papers after interpreting the publication. the 1st a part of the textual content is worried with measure-preserving adjustments of likelihood areas; recurrence houses, blending homes, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy conception are mentioned.
- Rational Algebraic Curves: A Computer Algebra Approach (Algorithms and Computation in Mathematics)
- Algebras, Rings and Modules: Lie Algebras and Hopf Algebras (Mathematical Surveys and Monographs)
- An Invitation to Arithmetic Geometry (Graduate Studies in Mathematics, Vol 9) GSM/9
- Guide to Geometric Algebra in Practice
- Algebraic Geometry for Scientists and Engineers (Mathematical Surveys and Monographs)
- Théorie des Intersections et Théorème de Riemann-Roch, 1st Edition
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).