By Ching-Li Chai, Visit Amazon's Brian Conrad Page, search results, Learn about Author Central, Brian Conrad, , Frans Oort
Abelian forms with complicated multiplication lie on the origins of sophistication box thought, they usually play a valuable position within the modern concept of Shimura forms. they're unique in attribute zero and ubiquitous over finite fields. This e-book explores the connection among such abelian kinds over finite fields and over arithmetically attention-grabbing fields of attribute zero through the examine of a number of ordinary CM lifting difficulties which had formerly been solved simply in particular situations. as well as giving entire options to such questions, the authors supply a number of examples to demonstrate the overall thought and current a close remedy of many basic effects and ideas within the mathematics of abelian forms, similar to the most Theorem of complicated Multiplication and its generalizations, the finer facets of Tate's paintings on abelian kinds over finite fields, and deformation concept. This booklet presents an awesome representation of the way sleek ideas in mathematics geometry (such as descent conception, crystalline equipment, and workforce schemes) might be fruitfully mixed with classification box idea to reply to concrete questions on abelian types. it will likely be an invaluable reference for researchers and complex graduate scholars on the interface of quantity thought and algebraic geometry
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Extra info for Complex multiplication and lifting problems
Moreover, G and G are S-ﬁnite if and only if G is S-ﬁnite. 4] for a generalization (using the fpqc topology). Proof. For any G -scheme T viewed as an S-scheme, let g ∈ G (T ) correspond to the given S-morphism T → G . Consider the set Eg (T ) that is the preimage under G(T ) → G (T ) of g . This is a sheaf of sets on the category of G -schemes equipped with the fppf topology, and as such it is a left G -torsor (strictly speaking, a left torsor for the G -group GG ) due to the given exact sequence.
6. Example. Consider a separable quadratic extension of ﬁelds K /K and a simple abelian variety A over K . Let σ ∈ Gal(K /K) be the non-trivial element, K × K via x ⊗ y → (xy, σ(x)y). Thus, the Weil restriction so K ⊗K K A := ResK /K (A ) satisﬁes AK A × σ ∗ (A ), so AK is not isotypic if and only if A is not isogenous to its σ-twist. 3. COMPLEX MULTIPLICATION 23 AK are obtained by taking A to be an elliptic curve over C with analytic model C/(Z ⊕ Zτ ) for τ ∈ C − R such that 1, τ, τ , τ τ are Q-linearly independent.
Ci fi kills A[ ]. For the purpose of Now consider c1 , . . , cn ∈ Z such that proving ci ∈ Z for all i, it is harmless to add to each ci any element of Z . Hence, ci fi : A → B makes sense and kills we may and do assume ci ∈ Z for all i, so A[ ]. 1] and ). Thus, ci fi = · h for some h ∈ Hom(A, B). Writing h = mi fi with mi ∈ Z, we get ci ⊗fi = · 1⊗mi fi in Z ⊗Z Hom(A, B). This implies ci = mi for all i, so we are done. A weakening of simplicity that is sometimes convenient is: 22 1. 2. Deﬁnition.
Complex multiplication and lifting problems by Ching-Li Chai, Visit Amazon's Brian Conrad Page, search results, Learn about Author Central, Brian Conrad, , Frans Oort