By Serge Lang (auth.)
Read Online or Download Abelian Varieties PDF
Best algebraic geometry books
This vintage publication comprises an creation to structures of l-adic representations, a subject matter of serious value in quantity concept and algebraic geometry, as mirrored through the outstanding fresh advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one reveals a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now referred to as Taniyama groups).
This e-book resulted from experiences (published in 1928 and 1932) of the Committee on Rational variations, confirmed through the nationwide learn Council. the aim of the stories used to be to provide a finished survey of the literature at the topic. every one bankruptcy is considered a separate unit that may be learn independently.
This booklet is dedicated to contemporary development within the examine of curves and abelian types. It discusses either classical features of this deep and lovely topic in addition to vital new advancements, tropical geometry and the idea of log schemes. as well as unique study articles, this ebook includes 3 surveys dedicated to singularities of theta divisors, of compactified Jacobians of singular curves, and of ""strange duality"" between moduli areas of vector bundles on algebraic types
This e-book offers a self-contained, obtainable creation to the mathematical advances and demanding situations as a result of using semidefinite programming in polynomial optimization. This fast evolving study quarter with contributions from the varied fields of convex geometry, algebraic geometry, and optimization is named convex algebraic geometry.
- De Rham Cohomology of Differential Modules on Algebraic Varieties
- Measure, Topology, and Fractal Geometry
- Integrable Systems in the realm of Algebraic Geometry
- Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra
- Nilpotence and Periodicity in Stable Homotopy Theory
- Knotted Surfaces and Their Diagrams
Extra info for Abelian Varieties
Here of course, we have worked birationally and biholomorphically, and we have [II] HISTORICAL NOTE 53 completely lost track of the integral nature of parameters on which the curve may depend. On the other hand, Chow's method gives rise to Igusa's compatibility principle which allows us to study algebraic systems of Jacobian varieties [39J, . For instance, if we are given an algebraic system of curves, together with their canonical mappings into their Jacobians, if (C, j, f) is a generic member, and if C is a specialization of C which is non-singular, then (f, f) has a uniquely determined specialization (f', f') which is the Jacobian variety of C.
Let u be a generic point of f(C) . g-u = I. Then II 2 (t(P i )) i~1 where the Pi are independent generic points of C, uniquely determined (uP to a permutation) by the relation u = If~lf(Pi)' We are not going to give here the full theory of the Pontrjagin products of f(C) and of the relations which one can find between these products and their intersections, especially with respect to numerical equivalence. A number of questions concerning this topic are not yet completely cleared up. We shall limit ourselves to proving a result which is sufficient to show that f(C) is biholomorphic to C.
In order t? have a completely canonical situation, we proceed as follows. We shall say that a rational map f : V X V -J> A of the product V X V into an abelian variety is admissible if we can find a field k' and a rational map f' : V -J> A such that f(P, Q) = f'(P)-f'(Q) for P, Q independent generic points of V over k'. In view of Theorems 2 and 3, this amounts to saying that f vanishes on the diagonal. , f(P, P) = o. Of course, f may be defined over a field k, while f' is not defined over k. We note that f' is uniquely determined by f up to an additive constant.
Abelian Varieties by Serge Lang (auth.)