Geometric Methods in Algebra and Number Theory

by Bogomolov, Fedor; Tschinkel, Yuri

• ISBN13: 9780817643492
• ISBN10: 0817643494
• Format: Hardcover
• Copyright: 2004-12-01
• Publisher: Birkhauser

Summary

The transparency and power of geometric constructions has been a source of inspiration to generations of mathematicians. The beauty and persuasion of pictures, communicated in words or drawings, continues to provide the intuition and arguments for working with complicated concepts and structures of modern mathematics. This volume contains a selection of articles exploring geometric approaches to problems in algebra, algebraic geometry and number theory. Key topics include: * Curves and their Jacobians * Algebraic surfaces * Moduli spaces, Shimura varieties * Motives and motivic integration * Number-theoretic applications, rational points * Combinatorial aspects of algebraic geometry * Quantum cohomology * Arithmetic dynamical systems The collection gives a representative sample of problems and most recent results in algebraic and arithmetic geometry; the text can serve as an intense introduction for graduate students and those wishing to pursue research in these areas. Contributors: I. Bauer, F. Bogomolov, N. Budur, F. Catanese, C.-L. Chai, R. Cluckers, C. De Concini, J.S. Ellenberg, F. Grunewald, B. Hassett, T. Hausel, F. Loeser, J. Pineiro, R. Pink, C. Procesi, M. Spitzweck, P. Swinnerton-Dyer, L. Szpiro, H. Tamvakis, Y. Tschinkel, T.J. Tucker, A. Venkatesh, and Y.G. Zarhin.

Table of Contents

Beauville surfaces without real structures	p. 1
Couniformization of curves over number fields	p. 43
On the V-filtration of D-modules	p. 59
Hecke orbits on Siegel modular varieties	p. 71
Ax-Kochen-Ersov theorems for p-adic integrals and motivic integration	p. 109
Nested sets and Jeffrey-Kirwan residues	p. 139
Counting extensions of function fields with bounded discriminant and specified Galois group	p. 151
Classical and minimal models of the moduli space of curves of genus two	p. 169
Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve	p. 193
Mahler measure for dynamical systems on P[superscript 1] and intersection theory on a singular arithmetic surface	p. 219
A combination of the conjectures of Mordell-Lang and Andre-Oort	p. 251
Motivic approach to limit sheaves	p. 283
Counting points on cubic surfaces, II	p. 303
Quantum cohomology of isotropic Grassmannians	p. 311
Endomorphism algebras of superelliptic jacobians	p. 339
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