gfarkas@math.utexas.eduDescription
Abelian varieties are central objects of increasing importance in areas of mathematics ranging from algebraic geometry and number theory to string theory. Abelian varieties can be thought of as the algebro-geometric analogue of compact complex Lie groups. 1-dimensional abelian varieties are the elliptic curves which played a crucial role in Wiles's proof of Fermat's Last Theorem. The importance of abelian varieties in algebraic geometry lies in the fact that there are natural ways to associate to each algebraic variety X an abelian variety A and one can investigate the geometry of X by looking at A. Examples of this are the Picard variety, the Albanese variety or some intermediate Jacobians. This class will present the general theory of abelian varieties. We plan to treat topics like complex tori, line bundles on abelian varieties and their cohomology. We will spend some time discussing the theory of theta functions. Special attention will be devoted to the study of linear systems on abelian varieties and their defining equations in various embeddings. We will then study in detail a special type of abelian varieties Jacobians of algebraic curves, which are important because every abelian variety can be represented as a quotient of a Jacobian. In the last part of the course we will concentrate on more recent developments involving moduli of abelian varienties and/or questions of arithmetic nature related to Mordell-Weil type Theorems. Some familiarity with the concepts of Algebraic Geometry is desirable although not absolutely necessary. I will not assume familiarity with the theory of schemes.
Essays
The students taking this class had to write a project on a topic (loosely) related to abelian varieties. Please note that these papers are student works which I have not edited in any way or checked for correctness or accuracy. Some of these essays might contain mistakes and should not be used as reference.Kevin Klonoff and D. B. McReynolds: The Jacobi triple product via representation theory (after Pressley and Segal)
Petar Ivanov and Paul Larsen: Prym varieties (after Mumford and Harris)
John Meth and Kelly Mc Kinnie: An algebraic construction of the Poincare bundle (following Mumford)
Matthias Ihl and Alex Kahle: Mirror symmetry for elliptic curves (after Polishchuk and Zaslow)
Brian Katz: Recognizing Jacobians and 2-theta functions (after Welters)
Mark Luxton: A compactification of A2 (after Alexeev and Hulek)
Silvia Adduci and Dave Jensen: A-discriminants for the reflexive polygons
Rohit Ghosh: Curves with many points (after van der Geer and van der Vlugt)
Paul Fili: Heights for elliptic curves (following Silverman)
Jiuyuan Li: Torelli's Theorem (following Andreotti)
Pippa Charters: Using quadratic forms to find curves with many points (after van der Geer and van der Vlugt)
Notes
These are the course notes in dvi and pdf format as taken and typed up by the students enrolled in this class. I have only done minimal editing. Please let me know if you find any typos or inaccuracies:January 19, 2006 dvi pdf (Notes taken by Brian Katz)
January 24, 2006 dvi pdf (Notes taken by Petar Ivanov)
January 26, 2006 dvi pdf (Notes taken by Paul Fili)
January 31, 2006 dvi pdf (Notes taken by Paul Larsen)
February 7, 2006 dvi pdf (Notes taken by Dave Jensen)
February 9, 2006 dvi pdf (Notes taken by Kelly McKinnie)
February 14, 2006 dvi pdf (Notes taken by Pippa Charters)
February 16, 2006 dvi pdf (Notes taken by Alex Kahle)
February 23, 2006 dvi pdf (Notes taken by Parker Lowrey)
February 28, 2006 dvi pdf (Notes taken by John Meth)
March 2, 2006 dvi pdf (Notes taken by Craig Michoski)
March 23, 2006 dvi pdf (Notes taken by Mark Luxton)
March 07, 2006 dvi pdf (Notes taken by Jiuyuan Li)