2004-05 VIGRE Number Theory Working Group

Organized by Brian Conrad and Chris Skinner

This year's topic is the theory of complex multiplication for abelian varieties. One goal is to produce a written account of the complete proof of the Main Theorem of Complex Multiplication using the techniques of modern algebraic geometry, and to determine the L-series of a CM abelian variety in terms of Hecke L-series. We hope to also treat some generalizations as required in the theory of Shimura varieties. As talks are TeXed up, they will be listed with a link below.

Outline for the year: pdf. ps.
Main Theorem for CM elliptic curves: pdf.
Analytic theory of abelian varieties: pdf ps.
Algebraic theory of abelian varieties: pdf.
CM abelian varieties: pdf.
Polarizations: pdf. ps.
Honda-Tate theory: pdf. ps.
The Shimura-Taniyama formula: pdf. ps.
Weil restriction and adelic points: pdf.
Reflex norms: pdf. ps.
Serre's tensor construction: pdf. ps.
Main Theorem of complex multiplication: pdf. ps.
L-functions for CM abelian varieties: dvi. ps.