Akshay on Shimura Varieties Notes/errors by Jeremy Booher and Sam Lichtenstein

Shimura Varieties: the Geometric Side of the Zeta Function

These are my notes for four talks at the Institute for Advanced Study, February 21,23,28, and March 2, 1995. Appearances to the contrary, they are rough notes. After giving a brief introduction to Shimura varieties, and in particular explaining how to realize them as moduli varieties, we give a heuristic derivation of a formula for the number of points on the reduction of a Shimura variety with...

Canonical models of Shimura curves

As an introduction to Shimura varieties, and, in particular, to Deligne’s Bourbaki and Corvallis talks (Deligne 1971, 1979), I explain the main ideas and results of the general theory of Shimura varieties in the context of Shimura curves. These notes had their origin in a two-hour lecture I gave on September 10, 2002. They are available at www.jmilne.org/math/. Please send corrections and comme...

Tate’s Isogeny Theorem for Abelian Varieties over Finite Fields

Shimura Varieties: the Adelic Point of View

S h → G R Ad → GL(g) Concretely, C× acts on gC via the characters z/z, 1, z/z. (2) ad(h(±i)) is a Cartan involution of G(R). (3) There is no Q-simple factor Gi of G such that S h → G R → G i,R is trivial. 1.2.2. Remark. Brian gave a slightly different defintion of a connected Shimura datum, using the continuous map u : S → G(R) obtained by restricting h(R) to the unit circle S ⊂ C× = S(R). This...

A GUIDE TO THE REDUCTION MODULO p OF SHIMURA VARIETIES

This report is based on my lecture at the Langlands conference in Princeton in 1996 and the series of lectures I gave at the semestre Hecke in Paris in 2000. In putting the notes for these lectures in order, it was my original intention to give a survey of the activities in the study of the reduction of Shimura varieties. However, I realized very soon that this task was far beyond my capabiliti...