STUDY GROUP ON SHIMURA VARIETIES

Warwick University.

Coordinated by Xenia Dimitrakopoulou, David Loeffler and Óscar Rivero

Main reference: Milne's book. https://www.jmilne.org/math/xnotes/svi.pdf

Auxiliary reference: Rotger's notes. https://web.mat.upc.edu/victor.rotger/docs/VarShiCRM.pdf

Complementary texts.

These notes of Kai-Wen Lan are also a nice introduction to the topic, with a plethora of examples. http://www-users.math.umn.edu/~kwlan/articles/intro-sh-ex.pdf

This paper of Christian Johansson (specially Section 3) discusses some conjectures connecting Shimura varieties with Galois representations: https://arxiv.org/pdf/1206.6730.pdf

See also the paper by Buzzard and Gee: http://wwwf.imperial.ac.uk/~tsg/Index_files/bgagsvn.pdf

Session 0 (David, April 30th). General overview. Here you can read David's notes.

Session 1 (Xenia, May 7th). Review of algebraic groups. Algebraic preliminaries, reductive and semisimple groups, the parabolic subgroup, roots, weights... Here you can read Xenia's notes.

Main references. Milne Chapters 1-3, Rotger Chapter 1, Warwick course on Algebraic Groups.

Session 2 (Diana, May 14th). Cartan's classification and connected Shimura varieties. Algebraic groups over the reals and Cartan's classification. Definition of connected Shimura datum. Examples, the upper half plane and generalizations of it.

Main references. Milne Chapter 4, Rotger Chapter 1.

Session 3 (Nuno, May 21st). Shimura varieties. The real points of algebraic groups. Shimura data and Shimura varieties. The structure of a Shimura variety. Examples.

Main references. Milne Chapter 5.

Session 4 (Steven, May 28th). The Siegel Shimura variety. Symplectic spaces and their Shimura datum. The Siegel modular variety. Complex abelian varieties. Modular descriptions.

Main references. Milne Chapter 6, Rotger Section 3.3.

Session 5 (Pak-Hin, June 4th). General Shimura varieties. Brief remarks on Shimura varieties of Hodge--Tate and PEL type. Abelian motives and Shimura varieties of abelian type (examples). Shimura varieties of non-abelian type (examples). Interpretation as moduli varieties.

Main references. Milne Chapter 9.

Session 6 (Muhammad, June 11th). Complex multiplication. CM fields. Abelian varieties of CM type. Notion of reflex field. Statement of the main theorem of CM (including brief review of CFT, if needed).

Main references. Milne Chapters 10-11 (mainly 11), Rotger Chapter 4.

Session 7 (Elvira, June 18th). Canonical models: definitions. Models of varieties. Special points. The canonical model. Examples (Shimura varieties defined by tori).

Main references. Milne Chapter 12, Rotger Chapter 4.

Session 8 (Chris, June 25th). Results on canonical models: existence and uniqueness. Existence and uniqueness of canonical models. General overview of the main results and ideas of the proofs. The Siegel case.

Main references. Milne Chapter 6, Rotger Chapter 4.

Session 9 (Oscar, July 2nd). Construction of Galois representations. Galois representations attached to Shimura varieties. General conjectures. Examples (symplectic and unitary groups).

Main references. The papers of Buzzard--Gee and Johansson listed above.