Workshop on modular curves

Date: 17th-19th December 2012, each lecture will last 75 minutes.

Place: room number 403 in IMPAN, ul. Śniadeckich 8, Warsaw, Poland.

See also the other webpage.

Description of the workshop:

The goal of the workshop is to review basic geometric and representation-theoretic properties of modular curves which are fundamental to modern algebraic geometry and number theory. We will discuss geometry of modular curves and their local counterpart, Lubin-Tate spaces. We will describe the cohomology (Betti and l-adic) of modular curves and show how naturally a notion of an automorphic form arises in this setting. One of the main problem will be then to construct Galois representations associated to automorphic forms. This will lead to a complete description of cohomology groups and will be a first step into Langlands program.

References:

[De1] P. Deligne, "Formes modulaires et representations l-adiques", Sem. Bourbaki, exp. 355, 1969

[De 2] P. Deligne, letter to I. Piatetski-Shapiro, 1973

[DS] F. Diamond, J. Shurman "A First Course in Modular Forms", book

[Hi] H. Hida, "Geometric modular forms and elliptic curves", book

[KM] N. Katz, B. Mazur "Arithmetic moduli of elliptic curves", book

[Si] J. Silverman, "The Arithmetic of Elliptic Curves", book

[Yo] T. Yoshida, lecture course in Cambridge, 2008

Timetable:

17th December:

Lecture 1 (Maciej Ulas), 10.30 - 11.45:

Contents: General definition of elliptic curves and basic facts about them. Reminder on algebraic number theory. Supersingular and ordinary elliptic curves. Hasse invariant. Definition of modular curves.

Lecture 2 (Joachim Jelisiejew), 13.00 - 14.15:

Contents: Formal groups as a local analogue of elliptic curves. Formal groups associated to elliptic curves. Examples with Lubin-Tate extensions. Lubin-Tate spaces.

Lecture 3 (Krzysztof Górnisiewicz), 14.35 - 15.50:

Contents: Modular curves as a moduli problem. Vector bundles on modular curves and their sections - automorphic forms. Basic facts about the geometry of modular curves. Connections between Lubin-Tate spaces and modular curves.

18th December:

Lecture 4 (Adrian Langer), 10.30 - 11.45:

Contents: Complex and p-adic uniformisation of elliptic curves. Tate curve. Introduction to rigid analytic geometry of Tate.

Lecture 5 (Michal Zydor), 13.00 - 14.15:

Contents: Why automorphic forms appear naturally in the context of Betti cohomology of modular curves. About Matsushima formula.

Lecture 6 (Przemyslaw Chojecki), 14.35 - 15.50:

Contents: Reminder on etale cohomology in the context of elliptic and modular curves. Applications of l-adic cohomology in Matsushima formula.

19th December:

Lecture 7 (Grzegorz Banaszak), 10.30 - 11.45:

Contents: Why do we want to construct Galois representations? Applications.

Lecture 8 (Bartosz Naskrecki), 13.00 - 14.15:

Contents: Construction of Galois representations associated to modular forms of weight 2.

Lecture 9 (Przemyslaw Chojecki), 14.35 - 15.50:

Contents: Cohomology of modular curves in the context of Langlands program.