Workshop on modular curves
17 - 19 December, 2012, Warsaw
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 problems will be then to construct Galois representations associated to automorphic forms. This will lead to a complete description of cohomology groups and it will be a first step into the Langlands program.
For more details please, see the workshop's website.
Speakers of the Workshop will include: G. Banaszak, P. Chojecki, K. Górnisiewicz, J. Jelisiejew, A. Langer, B. Naskręcki, M. Ulas, M. Zydor.
Organizing Committee
- Przemysław Chojecki
- Adrian Langer
Programme
| Monday, 17th of December |
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Lecture 1 10.30 - 11.45 (Maciej Ulas) |
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. |
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Lecture 2 13.00 - 14.15 (Joachim Jelisiejew) |
Content: Formal groups as a local analogue of elliptic curves. Formal groups associated to elliptic curves. Examples with Lubin-Tate extensions. Lubin-Tate spaces. |
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Lecture 3 14.35 - 15.50 (Krzysztof Górnisiewicz)
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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. |
| Tuesday, 18th of December | |
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Lecture 4 10.30 - 11.45 (Adrian Langer) |
Contents: Complex and p-adic uniformisation of elliptic curves. Tate curve. Introduction to rigid analytic geometry of Tate. |
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Lecture 5 13.00 - 14.15 (Michał Zydor) |
Contents: Why automorphic forms appear naturally in the context of Betti cohomology of modular curves. About Matsushima formula. |
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Lecture 6 14.35 - 15.50 (Przemysław Chojecki) |
Contents: Reminder on etale cohomology in the context of elliptic and modular curves. Applications of I-adic cohomology in Matsushima formula. |
| Wednesday, 19th of December | |
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Lecture 7 10.30 - 11.45 (Grzegorz Banaszak) |
Contents: Why do we want to construct Galois representations? Applications. |
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Lecture 8 13.00 - 14.15 (Bartosz Naskręcki) |
Contents: Construction of Galois representations associated to modular forms of weight 2. |
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Lecture 9 14.35 - 15.50 (Przemysław Chojecki) |
Contents: Cohomology of modular curves in the context of Langlands program. |