This is the web page for the seminar "moduli spaces and modular forms'' organised by Peter Bruin and Marco Streng at Universiteit Leiden during the fall semester of 2007.
Location: room 403 of the Snellius building, Universiteit Leiden.
Description
The seminar is aimed at Ph. D. students and possibly also Master students with the goal of understanding moduli spaces and modular forms. The exact program will depend on the wishes of the participants, but we will start with moduli spaces of elliptic curves and with modular forms, both in a complex analytic and a more algebraic geometric setting.
If you are interested in participating, then please let us know. We also welcome all suggestions for the content of the seminar.
Schedule
September 12:
- 13:30-15:00 Marco Streng: Complex analytic introduction: lattices, level structures and modular curves (summary)
- 15:00-15:45 Discussion of the program / break
- 15:45-17:15 Peter Bruin: Algebraic introduction: algebraic moduli of elliptic curves (notes)
- Monday, September 17, 16:30-17:00 Maite Aranes: Modular Forms and Elliptic Curves over Imaginary Quadratic Fields
- Thursday, September 20, 12:30–13:00 Burcu Baran: A Modular Curve of Level 9 and The Class Number One Problem
- Thursday, September 20, 15:30-16:00 Xavier Taixes: An algorithm to compute congruences of representations attached to modular forms
- Friday, September 21, 15:00-15:45 Gabor Wiese: Modular Forms in Inverse Galois Theory
- 13:30-15:15 Johan Bosman: Modular forms of level one (handout)
- 15:30-17:15 Jeroen Sijsling: Genera of modular curves and dimensions of spaces of modular forms (notes)
- 13:30-15:15 Erwin Dassen: Compactification and modular forms for congruence subgroups
- 15:30-16:15 Arjen Stolk: Compactification in the algebraic setting
- 16:30-17:15 Lenny Taelman: Application: Quadratic forms and theta functions (notes + picture)
- 13:30-15:15 Peter Bruin: Hecke operators, eigenforms and the Petersson inner product
- 15:30-17:15 Maarten Hoeve: Moduli of curves (notes)
- 16:00-17:00, room 174, Bas Edixhoven: How to count vectors with integral coordinates and given length in R^{n} (abstract)
- 13:30-14:15 Phyllis Joris: Jacobians
- 14:30-15:15 Arjen Stolk: The Tate curve (notes)
- 15:30-17:15 Zongbin Chen: The Eichler-Shimura isomorphism
- 13:30-15:15 Marco Streng: Galois representations
- 15:30-17:15 Jeroen Sijsling: Examples of moduli of curves (notes)
- 13:30-15:15 Johan Bosman: The NWO Vici project Arithmetic geometry, motives: computational aspects
- 15:15 Early end of the day's program because of Sinterklaas
- 15:00-16:45, room 312, Eyal Goren (McGill University): Class invariants for CM fields of degree 4 (abstract)
- 13:30-15:15 Arjen Stolk: Deformations and smoothness
- 15:30-17:15 Bas Edixhoven: Fermat's Last Theorem
Literature
For the basic theory of elliptic curves, we refer to [Sil86] and for algebraic geometry to [Liu06] and [Har77].
Introductions to modular forms include [DS05], [Lan76] and [Gun62].
Further good references include the survey article [DI95], which contains many references itself, and Bas Edixhoven's notes [Edi97].
For an application of modular functions and complex multiplication to representability of primes by quadratic forms, see [Cox89]. For an application of modular forms and L-series to a problem from elementary number theory, see [Kob84].
[Cox89] | David A. Cox, Primes of the form x^2+ny^2 (Fermat, Class Field Theory, and Complex Multiplication), 1989 |
[DI95] | Fred Diamond and John Im, Modular forms and modular curves, in V. Kumar Murty, Seminar on Fermat's last theorem, 1995. A scanned copy of the article (36 MB and bad quality) is available here. |
[DS05] | Fred Diamond and Jerry Shurman, A First Course in Modular Forms, 2005 |
[Edi97] | The notes of Bas Edixhoven's series of lectures The modular curves X_0(N) at the ICTP Summer school on rational torsion of elliptic curves over number fields in Trieste, 1997 |
[Gun62] | R.C. Gunning, Lectures on Modular Forms, 1962 |
[Har77] | Robin Hartshorne, Algebraic Geometry, 1977 |
[Kob84] | Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 1984 |
[Lan76] | Serge Lang, Introduction to Modular Forms, 1976 |
[Liu06] | Qing Liu, Algebraic Geometry and Arithmetic Curves, 2006 |
[Sil86] | Joseph H. Silverman, The Arithmetic of Elliptic Curves, 1986 |
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