This is the web page for the seminar "complex multiplication'' organised by Peter Bruin and Marco Streng at Universiteit Leiden during the fall semester of 2009.
Location: room 401 of the Snellius building, Universiteit Leiden.
Description
The seminar is aimed at PhD students and possibly also Master students with the goal of understanding the theory and applications of complex multiplication. The exact program depends on the wishes of the participants, but we try to focus on the proof and applications of complex multiplication of abelian varieties. Applications include constructive class field theory and constructions of curves and abelian varieties with a given number of points.
As most of the theory is already interesting in the dimension-1 case (elliptic curves), it should be possible to participate with knowledge only of algebraic number theory and algebraic curves. Some prior contact with class field theory or abelian varieties helps, but is not strictly necessary.
If you are interested in participating, then please send us an email. We also welcome suggestions for the content of the seminar.
We plan to meet for three hours every Wednesday afternoon (if you prefer a different time, let us know). If participants from outside Leiden are interested, then we may consider longer sessions with a lower frequency.
Schedule
Wednesday October 14:
- 14:30 -- 16:15 Marco Streng: Introduction to complex multiplication, the main theorems (notes)
- 16:30 -- 17:15 Peter Bruin: Endomorphisms of Abelian varieties and their representations (I) (notes)
Wednesday October 21:
- No lectures.
Thursday October 29 (different day, different time!):
- 11:00 -- 12:45 Jeroen Sijsling: Polarizations (notes)
Wednesday November 4:
- 14:30 -- 16:00 Peter Bruin: Endomorphisms of Abelian varieties and their representations (II) (notes)
- 16:15 -- 17:15 Richard Moloney: The tangent space and the CM type
Thursday November 12 (different day, different time!):
- 11:00 -- 11:45 René Pannekoek: Class field theory
- 12:00 -- 12:45 David Freeman: Existence and classification of CM abelian varieties over C (notes)
Wednesday November 18:
- No lectures
Wednesday November 25:
- 14:30 -- 16:15 Jeroen Sijsling: Multiplication by ideals: a-multiplications (notes)
- 16:30 -- 17:15 Gabriele Dalla Torre: The Frobenius morphism and its relation to the tangent space
Wednesday December 2:
- No lectures.
Wednesday December 9:
- 14:30 -- 15:45 René Pannekoek: Reduction
- 16:00 -- 17:15 Marco Streng: The reflex field and the type norm (notes)
Wednesday December 16 (one hour earlier!):
- 13:30 -- 14:15 Marco Streng The reflex type (notes)
- 14:30 -- 15:00 Jeroen Sijsling: Specialization and why CM abelian varieties are defined over Q ([Mil06] Proposition 7.9)
- 15:15 -- 16:30 Peter Bruin The Frobenius as an a-transform
Literature
For elliptic curves, we refer to [Sil86], for algebraic geometry to [Liu06] and [Har77], and for algebraic number theory to [ANT].
For class field theory, see [CFT].
Complex multiplication of elliptic curves is explained in Chapter II of [Sil94], which also contains many further references. Alternatively, check out [Cox89], which also gives a very nice application to representability of primes by quadratic forms.
For the basic theory of abelian varieties, see [vdGM] or [Mum70].
For complex multiplication of abelian varieties, we refer to [Shi98], [Lan83], and [Mil06].
For Honda-Tate theory, see [Tat71].
[ANT] | Peter Stevenhagen, lecture notes Number Rings available online. |
[CFT] | Peter Stevenhagen, lecture notes Class Field Theory available online. For exercises by Hendrik W. Lenstra, look here. |
[Cox89] | David A. Cox, Primes of the form x^{2}+ny^{2} (Fermat, Class Field Theory, and Complex Multiplication), 1989 |
[vdGM] | Gerard van der Geer and Ben Moonen, Abelian Varieties, available online. |
[Har77] | Robin Hartshorne, Algebraic Geometry, 1977. |
[Lan83] | Serge Lang, Complex Multiplication, 1983. |
[Liu06] | Qing Liu, Algebraic Geometry and Arithmetic Curves, 2006. |
[Mil06] | J.S. Milne, Complex Multiplication, 2006, available online. |
[Mum70] | David Mumford, Abelian Varieties, 1970. |
[Shi98] | Goro Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, 1998 (Sections 1 -- 16 essentially appeared before in Goro Shimura and Yutaka Taniyama, Complex Multiplication of Abelian Varieties and Its Applications to Number Theory, 1961) |
[Sil86] | Joseph H. Silverman, The Arithmetic of Elliptic Curves, 1986 |
[Sil94] | Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, 1994 |
[Tat71] | John Tate, Classes d'isogénie des variétés abéliennes sur un corps fini (d'après Z. Honda). Sémin. Bourbaki 1968/69, No.352, 95-110, 1971, available online. |
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