In Fall 2003, Princeton will host a special semester devoted to the Birch-Swinnerton-Dyer conjecture. Henri Darmon, Ravi Ramakrishna, and Chris Skinner are visiting the department; Darmon and Skinner will give semester-long seminars (see descriptions below) and Andrew Wiles will give a graduate course.
From 5-8 November there will be an AIM workshop at Princeton detailing recent progress on topics related to B-S-D, organized by Andrew Wiles, Henri Darmon, Jordan Ellenberg, and Chris Skinner.
5 Nov
10:30 Andrew Wiles: Introduction
11:30 Shin-ichi Kobayashi: "Iwasawa theory for elliptic curves at supersingular primes."
2:30 Robert Pollack: "The main conjecture for CM elliptic curves at supersingular primes."
4:00 Henri Darmon: "Stark-Heegner points and Kronecker's solution to Pell's equation."
6 Nov
10:00 Chris Skinner: "L-values for GL(2) and an Eisenstein ideal for GU(2,2)."
11:30 Eric Urban: "On the main conjecture for ordinary elliptic curves."
2:30 Vinayak Vatsal: "Special values of Rankin L-functions."
4:00 Shou-Wu Zhang: "On the Gross-Zagier formula."
7 Nov
10:00 Douglas Ulmer: "What is known over function fields."
11:30 Susan Howson: "Non-abelian Iwasawa theory with applications to the arithmetic of elliptic curves."
2:30 Keith Conrad: "Partial Euler products on the critical line."
4:00 Emmanuel Kowalski: "Variation of the rank and related invariants in families of elliptic curves."
8 Nov
10:00 Noam Elkies: "Heegner point constructions for curves x^3 + y^3 = k."
11:30 William Stein: "Connections between the visibility of Shafarevich-Tate groups and the Birch-Swinnerton-Dyer conjecture."
All talks will take place at Taplin Auditorium in Fine Hall (Princeton Math Department.) The conference will begin on the morning of 5 Nov and conclude with lunch on 8 Nov.
Other confirmed participants include A.Agashe, M. Bertolini, P. Green, M. Harris, B. Howard, A.Iovita, M. Kurihara, A.Logan, L. Merel, K. Rubin, D. Savitt, O. Venjakob, T. Weston.
The most convenient airport to Princeton is Newark (EWR). More detailed information about traveling to Princeton can be found here. Many visitors are staying at the Holiday Inn, from which a free shuttle will take you to the conference each morning and to which it will return you in the evening; if you're interested in staying there, please e-mail Scott Kenney at skenney@math.princeton.edu to assure yourself of the conference rate.
Funding may be offered, subject to availability.
Weds, 3-4:30 pm, Fine 1201
starting 1 Oct
In this seminar, I will explain an approach to proving the Main Conjecture of Iwasawa Theory for elliptic curves and for elliptic modular forms in general. This conjecture relates the p-adic properties of the values L(E,\chi,1), E an elliptic curve with multiplicative or ordinary reduction at p and \chi a character of p-power conductor, to those of the p-part of certain Selmer groups (Galois cohomology groups) attached to E. This approach makes use of the arithmetic of Eisenstein series and p-adic modular forms for the unitary group U(2,2) and of Galois representations associated to cuspforms for this group. The results obtained through this approach, when combined with work of Kato and assuming some expected properties of the Galois representations, prove many instances of the Main Conjecture. I will assume familiarity with the theory of elliptic modular forms and some acquaintance with automorphic forms on higher rank unitary groups* (such as Hermitian modular forms) from both the classical and adelic points of view. *As discussed in Shimura's book "Euler products and Eisenstein series," for example.
Thursday, 1-2:30 pm, Fine 801
Starting 25 Sep
The goal of this seminar is to discuss the notion of Heegner points on modular elliptic curves as well as certain conjectural variants, and the information they provide on the Birch and Swinnerton-Dyer conjecture. Topics to be covered (roughly in chronological order) will include:
1. Classical Heegner points attached to imaginary quadratic fields.
2. Proof of Kolyvagin's theorem and its application to the Birch and Swinnerton-Dyer conjecture.
3. The rudiments of rigid analysis and p-adic uniformisation.
4. The notion of "Stark-Heegner" points introduced in [1].
5. Relation with p-adic L-functions attached to Hida families.
6. A proof of some cases of the main conjecture of [1] (work in progress with M. Bertolini).
References:
[1] H. Darmon. Integration on ${\cal H}_p\times{\cal H}$ and arithmetic applications. Annals of Mathematics 154 (2001) 589-639.
[2] H. Darmon. Rational points on modular elliptic curves. NSF-CBMS Lectures, August 8-12 2001, to appear. Can be downloaded from: http://www.math.mcgill.ca/darmon/pub/pub.html (Note: It is better to download the .ps version if you want some figures to print correctly.)