CMI Workshop:
Sage Days 18: Computations related to the Birch and Swinnerton-Dyer Conjecture
December 1 - 5, 2009
The goal of this workshop is to study explicit computation with mathematical objects in connection with the Birch and Swinnerton-Dyer conjecture, including Mordell-Weil groups, complex and p-adic L-series, Heegner points, Kolyvagin classes, Stark-Heegner points, Iwasawa modules, and analogous objects for elliptic curves over function fields. We also intend to create and discuss practical implementations of associated algorithm in the open source computer software Sage.
Topics may include:
- Computing Heegner points on elliptic curves and abelian varieties
- Computing Stark-Heegner points
- Equidistribution of Heegner points
- High precision computation of complex L-series
- Computing p-adic L-series
- Verifying the BSD conjecture for specific elliptic curves and abelian varieties
- Universal norms and Mordell-Weil "shadow lines"
- Computing p-adic heights and p-adic regulators
- Splitting behavior of Galois cohomology classes over solvable extensions
The workshop will be held at the Clay Mathematics Institute Headquarters in Cambridge, Massachusetts.
Schedule
Tuesday, December 1, Research Day
| 9:00-9:30 |
Registration |
| 9:30-12:30 |
Morning working sessions |
| 12:30-2:00 |
Lunch |
| 2:00-3:00 |
William Stein, The Kolyvagin-Gross-Zagier approach to the Birch and Swinnerton-Dyer conjecture |
| 3:10-4:10 |
Robert Miller, Computationally verifying BSD for individual curves |
| 4:20-5:20 |
Drew Sutherland, Images of Galois representations associated to elliptic curves |
| 5:30-6:00 |
Project organization |
Wednesday, December 2, Research Day
| 9:00-12:30 |
Morning working sessions |
| 12:30-2:00 |
Lunch |
| 2:00-3:00 |
William Stein, Computing Kolyvagin classes |
| 3:10-4:10 |
Robert Bradshaw, Computing motivic L-functions |
| 4:20-5:20 |
Salman Baig, Elliptic curves over function fields |
| 5:30-6:00 |
Project status reports |
Thursday, December 3, Research Day
| 9:00-12:30 |
Morning working sessions |
| 12:30-2:00 |
Lunch |
| 2:00-3:00 |
Karl Rubin, Kolyvagin systems and refined class number formulas |
| 3:10-4:10 |
Jared Weinstein, Distribution of Kolyvagin classes |
| 4:20-5:20 |
Robert Pollack, Computing p-adic L-functions |
| 5:30-6:00 |
Project status reports |
Friday, December 4, Research Day
| 9:00-12:30 |
Informal working sessions |
| 12:30-2:00 |
Lunch |
| 2:00-3:00 |
Dimitar Jetchev, Equidistribution of Heegner points and ternary quadratic forms |
| 3:10-4:10 |
Mirela Ciperiani, Solvable points and Kolyvagin's Euler system |
| 4:20-5:20 |
Matthew Greenberg, Fundamental domains for Shimura curves and computation of Stark-Heegner points |
| 5:00-6:00 |
Project status reports (final wrap up) |
Saturday, December 5, Education Day
| 9:00-9:30 |
Registration |
| 9:30-9:45 |
Introduction and Announcements |
| 9:45-10:30 |
Using Sage in the classroom |
| 10:30-11:00 |
Math software capabilities: what is and isn't in Sage |
| 11:00-11:10 |
Break |
| 11:10-11:30 |
Introduction to Lurch (Lite) |
| 11:30-11:50 |
Math and word processing: what works, what doesn't, and the state of Lurch and Sage |
| 11:50-12:20 |
Sage for newbies: logistics, browsers, portability, where documentation really is, etc |
| 12:20-1:30 |
Lunch |
| 1:30-2:30 |
Session on undergraduate teaching with Sage |
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