Friday, October 31, 2008

Annals, Crypto, and Modular Forms

This morning I read Iwasawa's original paper where he introduces p-adic L-series of Dirichlet characters. I basically read this on a whim, since I saw it in a friend's collection of scans.

I also skimmed through all the number theory related papers published by Annals of Math during the last few years. Probably the most interesting were Kartik Prasanna's paper on integrality of Petersson norms (vol. 163, 2006, 901-967), in which he remarks "I am grateful to my advisor Andrew Wiles for suggesting the problem mentioned above and for his guidance and encouragement. The idea that one could use Iwasawa theory to prove the integrality of the Rankin-Selberg L-value is due to him and after his oral explanation I merely had to work out the details." Very humble. Another very nice paper is the fourth in Manjul Bhargava's sequence of papers about parametrizing quintic rings. There's also a nice paper by H. Darmon about his p-adic version of Heegner points.

After that I read through half my paper on generalizing the Gross-Zagier formula, and marked it up with comments.

Then I met with some "Dutch crypto guys" from Microsoft Research and had lunch. After that I went to Clement Pernet's class, where he explained Weiedemann's algorithm for computing minimal polynomials of pairs (A,v) where A is a sparse matrix and v is a vector.

Next I went to the new Number Theory and Computation seminar that Dan Shumow and I started, and saw a nice talk by Dustin Moody on how pairings provide new techniques for algebraic crypto. Next Craig Citro gave an exciting and erudite talk on enumeration of mod p modular forms, which develops in great details some ideas I discussed with him at Sage Day 7 at IPAM. I gave Craig a hard time during his talk though :-).

Now I'm dressed up as a skateboarder for Halloween.

Thursday, October 30, 2008

Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves

I spent last night and this morning making changes to my paper Verification of the Birch and Swinnerton-Dyer Conjecture for Specific Elliptic Curves. I had been sitting on the referee report for several months, since there are four co-authors, and I naively hoped one of them would fix all the problems.

This is a paper I wrote nearly four years ago really as a summer student project: SAGE -- Summer of Arithmetic Geometry Experience -- that I ran at Harvard for a group of undergrads. Five undergrads there all independently requested to work with me on research over the summer, and this paper was one of the results of that summer.

This paper was also the first application of the Sage Mathematics Software. I tried to repeat some calculations from the paper when making improvements suggested by the referee, and was scared that I would find bugs that prevent me from doing them. Indeed, within a few seconds I was disappointed to find a showstopper bug, which fortunately I was able to fix in a few minutes.

The referee remarks about the paper were almost all very useful; they definitely improve the quality of the paper a lot.

The paper itself proves most of the BSD conjectural formula for elliptic curves of conductor up to 1000. The obvious next steps with that paper done are:

  1. Mostly prove BSD for far more curves (say up to conductor 130,000). This would be subtle and interesting, since all kinds of problems that don't quite happen for conductor up to 1000 would certainly happen here.
  2. Completely prove BSD for all curves up to conductor 1,000, except the 18 curves of rank 2. This will rely on my joint paper with Wuthrich, which will make this almost completely straightforward, except perhaps in a handful of surprising cases.
  3. Prove the prediction of BSD about Sha at a bunch of primes for curves of rank 2. This can also be done using my joint paper with Wuthrich.

Wednesday, October 29, 2008

A Reading Seminar, Schemes, and Petersson norms

Today I had lunch with Craig Citro, Robert Miller, Robert Bradshaw, and Dan Shumow, and we mainly discussed starting an "advanced number theory reading seminar". Craig's the main impetus behind it, and it will be mainly about talking about papers like Gross-Zagier, etc.

After that I met with Tom Boothby and Sourav San Gupta and defined schemes for them, and explained how to think of maps between rings of integers as morphisms of schemes, and factorization of primes in integer rings as computation of fibers. We also explicitly computed the morphism from y=x^2 to y=0 in terms of maps on affine rings.

After that, I met with Robert Bradshaw (one of my Ph.D. students), and looked at a bunch of papers that are related to computing Petersson norms and special values of symmetric power L-functions. The problem we're looking at currently is: efficiently compute L(Sym^2(f), 2) for general newform f, and compute the Petersson norm of f. This is "essentially done," as explained in (Delaunay, 2003), when the level of f is square free. When the level is not square free, things are more interesting, and we discussed at least two ideas for dealing with this more general case: (1) try all possibilities for the bad factors and see which leads to a consistent functional equation, or (2) compute the modular degree via modular symbols and linear algebra, and use a formula as in Cremona or Zagier's papers on modular degree. We also looked at an old paper of Rankin that Kevin Buzzard sent us that computes Petersson inner products for level 1 forms -- it is horrendous, only level 1, and doesn't use symmetric square L-functions at all, and is clearly at least O(N^3) complexity (or worse).

When reading through these papers, we ran into numerous cases where we couldn't get relevant references because scans are not available online (or cost $30/each, and our university subscription doesn't go back far enough), and I don't have a scan or photocopy, and of course the library is closed. It's extremely annoying how frustrating the situation still still is regarding having access to the literature in number theory.