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.

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