Wednesday, November 12, 2008

L-functions: a seminar and the Gross-Zagier L-function

Today we had the first meeting of the Advanced Grad Student Number Theory Research Reading Seminar at UW. The participants are me, Robert Miller, Robert Bradshaw, and Craig Citro, and we intend to talk about papers of Dokchitser, Gross-Zagier, Watkins, Delauney, Kartik, Bosman, Kolyvagin, Ghate, and Hida. The unifying theme is mostly about explicit applications of L-functions and modular forms to very hard problems in algebraic number theory. We first had an organizational meeting and decided what we'll be talking about, then I gave a talk on Dokchitser's paper (basically following the talk I gave at UT Austin a few days ago, but greatly abbreviated due to lack of time).

After that, I went out to lunch with Tom Boothby, Sourav San Gupta, Craig Citro, and Robert Bradshaw at Cedars, where we talked about a reading course Boothby and Gupta are doing on algebraic number theory (basically, exercises from Marcus's book).

Next, Robert Bradshaw, Craig Citro, and I sat down in my office and figured out precisely how to use Dokchitser's algorithm to compute the Gross-Zagier L-function L_A(f,s). This involved using the Legendre duplication formula (page 217 of Ahlfors), explicitly computing the representation numbers r_A(n) using quadratic forms (Robert Bradshaw has a super-fast Cython implementation of this which is 20 times faster than Magma's, which uses careful bounds on the ellipse one enumerates over), and explicit computation of the product of two Dirichlet series as a Dirichlet series, which again Robert coded up very quickly by noting that one of the series is very sparse. Then we got all this to work via the Sage/Dokchitser interface, and it computed the power series of L_A(f,s) at 1 for 37a and D=-40. Robert will be making this all much more systematic as a patch to Sage, and extend it to work for any newform of any weight or level (for Gamma0(N)), so that we can do many systematic computations. In particular, I'm very interested in what happens when ord_{s=1} L_A(f,s) >= 3 (it's always odd), and also what happens when f isn't attached to an elliptic curve.

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