Status Reports

Tue September 15, 2009

• Amod Agashe: Getting here; will arrive at ferry terminal at 3:20pm and needs a ride!
• Tom Boothby: implement an "argument fixer" class (fix for review); also working on disk caching
• Robert Bradshaw: L-functions precision issues with his implementation of Tim's algorithm
• Sal Butt: Cleaned up database and make sure all online; reading Kedlaya-Sutherland.
• Craig Citro: pick a database and go!
• Tim Dokchitser: L-functions precision issues with his algorithm
• Randy Heaton: Petersson inner product -- nobody remembers the formula...
• Robert Miller: keeping his mother entertained
• Victor Miller: picking up Amod; working through paper of Yang "Defining Equations of modular curves"
• Rishi: Polishing lcalc wrapper (see John Cremona's review); association from modular forms to L-functions (needs assistance).

• William Stein: Galois action on cuspidal subgroup trac #5969

• Kevin Steuve: compressing the function f(x) = Li(x)-pi(x)
• Soroosh Yazdani: real component groups
• Jared Weinstein: Heegner points mod ell.

Wed September 16, 2009

• Amod Agashe: did: arrived safely; testing level lower conjecture when no p-torsion up to level 800. plan to do: run code further; investigate Soroosh counterexample at 13; way to capture congruences only with old forms; craig and congruences with old forms.
• Tom Boothby: did: nearly finished disk caching and parellelizing decorator; plan to do: math -- million digits of Sha?
• Robert Bradshaw: did: dokchitser for computing L-functions. figured out problem, but found others; plan to do: millions digits of Sha.
• Sal Butt: did: worked through kedlaya-sutherland about statistics of L-polynomials (Euler factors at places of good reduction? over QQ. "the symplectic group") and now taking their ideas to do what I want to do. I'm doing this in the function field case. Now I'm looking at L-polynomial in function field case and doing analogue. Written sage code, but there are bugs. plan to do: debug and do calculation.
• Craig Citro: creates inputs to Dokchitser algorithm for a bunch of L-function; plan to do: continue
• Tim Dokchitser: helped Robert figure out precision issue; talk to people. plan to do: look at Amod's conjecture.
• Randy Heaton: wrote two small patches to automize stuff I do a lot with modular forms. worked out almost everything to compute petersson inner product. Need: (1) prove small result, (2) talk to Robert and Craig about computing L-functions, (3) talk to Craig about a level raising constant relating Petersson of old form and new form -- what is the Petersson norm of an image. plan to do: the above, plus todos in the source code, plus newforms don't no they're newforms.
• Robert Miller: got mom to airport and got a cable for TV!; plan to do: binary codes into database, working on descent, or anything else.

• Victor Miller: tearing hair out looking at paper of Yang, and got some of it implemented. Yang defines a bunch of modular functions with character on Gamma(N) with divisors supported on the cusps. He gets generators by taking products that kill action of character. He makes claims about them generating... but then he uses only a much simpler class of functions. I'm implementing both. Question: is there support for Puiseaux series. I noticed also that power series over cyclotomic fields: x^10000 takes forever; making it takes forever. Plan to do: Implement other class of functions; lattice reduction to find good multiplicative basis. (The more I read this paper, the more I am annoyed!) Get the nitty gritty of classes in Sage right. This is all a generalization of David Loeffler's eta products code.

• Rishi: Did everything Cremona suggested in his review, except Cremona's class hierarchy is impossible. Implemented a Dirichlet L-function. Today: Given a newform find the corresponding L-series that does everything Rubinstein's library provides.
• William Stein: Heegner points (explained to Jared and came up with a fascinating conjecture plus a great consistency check), real component group (compute action of Atkin-Lehner and Hecke), cuspidal torsion (lots of conjectures, counterexamples). Today: try to prove real component group conjecture by following Ling-Oesterle's result on Shimura subgroup. Make and organize more modular forms tables.

• Kevin Steuve: Compressing tables of differences between Li(x) and pi(x) by looking at differences of errors. Using lza only save 1/8 th disk space (thought we would get more). Also made my code use multi-core above 10^{12}. Today: working on nth prime function using Victor Miller's linear interpolation method.

• Jared Weinstein: As William said, we identify pattern in vanishing of Kolyvagin classes associated to elliptic curve of rank 2. We found conjecture that predicts a sufficient condition for them to vanish. TODO: Want to find a necessary condition for vanishing.
• Soroosh Yazdani: Look at cuspidal torsion, and ideas for proving it. Discussed example of multiplicity one failing for Eisenstein primes. Trying to understand action of Atkin-Lehner on real components of J0(N) (an F2 vector space). When there are three prime divisor of N, there is a basis such that action is straightforward, but for 4 primes not clear. Also, full Atkin-Lehner involution acts trivially on component group. Today: But Amod has other ideas about extending Mazur.

Meeting about making online tables at 2:30 clock to discuss modular forms database.

Somebody: Implement Shimura subgroup

Friday September 18, 2009

• Jared Weinstein:
• Amod Agashe: Checked my hunch that if an odd prime p divides a Tamagawa number, but does not divide the order of the torsion subgroup, then one can lower level by p for level up to 1010; discussed my strategy for computing congruences with you and Randy; discussed torsion and cuspidal subgroups with William and Soroosh. The goal for Thursday was to get back safely, which was achieved!
• Tom Boothby:
• Robert Bradshaw:
• Sal Butt:
• Craig Citro:
• Tim Dokchitser:
• Randy Heaton:
• Robert Miller:
• Victor Miller:
• Rishi:
• William Stein:
• Kevin Steuve:
• Soroosh Yazdani: