MSRI 2006 Summer Graduate Workshop
in Computational Number Theory
L-series of elliptic curve of rank 2
- Level 1 Charpolys: Gather data about the following
question, which Ralph Greenberg asked during a talk I recently gave:
Is the characteristic polynomial of every Hecke operator
Tp irreducible on the 2-dimensional
space of cusp forms of level 1 and weight 24?
(I've checked that it is for all primes up to 800 and for p=144169.)
Koopa Koo has already done some things with the
problem about level 1 charpolys, but there are a number of
interesting generalizations he hasn't thought about much yet.
- Gabor Wiese's Challenge Problems:
Big images and non-liftable weight one modular forms mod p.
- Hilbert modular forms database: Start a database
of Hilbert modular forms on real quadratic fields of small
discriminants. It would be very nice, for example, to find equations
for modular abelian surfaces of small levels and used them to
investigate BSD for real quadratic fields.
- Compute every elliptic curve over Q of
conductor 234446 = 2 * 117223. This is the smallest
known conductor of a rank 4 elliptic curve. It would
be very interesting to find all curves of this conductor
just to demonstrate that it's possible. (Note finding
all curves of prime conductor < 234446 has been done.)
Much more difficult -- compute all semistable
elliptic curves (up to isogeny) of conductor < 234446.
- Hilbert-Siegel modular forms: Use Brandt modules to find
examples of Hilbert-Siegel modular forms and, if possible, study the
corresponding Galois representations.
- p-adic Heights: Design and implement an algorithm for
computing anticyclotomic p-adic height pairings on elliptic curves.
This is of intense interest to Barry Mazur and Karl Rubin, and could
be done efficiently using ideas of a recent paper of Mazur-Stein-Tate.
Then carry out the specific calculations Mazur keeps asking me (Stein)
- Level lowering/raising: Do computations and formulate
conjectural analogues of Ken Ribet's level lowering and level raising
theorems but modulo a power of a prime ideal. This is relevant for
much work on the Birch and Swinnerton-Dyer conjecture.
- Invariants of modular abelian varieties: