MSRI 2006 Summer Graduate Workshop in Computational Number Theory

Lseries of elliptic curve of rank 2

Challenge Problems
 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
T_{p} irreducible on the 2dimensional
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 nonliftable 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.
 HilbertSiegel modular forms: Use Brandt modules to find
examples of HilbertSiegel modular forms and, if possible, study the
corresponding Galois representations.
 padic Heights: Design and implement an algorithm for
computing anticyclotomic padic 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 MazurSteinTate.
Then carry out the specific calculations Mazur keeps asking me (Stein)
to do.
 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 SwinnertonDyer conjecture.
 Invariants of modular abelian varieties:
