CoCalc Shared Fileswww / msri06 / challenges.html
Author: William A. Stein
]

# Computing With Modular Forms

L-series of elliptic curve of rank 2

# Challenge Problems

1. 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.
2. Gabor Wiese's Challenge Problems: Big images and non-liftable weight one modular forms mod p.
3. 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.
4. 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.
5. Hilbert-Siegel modular forms: Use Brandt modules to find examples of Hilbert-Siegel modular forms and, if possible, study the corresponding Galois representations.
6. 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) to do.
7. 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.
8. Invariants of modular abelian varieties: