Status update -- June 28, 2010
A. Find curves whose mod p representations match those of modular abelian varieties.
* Data compiled by (Jon C. et al):
See the .csv files in Jon's directory on William's server for data:
http://sage.math.washington.edu/home/joncass/
B. Elliptic curves from S_4=PGL(2,Z/3) polynomials
* Although all Galois representations into GL(2,3) with cyclotomic determinant come from elliptic curves, not every such projective representation. There is a lifting obstruction described by Serre in "Topics in Galois theory". Noam has identified many obstructed polynomials from the Bordeaux tables of quartic fields. We would like to find elliptic curves yielding the splitting fields of the nonobtructed polynomials for a large range of fields in the tables.
Mike -- post some info on the interesting examples you found.
C. mod 7 representations on elliptic curves
* We are still hunting for Galois representations into GL(2,Z/7Z) which do not arise from elliptic curves, but not for local reason like the exponent of the conductor being too large. To get a handle on such representations, we are playing with Edixhoven's algorithm for computing Galois representations attached to newforms. The first step (Sam L.) is to implement period lattices. Group 5: How's the code for computing modular symbols by complex integration coming?
D. Elliptic curves with Galois representation surjective mod 2 (resp. 4) but not mod 4 (resp. mod 8)
* Gagan S. and Ben L. have been working on this and have computed the sizes of mod 2,4,8 Galois images for a lots of curves to get an idea how much variety is present. Sage doesn't play nicely with projective linear groups, and there is interest in improving this state of affairs.