Status update -- June 28, 2010
A. Find curves whose mod p representations match those of modular abelian varieties.
* Data:
norm2cond10000 norm3cond10000 norm5cond10000
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. Here are the obstructed quartics that Noam found with smallest field discriminant:
x4 - 2*x3 + 2*x^2 + 10*x + 1 x4 - 4*x2 - 4*x + 2 x4 + 2*x2 - 8*x - 19 x4 - 2*x3 + 5*x^2 + 6*x - 15 x4 + x2 - 6*x - 3 x4 + 2*x2 - 20*x - 19 x4 - x3 - 10*x^2 + 12*x + 12 x4 + 4*x2 - 12*x + 6 x4 - 2*x3 - 9*x^2 - 10*x + 5 x4 - 2*x3 - 4*x^2 - 5*x - 5