Existence of Nonelliptic mod Galois Representations for Every > 5 Luis

In [Shepherd-Barron and Taylor 97] it is shown that for = 3 and 5 every odd, irreducible, two-dimensional Galois representation of Gal(Q̄/Q) with values in F and determinant the cyclotomic character is “elliptic,” i.e., it agrees with the representation given by the action of Gal(Q̄/Q) on the -torsion points of an elliptic curve defined over Q. In this note we will show that this is false for every prime > 5, i.e., that for every such prime there exists a Galois representation verifying the above properties but “nonelliptic,” i.e., not corresponding to the action of Galois on torsion points of any elliptic curve defined over Q. We will show this by giving concrete examples of nonelliptic representations. For any prime > 7, the example will be constructed starting from a weight-4 classical modular form, corresponding to a rigid Calabi-Yau threefold. The case of = 7 will be treated separately in the next section. We consider the cuspidal modular form f ∈ S4(25) (i.e., of weight 4, level 25, and trivial nebentypus) which has all eigenvalues in Z and whose attached Galois representations ρf, agree (see [Schoen 86, Yui 03]) with the Galois representations on the third étale cohomology groups of the Schoen rigid Calabi-Yau threefold. This threefold is obtained (after resolving the singularities) from