Algebra Seminar

Modular forms which are Cusp forms and Hecke Eigen forms give rise to l-adic Galois Galois representations, and taking the forms mod l (l a prime) give rise to representations of the Galois group of Q in GL₂(k), where k is some finite extension of the finte field with l elements. Ribet proved that for a given modular form, the corresponding modular mod l Galois representaions almost always have image "as large as possible" in PGL₂(k) (i.e., the projectivization of the image). The exceptional cases are where the projective image is cyclic, dihedral, or isomorphic to A₄, S₄, or A₅. I will discuss joint work with Ian Kiming on how to determine and prove when the image is one of the latter 3 types, and give bounds on how large l can be for such an image to occur.