
School of Mathematics and Statistics
Helena Verrill
Cambridge University
Images of modular mod l Galois representations.
Friday 25th May, 121pm,
Carslaw 159.
Modular forms which are Cusp forms and Hecke Eigen forms give rise
to ladic Galois Galois representations, and taking the
forms mod l (l a prime) give rise to
representations of the Galois group of Q in
GL_{2}(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 PGL_{2}(k) (i.e., the projectivization
of the image). The exceptional cases are where the projective image
is cyclic, dihedral, or isomorphic to A_{4},
S_{4}, or A_{5}. 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.
