On the image of lambda-adic Galois representations.

摘要

We explore the question of how big the image of a Galois representation attached to a Λ-adic modular form with no complex multiplication is and show that for a "generic" set of Λ-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all but a density 0 subset have large image.; The questions motivating this thesis go back to Class Field Theory (or 1-dimensional representations). In the 1-dimensional (thus abelian) setting, there is a correspondence between Hecke characters and Galois representations (characters, really). However, when one tries to extend the theory to 2-dimensional (GL2) representations, one quickly realizes that there is more complexity there. By work of Serre, Swinnerton-Dyer, Ribet, and Momose we know that when the Galois representation arises from a modular form with no complex multiplication, its image is large (in the sense of containing an SL2 subgroup) and thus non-abelian (a nice overview of these developments is available in [Rib85]). All of the preceding work on the question of the size of the image of a modular Galois representation has focused on proving that except at finitely many primes, the image is large. Thus we can view the process of varying the prime at which the representation is defined as not changing the representation in a fundamental way. However, no work is known to us that handles the varying of the underlying modular form (or representation) within a family. In this thesis we prove that for ordinary families (Λ-adic modular forms), varying the form in the family still doesn't change the size of the image significantly (except at finitely many primes). This provides both a new interpretation of the existing feeling that non-CM representations should have large image wherever they arise, and a verification that the notion of ordinary families of representations fit in with the rest of what is known about classical p-adic representations.