On degree 2 Galois representations over F4

Octahedral Galois Representations Arising from Q-curves of Degree 2

Generically, one can attach to a Q-curve C octahedral representations ρ : Gal(Q/Q) −→ GL 2 (F 3) coming from the Galois action on the 3-torsion of those abelian varieties of GL 2-type whose building block is C. When C is defined over a quadratic field and has an isogeny of degree 2 to its Galois conjugate, there exist such representations ρ having image into GL 2 (F 9). Going the other way, we ...

Serre’s Conjecture on 2-dimensional Galois representations

Deformation of Outer Representations of Galois Group

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...

On three-dimensional Galois representations

We give an irreducibility criterion for the Galois images of compatible l-adic systems of Galois representations. We apply this to the l-adic realizations of the van Geemen-Top motive M and prove an analogue of Serre’s theorem for M.

On the Modularity of Certain 2-adic Galois Representations

We prove some results of the form “r residually irreducible and residually modular implies r is modular,” where r is a suitable continuous odd 2-dimensional 2-adic representation of the absolute Galois group of Q. These results are analogous to those obtained by A. Wiles, R. Taylor, F. Diamond, and others for p-adic representations in the case when p is odd; some extra work is required to overc...