In this thesis, we consider the method of Harris-Soudry-Taylor et al. of attaching 2-dimensional l-adic Galois representations to cuspidal automorphic representations for GL(2) over imaginary quadratic fields. In this process, one lifts an automorphic representation for GL(2) over an imaginary quadratic field to an automorphic representation for GSp(4) over the rationals; the latter automorphic representation has associated a 4-dimensional l-adic Galois representation, which turns out to be induced from a 2-dimensional representation of the absolute Galois group over the imaginary quadratic field. We aim in using this method to transfer level lowering results for GSp(4) over the rationals to level lowering results for GL(2) over an imaginary quadratic field.
Firstly, we study in detail the conductors of irreducible admissible non-supercuspidal and non-generic supercuspidal representations of GSp(4) over a non-archimedean local field, and we obtain a result in the sense of Carayol and Livne on how the conductors degenerate modulo a prime number. In particular, when we have a corresponding mod l Galois representation and an l-adic lift of it, we list all the cases where the conductors differ.
Having this in our machinery, together with an explicit local theta correspondence between irreducible admissible representations of GL(2,L) and irreducible admissible representations of GSp(4,F) (here L is either a degree 2 field extension over F, or L is isomorphic to FxF, with F a non-archimedean local field), we obtain a conditional result on level lowering for automorphic representations of GL(2) over an imaginary quadratic field of prime discriminant. The result is conditional in the sense that we assume a level lowering result for representations of GSp(4) over the rationals.
Finally, we prove a level lowering result by twisting particular automorphic representations over imaginary quadratic fields by grossencharacters
