Congruences between Ordinary Symplectic Galois Representations

We prove the existence of congruences between ordinary symplectic Galois representations in two different settings. First, we calculate lower bounds on the degree of the weight space map for Hida families given assumptions on the p-adic L-invariant (or the adjoint L-invariant) of a weight (3,3) automorphic representation on the Hida family when such an L-invariant is defined using theorems of Giovanni Rosso. Second, we set up a Galois deformation problem for a fixed absolutely irreducible Galois representation which is odd, ordinary and indecomposable at p, and unramified everywhere else. Under a mild local hypothesis, we prove the existence of at least two characteristic zero lifts of our fixed Galois representation.