Descending congruences of theta lifts on GSp4

We study the question of when a congruence between two theta lifts on descends to a congruence on modular forms on over a quadratic field. In order to accomplish that, we use the theory of the local theta correspondence between similitude orthogonal groups and the similitude symplectic group , together with a classification for the degeneration modulo a prime of conductors for the L-parameters of irreducible admissible representations of over a non-archimedean local field. We explain that this is unlikely to be used in conjunction with existing results on congruences for to deduce a theory of congruences over imaginary quadratic fields. On the other hand, we prove a result which does give some such congruence results by twisting.