In this article, we explore a beautiful idea of Skinner and Wiles in the context of GSp(4) over a totally real field. The main result provides congruences between automorphic forms which are Iwahori-spherical at a certain place ω, and forms with a tamely ramified principal series at ω, Thus, after base change to a finite solvable totally real extension, one can often lower the level at ω. For the proof, we first establish an analogue of the Jacquet–Langlands correspondence, using the stable trace formula. The congruences are then obtained on inner forms, which are compact at infinity modulo the centre, and split at all the finite places. The crucial ingredient allowing us to do so, is an important result of Roche on types for principal series representations of split reductive groups.
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