On the Control Theorem for the Symplectic Group

We build a theory of -adic Siegel modular forms related to the Klingen parabolic subgroup of GSp(4). These correspond to families of cohomology classes of increasing levels whose Hecke eigenvalues enjoy strong congruence properties. In the spirit of Hida's theory, a control theorem to relate the family to finite-level members is proved for almost all primes p; in particular we show that the error term appearing in degree one cohomology is killed by the ordinary idempotent.