Abstract. We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of SU 3 (Qp). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

We show the analytic continuation of certain Siegel Poincaré series to their critical point for weight three in genus two. We proof that this continuation posesses a nonhomomorphic part and describe it. We show that Sturm's operator also produces a nonhomorphic share for weight three, we call it a phantom term. Weight three is the distinguished weight for genus two where this phenomenon arises.

We give a full set of Casimir operators for the symplectic group of arbitrary genus in terms of a basis chosen such that the action on representations of known K-type becomes transparent. We give examples for the latter.

Abstract. We define non-holomorphic Poincaré series of exponential type for symplectic groups Sp m (R) and continue them analytically in case m = 2 for the small weight (4, 4). For this we construct certain Casimir operators and study the spectral properties of their resolvents on L 2 (Γ\ Sp 2 (R)). Using the holomorphically continued Poincaré series, the holomorphic projection is described in terms of Fourier coefficients using Sturm's operator.

Abstract. In contrast to the wellknown cases of large weights, Sturm's operator does not realize the holomorphic projection operator for lower weights. We prove its failure for arbitrary Siegel genus m ≥ 2 and scalar weight κ = m + 1. This generalizes a result for genus two in [4].