On a Theorem of Ihara

Let p be a prime number and let n be a positive integer prime to p. by an lhara-result, one means the existence of an injection with torsion-free cokernel, from a full lattice, in the space of p-old modular forms, into a full lattice, in the space of all modular forms of level np. in this paper, lhara-results are proven for genus two siegel modular forms, siegel-jacobi forms and hilbert modular forms. ihara did the genus one case of elliptic modular forms [1]. a geometric formulation is proposed for the notion ofp~old siegel modular forms of genus two, using clarifying comments by r. schmidt [2] and, then, following suggestions in an earlier paper [3] on how to prove ihara results. the main theorem in [3] is used, where an argument by g. pappas has been extended to prove the torsion-freeness of certain cokernel, using the density of hecke-orbits in the moduli space of principally polarized abelian varieties and in the hilbert-blumenthal moduli space, which was proved by c. chai [4].