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].