On a Theorem of Ihara

Let $p$ be a prime number and let $n$ be a positive integer prime to $p$. By an Ihara result we mean 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, we prove Ihara results for genus two Siegel modular forms, Siegel-Jacobi forms and Hilbert modular forms. We also propose a geometric formulation for the notion of $p$-old Siegel modular forms of genus two.