Siegel modular forms and Ihara lifts to algebraic modular forms

Around in 1963, Ihara gave a theory of lifts from elliptic modular forms to automorphic forms
of compact symplectic group of complex rank two. This can be regarded as a compact version
of Saito-Kurokawa lift and Yoshida lift, which were found much later. In this talk, we give a
precise conjectural global correspondence (of Langlands type) between Siegel modular forms
and automorphic forms on compact symplectic group, for discrete subgroups which are parahoric
subgroups locally, based on exact dimension formulas of automorphic forms and a lot of numerical
examples. As a byproduct, we propose a conjecture on precise images of the Ihara lift.