  We introduce a higher dimensional Atkin-Lehner theory for Siegel-Parahoric
congruence subgroups of $GSp(2g)$. Old Siegel forms are induced by geometric
correspondences on Siegel moduli spaces which commute with almost all local
Hecke algebras. We also introduce an algorithm to get equations for moduli
spaces of Siegel-Parahoric level structures, once we have equations for prime
levels and square prime levels over the level one Siegel space. This way we
give equations for an infinite tower of Siegel spaces after N. Elkies who did
the genus one case.
