On Atkin-Lehner correspondences on Siegel spaces

We introduce a higher dimensional Atkin-Lehner theory forý ýSiegel-Parahoric congruence subgroups of GSp(2g)ý. ýOldý ýSiegel forms are induced by geometric correspondences on Siegelý ýmoduli spaces which commute with almost all local Hecke algebrasý. ýWe also introduce an algorithm to get equations for moduli spaces ofý ýSiegel-Parahoric level structuresý, ýonce we have equations for prime levels and square prime levelsý ýover the level one Siegel spaceý. ýThis way we give equations for an infinite tower of Siegel spacesý ýafter Ný. ýElkies who did the genus one caseý.