On Atkin-Lehner correspondences on Siegel spaces

Abstract

‎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.