On The Notion Of $p$-old Siegel Modular Forms

Details On The Notion Of $p$-old Siegel Modular Forms On The Notion Of $p$-old Siegel Modular Forms

2004-06-11

arxiv.org/abs/math/0406232v1

Mathematics

Number Theory

14 pages

Scientific paper

We introduce a higher dimensional Atkin-Lehner theory for Siegel-parahoric and Borel 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. Any satisfactory local definition of $p$-old Siegel forms which guarantees that, any new eigenform of almost all local Hecke algebras is an eigenform of all local Hecke algebras prime to the level, will imply that our correspondences introduce a geometric formulation for the notion of $p$-old Siegel forms inside all Siegel forms of Siegel parahoric level $pn$. We also prove an injection result for Siegel modular forms in finite characteristics.

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