Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms

Details Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms

2003-08-31

arxiv.org/abs/math/0309006v1

Mathematics

Number Theory

28 pages; submitted to the Journal of Number Theory. Based on chapter 3 of math.NT/0306224, reworked and corrected. Comments a

Scientific paper

In a letter to Tate (published in Israel J. Math. in 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space constructed from the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p) are the same as the ones given by algebraic modular forms (mod p) on a quaternionic unitary group, as defined by Gross in Israel J. Math. in 1999. The correspondence is obtained by restricting to the superspecial locus of the moduli space of abelian varieties.

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