Action of Hecke operators on Siegel theta series, II

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We apply the Hecke operators T(p)2 and (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)2.
Translated title of the contributionActioin of Hecke operators on Siegel theta series, II
Original languageEnglish
Pages (from-to)981 - 1008
Number of pages28
JournalInternational Journal of Number Theory
Issue number6
Publication statusPublished - Dec 2008

Bibliographical note

Publisher: World Scientific


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