Theta correspondence and Hecke operators relative to a quadratic extension

Action of Hecke Operators on Theta-functions with Rational Characteristics

The explicit formulas for the transformation of theta-functions of integral positive definite quadratic forms under the action of regular Hecke operators, obtained in the author’s earlier paper (1996), are converted to transformation formulas for the theta-functions with rational characteristics (the theta-series) viewed as Siegel modular forms. As applications, sequences of invariant subspaces...

Action of Hecke Operators on Siegel Theta Series Ii

We apply the Hecke operators T (p) and T ′ j (p) (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 1 p L, thereby generalizing the Eichler Commuta...

Action of Hecke Operators on Siegel Theta Series I

We apply the Hecke operators T (p) and T̃j(p 2) (1 ≤ j ≤ n, p prime) to a degree n theta series attached to a rank 2k Z-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the theta series under these Hecke operators can be realized as a sum of theta series attached to certain closely related lattices, thereby generali...

Hecke Operators

Action of Hecke Operators on Maass Theta Series and Zeta Functions

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construc...