Hecke operators on differential modular forms mod p

Hecke Operators on Hilbert–siegel Modular Forms

We define Hilbert–Siegel modular forms and Hecke “operators” acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel d...

Lectures on Modular Forms and Hecke Operators

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

In his letter (Serre, 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 A×B/B × → Fp, where B is 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)...

Hecke Operators and Hilbert Modular Forms

Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H(Γ ;C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way...

UPPER BOUND ON THE NUMBER OF SYSTEMS OF HECKE EIGENVALUES FOR SIEGEL MODULAR FORMS (MOD p)

. The constants are effectively computable. Proof. Part (a) follows from the fact that the algebraic group GSp2g has dimension 2g +g+1. Part (b) is obvious. Combined with Theorem 1.1 of [Ghi04], Theorem 1 gives Corollary 3 (Algebraic modular forms). Let B/Q be the quaternion algebra ramified at p and ∞. The number of systems of Hecke eigenvalues coming from algebraic modular forms (mod p) of le...