On mod p properties of Siegel modular forms

Abstract

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p − 1 which is congruent to 1 mod p. Second, we define a theta operator Θ on q-expansions and show that the algebra of Siegel modular forms mod p is stable under Θ, by exploiting the relation between Θ and generalized Rankin-Cohen brackets.

A theta operator on Picard modular forms modulo an inert prime

“…Consider the differential forms dζ 1 , dζ 2 and dζ 3 . As their periods along any l ∈ L x vary holomorphically in z and u, the five coordinates ζ 1 , ζ 2 , ζ 3 , z, u form a local system of coordinates on the family A → X. Identifying A with A allows us to put the desired complex structure on the family A. Alternatively, we may define A as the quotient of C 3 × X by ζ → ζ + l(z, u) where l(z, u) varies over the holomorphic lattice-sections.…”

“…For unitary Shimura varieties, this has been done by Eischen [9,10], if p splits in the quadratic imaginary field [and the signature is (n, n)]. Böcherer and Nagaoka [3] defined theta operators on Siegel modular forms by studying their q-expansions.…”

Congruences for Siegel modular forms

“…Consider an arbitrary coefficient c(n, r). For any positive integer t set n := n + rt + mt 2 and r := r + 2mt. Then 4nm − r 2 = 4n m − r 2 and r = r (mod 2m), and hence c(n, r) = c(n , r ).…”

“…As before, let p 5 be a prime. Guerzhoy [7], Nagaoka [14,15], and Böcherer and Nagaoka [2] investigate Siegel modular forms modulo p. Set…”

Congruence properties of Hermitian modular forms

“…For example, the Siegel-Eisenstein series E (n) p−1 of weight p − 1 is no longer a solution in general. In [2], S. Boecherer and the second author studied this problem and gave some criteria for the existence problem in the case of Siegel modular forms.…”