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.
We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod p singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that p is a rational prime. Moreover, we construct some examples of mod p singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori.
We show that Siegel modular forms of level Γ 0 (p m ) are padic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vectorvalued modular forms. In our approach to p-adic Siegel modular forms we follow Serre [18] closely; his proofs however do not generalize to the Siegel case or need some modifications.
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod p by using theta series attached to positive definite quadratic forms.
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