The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

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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.

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Abstract. We compute the central critical value of the triple product L-function associated to three cusp forms f 1 , f 2 , f 3 with trivial character for groups Γ 0 (N i ) with square free levels N i not all of which are 1 and weights k i satisfying k 1 ≥ k 2 ≥ k 3 and k 1 < k 2 + k 3 . This generalizes work of Gross and Kudla and gives an alternative classical proof of their results in the case N 1 = N 2 = N 3 with k 1 = k 2 = k 3 = 2.