Special values of L-functions for Saito–Kurokawa lifts

Abstract

Abstract. In this paper we obtain special value results for L-functions associated to classical and paramodular Saito-Kurokawa lifts. In particular, we consider standard L-functions associated to Saito-Kurokawa lifts as well as degree eight L-functions obtained by twisting with an automorphic form defined on GL(2). The results are obtained by combining classical and representation theoretic arguments.

On ratios of Petersson norms for Yoshida lifts

“…As an easy corollary of Theorem 3, we have the following strengthening of Lemma 3.16. 15 In the literature, one can find several results similar to Theorem 3; however they all appear to be weaker or less general than our formulation in some respect or another. for almost all primes p. Assume that there is a constant c such that all the Fourier coefficients of cF lie in Q CM .…”

“…We now state our main result on arithmeticity. 15 Theorem 3. Let N , k, n be positive integers such that k ≥ n and nk is even.…”

“…For the second, we use the fact that by (15), the various local Hecke algebras comprise a system of commuting normal operators on S k (Γ (n), * (N ), χ) and hence can be simultaneously diagonalized.…”

Congruence Primes for Siegel Modular Forms of Paramodular Level and Applications to the Bloch–kato Conjecture

“…COROLLARY 6.4. [10,Corollary 4.6] Let φ ∈ S new 2k−2 ( 0 (M)) be a newform, D be a negative fundamental discriminant so that M D and D is a square modulo 4M, and f φ,D ∈ S k ( para (M)) as above. Then,…”

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“…At primes p | N , we define the local spin L-factors for F P S p2q k pN q new,SK as in [11]. Let F P S p2q k pN q be an eigenform of the Hecke algebra at all p N with eigenvalues λ p p k´3{2 .…”