On the Central Critical Value of the Triple Product L–Function

Abstract

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.

Special Cycles and Derivatives of Eisenstein Series

“…If we assume that the f i 's are newforms with associated cuspidal automorphic representations π i , i = 1, 2, 3, then the integral in the last line is 11 the integral representation of the triple product L-function, [23], [66], [30], [2],…”

“…In the case of a rational quadratic space V of signature (1,2), the varieties of part I are the classical Shimura curves. Let B be an indefinite quaternion algebra over Q, and let…”

“…In this case, G = GSpin(V ) = GL(2), the homomorphism G → SO(V ) ≃ PGL(2) is defined via the conjugation action on V , D ≃ H + ∪ H − , the union of the upper and lower half planes, and the M K 's are the usual modular curves. Note that it is more convenient to work with GL(2) rather than PGL (2), and this leads to the choice of GSpin(V ) rather than SO(V ) or the disconnected group O(V ) in general. More details about the case of signature (1,2) are given in the Appendix below.…”

“…Note that it is more convenient to work with GL(2) rather than PGL (2), and this leads to the choice of GSpin(V ) rather than SO(V ) or the disconnected group O(V ) in general. More details about the case of signature (1,2) are given in the Appendix below.…”

p-adic Banach Modules of Arithmetical Modular Forms and Triple Products of Coleman's Families

“…The basic idea how a Dirichlet character χ is incorporated in the integral representation [Ga87,BoeSP] is somewhat similar to the one used in [Boe-Schm], but (surprisingly) more complicated to carry out.…”