TAILIEUCHUNG - Đề tài " Integrality of a ratio of Petersson norms and level-lowering congruences "

To Bidisha and Ananya Abstract We prove integrality of the ratio f, f / g, g (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and , denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications, the first to proving the integrality of a. | Annals of Mathematics Integrality of a ratio of Petersson norms and level-lowering congruences By Kartik Prasanna Annals of Mathematics 163 2006 901 967 Integrality of a ratio of Petersson norms and level-lowering congruences By Kartik Prasanna To Bidisha and Ananya Abstract We prove integrality of the ratio J f g g outside an explicit finite set of primes where g is an arithmetically normalized holomorphic newform on a Shimura curve f is a normalized Hecke eigenform on GL 2 with the same Hecke eigenvalues as g and denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain level-lowering congruences satisfied by f and to the central values of a family of Rankin-Selberg L-functions. Finally we give two applications the first to proving the integrality of a certain triple product L-value and the second to the computation of the Faltings height of Jacobians of Shimura curves. Introduction An important problem emphasized in several papers of Shimura is the study of period relations between modular forms on different Shimura varieties. In a series of articles see for . 34 35 36 he showed that the study of algebraicity of period ratios is intimately related to two other fascinating themes in the theory of automorphic forms namely the arithmeticity of the theta correspondence and the theory of special values of L-functions. Shimura s work on the theta correspondence was later extended to other situations by Harris-Kudla and Harris who in certain cases even demonstrate rationality of theta lifts over specified number fields. For instance the articles 12 13 study rationality of the theta correspondence for unitary groups and explain its relation on the one hand to period relations for automorphic forms on unitary groups of different signature and on the other to Deligne s conjecture on critical values of L-functions attached to motives that occur in the cohomology of the associated Shimura varieties. To understand these

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