tailieunhanh - Đề tài " The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points "

In this paper we solve the subconvexity problem for Rankin-Selberg L-functions L(f ⊗ g, s) where f and g are two cuspidal automorphic forms over Q, g being fixed and f having large level and nontrivial nebentypus. We use this subconvexity bound to prove an equidistribution property for incomplete orbits of Heegner points over definite Shimura curves. L(f, s), | Annals of Mathematics The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points By P. Michel Annals of Mathematics 160 2004 185 236 The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points By P. Michel A Delphine Juliette Anna and Samuel Abstract In this paper we solve the subconvexity problem for Rankin-Selberg L-functions L f g s where f and g are two cuspidal automorphic forms over Q g being fixed and f having large level and nontrivial nebentypus. We use this subconvexity bound to prove an equidistribution property for incomplete orbits of Heegner points over definite Shimura curves. Contents 1. Introduction 2. A review of automorphic forms 3. Rankin-Selberg L-functions 4. The amplified second moment 5. A shifted convolution problem 6. Equidistribution of Heegner points 7. Appendix References 1. Introduction . Statement of the results. Given an automorphic L-function L f s the subconvexity problem consists in providing good upper bounds for the order of magnitude of L f s on the critical line and in fact bounds which are stronger than ones obtained by application of the Phragmen-Lindelof convexity principle. During the past century this problem has received considerable This research was supported by NSF Grant DMS-97-29992 and the Ellentuck Fund by grants to the Institute for Advanced Study by the Institut Universitaire de France and by the ACI Arithmetique des fonctions L . 186 P. MICHEL attention and was solved in many cases. More recently it was recognized as a key step for the full solution of deep problems in various fields such as arithmetic geometry or arithmetic quantum chaos for instance see the end of the introduction of DFI1 and more recently CPSS Sa2 . For further background on this topic and other examples of applications we refer to the surveys Fr IS or M2 . In this paper we seek bounds which are sharp with respect to the conductor of the automorphic form f. For rank one

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