Geometric Waldspurger periods II

Abstract

In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Compos. Math. 144 (2008), no. 2, 377–438] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from PGL 2 \operatorname {PGL}_2 to the metaplectic group SL ~ 2 \widetilde {\operatorname {SL}}_2 ; they agree with our conjectures from [Geometric Whittaker models and Eisenstein series for M p 2 \mathrm {Mp}_2 , arXiv:1221.1596]. In the process of the proof, we construct some new automorphic sheaves for GL 2 {\operatorname {GL}_2} in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair ( SL ~ 2 , PGL 2 ) (\widetilde {\operatorname {SL}}_2, \operatorname {PGL}_2) .