Geometric Waldspurger periods II

TL;DR

This paper extends the calculation of geometric Waldspurger periods to ramified coverings, explores their applications to automorphic sheaves and theta-lifting, and proposes stronger conjectures in the field.

Contribution

It advances the understanding of geometric Waldspurger periods in ramified settings and introduces new automorphic sheaves and conjectures.

Findings

Waldspurger periods are computed for ramified coverings.

Applications to Whittaker coefficients and theta-lifting are demonstrated.

New automorphic sheaves for GL_2 are constructed in ramified cases.

Abstract

In this paper we extend the calculation of the geometric Waldspurger periods from our paper math/0510110 to the case of ramified coverings. We give some applications to the study of Whittaker coefficients of the theta-lifting of automorphic sheaves from PGL_2 to the metaplectic group Mp_2, they agree with our conjectures from arXiv:1211.1596. In the process of the proof, we get some new automorphic sheaves for GL_2 in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair (SL_2, Mp_2).