Rigid meromorphic cocycles for orthogonal groups

TL;DR

This paper introduces rigid meromorphic cocycles for orthogonal groups, constructed via $p$-adic analogues of Borcherds' theta lift, with conjectures linking their special values to class fields, advancing explicit class field theory.

Contribution

It develops a new framework for rigid meromorphic cocycles in orthogonal groups and explores their potential in explicit class field theory beyond CM fields.

Findings

Construction of rigid meromorphic cocycles using $p$-adic theta lifts

Conjecture that special values generate class fields of reflex fields

Extension of class field theory to non-CM settings

Abstract

Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a $p$-adic analogue of Borcherds' singular theta lift. The values of rigid meromorphic cocycles at special points of an associated $p$-adic symmetric space are then conjectured to belong to class fields of suitable global reflex fields, suggesting an eventual framework for explicit class field theory beyond the setting of CM fields explored in the treatise of Shimura and Taniyama.