Arithmetic intersections of modular geodesics

Abstract

The arithmetic, p-arithmetic, and incoherent intersections between pairs of closed geodesics on a modular or Shimura curve are defined, and some of their expected algebraicity and factorisation properties are examined. These properties follow in a special case from the conjectures of [DV] on the RM values of rigid meromorphic cocycles, and are inspired from the recent work of James Rickards [Ri] and Xavier Guitart, Marc Masdeu and Xavier Xarles [GMX] in the quaternionic setting.