Overconvergent Quaternionic Forms and Anticyclotomic $p$-adic $L$-functions

Abstract

We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle $(i\infty) - (0)$ on the Poincaré upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic $p$-adic $L$-functions for modular forms of non-critical slope following the overconvergent strategy à la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess, and Chida-Hsieh and shows a certain integrality of the interpolation formula even for non-ordinary forms.