Overconvergent quaternionic forms and anticyclotomic p-adic L-functions

We reinterpret the explicit construction of Gross points given by Chida–Hsieh as a non-Archimedian analogue of the standard geodesic cycle (i∞)−(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.