# Overconvergent quaternionic forms and anticyclotomic p-adic L-functions

## TL;DR

This paper develops a geometric and p-adic analytic framework for anticyclotomic L-functions associated with modular forms, extending previous constructions and providing new interpretations of Gross points.

## Contribution

It introduces a novel geometric interpretation of Gross points and constructs anticyclotomic p-adic L-functions for non-critical slope modular forms using overconvergent methods.

## Key findings

- Generalizes previous constructions of Gross points and p-adic L-functions.
- Provides a geometric interpretation of Gross points for weight two forms.
- Constructs anticyclotomic p-adic L-functions for non-critical slope modular forms.

## Abstract

We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle from zero to the infinity on the Poincare 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 a 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.