$p$-adic interpolation of square roots of central $L$-values of modular forms

TL;DR

This paper constructs a meromorphic function on the eigencurve that interpolates square roots of ratios of central L-values of quadratic twists of modular forms, advancing understanding of p-adic properties of these special values.

Contribution

It introduces a novel p-adic interpolation method for square roots of ratios of central L-values on the eigencurve, linking modular forms and p-adic L-functions.

Findings

Successfully constructs a meromorphic interpolating function.

Provides new insights into p-adic properties of L-values.

Establishes connections between modular forms and p-adic L-functions.

Abstract

We construct a meromorphic function on the eigencurve that interpolates a square root of the ratio of the central values of two quadratic twists of the $L$-function at classical points.