A p-adic L-function for non-critical adjoint L-values

TL;DR

This paper constructs a $p$-adic L-function interpolating non-critical twisted adjoint L-values for modular forms over an imaginary quadratic field, using $ ext{Lambda}$-adic modular symbols and cohomology with $p$-adic measures.

Contribution

It introduces a new $p$-adic L-function for non-critical adjoint L-values in the context of Hida families, extending previous methods to non-critical cases.

Findings

Constructed an analytic $p$-adic L-function for non-critical values.

Interpolates twisted adjoint $L$-values across a Hida family.

Utilizes $ ext{Lambda}$-adic modular symbols and cohomology with $p$-adic measures.

Abstract

Let $K$ be an imaginary quadratic field, with associated quadratic character $\alpha$. We construct an analytic $p$-adic $L$-function interpolating the twisted adjoint $L$-values $L(1, \mathrm{ad}(f) \otimes \alpha)$ as $f$ varies in a Hida family; these special values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of $\Lambda$-adic modular symbols, which considers cohomology with values in a space of $p$-adic measures.