The kernel of newform Dedekind sums

TL;DR

This paper investigates the kernel of newform Dedekind sums, revealing their size properties and exploring Galois actions to enhance computational methods for these sums.

Contribution

It introduces the study of the kernel of newform Dedekind sums and analyzes their size, also providing insights into Galois actions for efficient computation.

Findings

Kernels are neither too large nor too small

Galois action enables more efficient numerical calculations

Provides foundational understanding of newform Dedekind sums

Abstract

Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums.