On linear relations for L-values over real quadratic fields

TL;DR

This paper develops a method to relate Fourier coefficients of Hilbert and classical modular forms over real quadratic fields, establishing connections between special L-values and arithmetic functions.

Contribution

It introduces a novel construction method for linear relations between Fourier coefficients and special L-values in the context of real quadratic fields.

Findings

Linear formulas relating Fourier coefficients of Hilbert and classical modular forms.

Relations between special L-values at zero and arithmetic functions.

Connections between sum of squares functions over quadratic fields and rationals.

Abstract

In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at $0,$ and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}$.