A space of weight one modular forms attached to totally real cubic number fields

TL;DR

This paper constructs weight 1 modular forms associated with totally real cubic fields and demonstrates that these forms uniquely determine the fields, forming a linearly independent set.

Contribution

It introduces a method to attach a unique weight 1 modular form to each totally real cubic field, establishing their linear independence.

Findings

Each cubic field corresponds to a unique modular form.

The set of forms attached to all such fields is linearly independent.

The forms have levels and nebentypus related to the field discriminant.

Abstract

Let $d$ be a positive fundamental discriminant, and let $\mathcal{C}_{d}$ be the set of isomorphism classes of cubic number fields of discriminant $d$. For each $K \in \mathcal{C}_{d}$, we construct a weight 1 modular form $f_{K}$ with level $3^{\pm 1}d$ and nebentypus $\left( \frac{-3^{\pm 1}d}{\cdot} \right)$. We show that the form $f_{K}$ completely determines the field $K$. Moreover, we show that $\{f_{K} : K \in \mathcal{C}_{d}\}$ is a linearly independent set.