Theta series and number fields: theorems and experiments

TL;DR

This paper constructs theta-series linked to number fields, proves their invariance for fields of degree up to 4, and investigates their linear independence, providing computational evidence for the conjecture.

Contribution

It introduces new theta-series invariants for number fields and explores their linear independence through theoretical results and computational experiments.

Findings

Theta-series are invariants for degree ≤ 4 number fields.

Computational evidence suggests theta-series are linearly independent for certain cases.

Heuristic analysis supports the conjecture on linear independence in general.

Abstract

We construct certain $\theta$-series associated to number fields and prove that for number fields of degree less than equal to 4, these $\theta$-series are number field invariants. We also investigate whether or not the collection of $\theta$-series associated to number fields of the same degree and discriminant are linearly independent. This is known to be true if the degree of the number field is less than or equal to 3. We do not prove in this paper that they are linearly independent in general but we do give computational and heuristic evidence that we would expect them to be.