# Numerical evidence for higher order Stark-type conjectures

## TL;DR

This paper develops a systematic numerical approach to test higher order Stark-type conjectures across various abelian extensions, and verifies these conjectures for nearly 20,000 real cubic extensions with small discriminant.

## Contribution

It introduces a general method for numerically testing higher order Stark-type conjectures applicable to any abelian extension of number fields.

## Key findings

- Numerical verification of conjectures for 19,197 fields with discriminant less than 10^{12}
- Method applicable regardless of signature and splitting types
- All tested cases support the conjectures in the examined range.

## Abstract

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension. We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields $K$ of this type with absolute discriminant less than $10^{12}$, for a total of $19197$ examples. The places that split completely in these extensions are always taken to be the two real archimedean places of $k$ and we are in a situation where all the $S$-truncated $L$-functions have order of vanishing at least two.