# Spectral Curves are Transcendental

## TL;DR

This paper explores the arithmetic properties of spectral curves, showing that for certain monopoles, the spectral curve cannot be defined over algebraic numbers if it is smooth, highlighting its transcendental nature.

## Contribution

It demonstrates that spectral curves of smooth Euclidean BPS monopoles are transcendental and not defined over algebraic numbers, revealing new arithmetic insights.

## Key findings

- Spectral curves of smooth monopoles are transcendental.
- Such curves are not defined over algebraic number fields.
- The work links spectral geometry with transcendental number theory.

## Abstract

Some arithmetic properties of spectral curves are discussed: the spectral curve, for example, of a charge $n\ge2$ Euclidean BPS monopole is not defined over $\overline{\mathbb{Q}}$ if smooth.