# On theta series attached to maximal lattices and their adjoints

## TL;DR

This paper studies theta series of adjoint lattices with specific properties, demonstrating their structure and extremality, and providing examples of dual extremal lattices for squarefree levels.

## Contribution

It introduces the Weierstrass property for theta series of adjoint lattices and constructs examples of dual extremal lattices at squarefree levels.

## Key findings

- Theta series of adjoint lattices have the Weierstrass property.
- Existence of dual extremal lattices at squarefree levels.
- Framework for defining extremality for arbitrary squarefree levels.

## Abstract

The space spanned by theta series of adjoints of maximal even lattices of exact level $N$ and determinant $N^2$ has the Weierstrass property and hence allows to define extremality for arbitrary squarefree level $N$. We find examples of such dual extremal lattices.