# On a symmetry of complex and real multiplication

## TL;DR

This paper establishes a correspondence between lattices with complex multiplication and pseudo-lattices with real multiplication, revealing a symmetry between these two types of algebraic structures.

## Contribution

It introduces a novel symmetry linking complex multiplication lattices to real multiplication pseudo-lattices, expanding understanding of their algebraic relationships.

## Key findings

- Each lattice with complex multiplication by $f\sqrt{-D}$ corresponds to a pseudo-lattice with real multiplication by $f'\sqrt{D}$.
- The integer $f'$ is explicitly defined by $f$, establishing a precise correspondence.
- The result uncovers a fundamental symmetry between complex and real multiplication structures.

## Abstract

It is proved that each lattice with complex multiplication by $f\sqrt{-D}$ corresponds to a pseudo-lattice with real multiplication by $f'\sqrt{D}$, where $f'$ is an integer defined by $f$.