# Modular systoles are extremal for the crossing number

## TL;DR

This paper investigates the crossing numbers of systoles on congruence surfaces, demonstrating that their intersection growth rate is minimal among all comparable curve sets on the same surface.

## Contribution

It establishes the optimal minimal growth rate of intersection numbers for systoles on congruence surfaces compared to other curve families.

## Key findings

- Systoles on congruence surfaces have the smallest possible intersection growth rate.
- The result applies to families of curves with the same cardinality on the same surface.
- The work characterizes extremal properties of systoles in topological surface theory.

## Abstract

We study crossing numbers for systoles of congruence surfaces. Taken as a family of curves on a family of surfaces, we show that the growth rate of their intersection is optimally small among all sets of curves of the same cardinality lying on the same topological surface.