On growth of systole along congruence coverings of Hilbert modular varieties

TL;DR

This paper investigates how the shortest non-contractible loops (systoles) in principal congruence coverings of Hilbert modular varieties grow as the coverings become larger, establishing a logarithmic lower bound related to volume.

Contribution

It provides a new logarithmic lower bound on systole growth for congruence coverings of Hilbert modular varieties, extending understanding of their geometric properties.

Findings

Systole grows at least logarithmically with volume.

The lower bound depends on the dimension of the variety.

The result is independent of the specific covering sequence.

Abstract

We study how the systole of principal congruence coverings of a Hilbert modular variety grows when the degree of the covering goes to infinity. We prove that given a Hilbert modular variety $M$ of real dimension $2n$, the sequence of principal congruence coverings $M_{I}$ eventually satisfies $$sys\pi_{1}(M_{I})\geq \frac{4}{3\sqrt{n}}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $M_{I}$.