On growth of systole along congruence coverings of Hilbert modular varieties

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 $M_k$ of real dimension $2n$ defined over a number field $k$, the sequence of principal congruence coverings $M_I$ eventually satisfies

$${sys}_1\left(M_I\right)\ge \frac{4}{3\sqrt{n}}log\left(vol\left(M_I\right)\right)-c,$$

where $c$ is a constant independent of $M_I$.