On growth of systole along congruence coverings of Hilbert modular varieties

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 $$sys{\pi }_1(M_I)\ge \frac{4}{3\sqrt{n}}\log (vol(M_I))-c,$$ where $c$ is a constant independent of $M_I$.