#### Volume 17, issue 5 (2017)

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On growth of systole along congruence coverings of Hilbert modular varieties

### Plinio G P Murillo

Algebraic & Geometric Topology 17 (2017) 2753–2762
##### 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 ${M}_{I}$.

##### Keywords
systole, arithmetic lattice, Hilbert modular varieties, congruence subgroups
##### Mathematical Subject Classification 2010
Primary: 22E40, 11R80
Secondary: 53C22
##### Publication
Revised: 6 April 2017
Accepted: 10 May 2017
Published: 19 September 2017
##### Authors
 Plinio G P Murillo Instituto de Matemática Pura e Aplicada Rio de Janeiro Brazil http://www.impa.br/~plinio