On growth of systole along congruence coverings of Hilbert modular varieties

Murillo, Plinio

doi:10.2140/agt.2017.17.2753

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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

DOI: 10.2140/agt.2017.17.2753

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 $2 n$ defined over a number field $k$ , the sequence of principal congruence coverings $M_{I}$ eventually satisfies

{sys}_{1} (M_{I}) \geq \frac{4}{3 \sqrt{n}} log (vol (M_{I})) - 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

References

Publication

Received: 9 June 2016

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

Volume 17, issue 5 (2017)