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

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 Mk of real dimension 2n defined over a number field k, the sequence of principal congruence coverings MI eventually satisfies

sys1(MI) 4 3nlog(vol(MI)) c,

where c is a constant independent of MI.

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systole, arithmetic lattice, Hilbert modular varieties, congruence subgroups
Mathematical Subject Classification 2010
Primary: 22E40, 11R80
Secondary: 53C22
Received: 9 June 2016
Revised: 6 April 2017
Accepted: 10 May 2017
Published: 19 September 2017
Plinio G P Murillo
Instituto de Matemática Pura e Aplicada
Rio de Janeiro