In this paper we prove that, for any arithmetic hyperbolic n-manifold M of the first type, the systole of most of the principal congruence coverings M I satisfywhere c is a constant independent of I. This generalizes previous work of Buser and Sarnak, and Katz, Schaps and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In an appendix together with Cayo Dória we prove that the constant 8 n(n+1) is sharp.

We provide an explicit lower bound for the systole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.

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We provide an explicit lower bound for the systole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.8 n(n+1) . In an appendix to this article, Dória and Murillo proved that this is the best possible constant in this case. A similar result for Hilbert modular varieties was obtained in [14].