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Plinio G. P. Murillo

Publications and source records attributed to Plinio G. P. Murillo.

9 recordsLinked to original sources
arXiv2026

Counting Salem numbers arising from arithmetic hyperbolic orbifolds

The relationship between Salem numbers and short geodesics has been fruitful in quantitative studies of arithmetic hyperbolic orbifolds, particularly in dimensions 2 and 3. In this article, we push these connections even further. The primary goals are: (1) to bound the proportion of Salem numbers of degree up to $n+1$ in the commensurability class of classical arithmetic lattices in any odd dimension $n$; (2) to improve lower bounds for the strong exponential growth of averages of multiplicities in the geodesic length spectrum of non-compact arithmetic orbifolds. In order to accomplish these goals, we bound, for a fixed square-free integer $D$, the count of Salem numbers with minimal polynomial $f$ satisfying $f(1)f(-1)=-D$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$. To do this, we make use of results on the distribution of Salem numbers, as well as classical methods for counting Pythagorean triples and Gauss' lattice-counting argument. To this end, we give a generalization of the count of Pythagorean triples and provide an elementary proof which may be of independent interest.

math.NT↗
arXiv2026

Salem numbers and commensurability classes of arithmetic hyperbolic manifolds

In this article we show that given a Salem number $λ$, a totally real number field $k\subseteq\mathbb{Q}(λ+λ^{-1})$, and a positive integer $n\geq\mathrm{deg}_k(λ)-1$, there exist infinitely many commensurability classes of arithmetic hyperbolic $n$-manifolds defined over $k$ which contain a geodesic of length $\logλ$.

math.GT↗
arXiv2025

On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds

In this article, we construct an arithmetic hyperbolic $6-$orbifold $\mathcal{O}$ such that, any square-rootable Salem number of degree at most $4$ over $\mathbb{Q}$ is realized as the exponential of the length of a closed geodesic in $\mathcal{O}$. We also prove that $n=6$ is the minimal dimension among arithmetic hyperbolic orbifolds of the first type where it can be obtained. In an appendix, we establish a general relation between the discriminant of a Salem number and the determinant of a quadratic space which realizes it. In particular, for any $m,d>0$ we present a geometric proof of the existence of Salem numbers of degree $2m$ with discriminant $(-1)^{m+1}d$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.

math.NT↗
arXiv2023

Hyperbolic manifolds with a large number of systoles

In this article, for any $n\geq 4$ we construct a sequence of compact hyperbolic $n$-manifolds $\{M_i\}$ with number of systoles at least as $\mathrm{vol}(M_i)^{1+\frac{1}{3n(n+1)}-ε}$ for any $ε>0$. In dimension 3, the bound is improved to $\mathrm{vol}(M_i)^{\frac{4}{3}-ε}$. These results generalize previous work of Schmutz for $n=2$, and Dória-Murillo for $n=3$ to higher dimensions.

math.GT↗
arXiv2021

Hyperbolic 3-manifolds with large kissing number

In this article we construct a sequence $\{M_i\}$ of non compact finite volume hyperbolic $3$-manifolds whose kissing number grows at least as $\mathrm{vol}(M_i)^{\frac{31}{27}-ε}$ for any $ε>0$. This extends a previous result due to Schmutz in dimension $2$.

math.GT↗
arXiv2020

Counting Salem numbers of arithmetic hyperbolic 3-orbifolds

It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic $3$-dimensional orbifold defines $c Q^{1/2} + O(Q^{1/4})$ square-rootable Salem numbers of degree $4$ which are less than or equal to $Q$. This quantity can be compared to the total number of such Salem numbers, which is shown to be asymptotic to $\frac{4}{3}Q^{3/2}+O(Q)$. Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic $3$-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions $2$ and $3$.

math.GT↗
arXiv2017

Systole of congruence coverings of arithmetic hyperbolic manifolds

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 $M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$ where $c$ 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 $\frac{8}{n(n+1)}$ is sharp.

math.DG↗
arXiv2016

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 $$sysπ_{1}(M_{I})\geq \frac{4}{3\sqrt{n}}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $M_{I}$.

math.DG↗