arXiv subjects

Otto Romero

Publications and source records attributed to Otto Romero.

4 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↗

arXiv2023

The Hodge Laplacian operator on 1-forms on $\mathbb{H}$ and 1-form $E_\mathfrak{a}^1$

As is well known, we can average the eigenfunction $y^s$ of the hyperbolic Laplacian on the hyperbolic plane by $Γ$ a lattice in $\mathbf{SL}(2,\mathbb{R})$ to obtain an automorphic form, the non-holomorphic Eisenstein series $E_\mathfrak{a} (z,s)$. In this note, we choose a particular eigenfunction $y^s dx$ of the Hodge-Laplace operator for 1-forms on the hyperbolic plane. Then, we average by $Γ$ to define a 1-form $E_\mathfrak{a}^1 \big( (z,v), s \big)$. We see that $E_\mathfrak{a}^1$ admits a Fourier expansion and calculates the corresponding coefficients. Also, we evaluate the integral $\int_γ E_\mathfrak{a}^1$ for when $γ$ is a lifting of horocycles and closed geodesics in the unit tangent bundle. Finally, we will obtain an analog to the Rankin-Selberg method for $E_\mathfrak{a}^1$.

math.NT↗

arXiv2019

A note on Mellin transform, Eisenstein Series and distribution $dε_{it}$ on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$

Let $f$ a smooth function with compact support defined on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$, we prove a formula for the Mellin transform of $f$, then we can define the micro-local lift $dε_{it}$ to $SL(2,\mathbb{C})$. We calculate $(f,dε_{it})$ for $f$ a cuspidal form and for $f$ an incomplete Eisenstein series. We also establish asymptotic estimates when $t$ tends to $\infty$. We conjecture that a new positive distribution $dε_{it}^F$, constructed with the Friedrichs' symmetrization technique, satisfies the same asymptotic estimates that $dε_{it}$. This would imply the quantum ergodicity for Eisenstein series on $PSL(2,\mathbb{Z}[i]) \backslash PSL(2,\mathbb{C})$

math.NT↗

arXiv2019

Eisenstein series and equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations of Bianchi 3-orbifolds

Inspired by the works of Zagier, we study the probability measures $ν(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=\mathbb{H}^3/\widetildeΓ_D$, of finite volume and with only one cusp. Here $Γ_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one.

math.DS↗