arXiv subjects

Lejla Smajlovic

Publications and source records attributed to Lejla Smajlovic.

15 recordsLinked to original sources

arXiv2026

Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

Let $(X,χ,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $χ$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,χ,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $χ$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $Δ_{2k}$ as well that of the scalar Laplacian $Δ_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\detΔ_{2k_n}$ for a sequence $(X_{n}, χ_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\detΔ_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_α$ which depends only on $α=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.

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arXiv2026

Determinants of twisted Laplacians and the twisted Selberg zeta function

Let $X$ be an orbisurface, meaning a compact hyperbolic Riemann surface possibly with a finite number of elliptic points, and let $X_1$ denote its unit tangent bundle. We consider the twisted Selberg zeta function $Z(s;ρ)$ associated to a representation $ρ: π_1(X_1) \to \text{GL}(V_ρ)$. We prove a relation between the twisted Selberg zeta function $Z(s;ρ)$ and the regularized determinant of the twisted Laplacian associated to $ρ$. These results can be viewed as a generalization of a result due to Sarnak who considered the trivial character. Yet our proof is different, as it is based on evaluation of the Laplace-Mellin type integral transformations. Going further, we explicitly compute the multiplicative constant, which we call the torsion factor, and express its dependence on parameters which determine the representation. We study the asymptotic behavior of the constant for a sequence of non-unitary representations introduced by Yamaguchi and prove that the asymptotic behavior of this constant as the dimension of the representation tends to infinity is the same as the behavior of the higher-dimensional Reidemeister torsion on $X_1$ (up to an absolute constant).

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arXiv2025

An explicit construction of heat kernels and Green's functions in measure spaces

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $σ$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.

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arXiv2024

On the functional equation of twisted Ruelle zeta function and Fried's conjecture

Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $χ$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,χ)$ be the associated Ruelle zeta function, and $φ(s,χ)$ the determinant of the scattering matrix. We prove the functional equation that $R(s,χ)φ(s,χ) = R(-s,χ)φ(s,χ)H(s,χ)$ where $H(s,χ)$ is a meromorphic function of order one explicitly determined using the topological data of $M$ and of $χ$, and the trigonometric function $\sin(s)$. From this, we determine the order of the divisor of $R(s,χ)$ at $s=0$ and compute the lead coefficient in its Laurent expansion at $s=0$. When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of $R(0,χ)$.

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arXiv2021

Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas

Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $μ$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a generalization of the so-called elliptic Eisenstein series which has been defined and studied for finite volume Riemann surfaces.

math.NT↗

arXiv2021

Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space

In Cogdell et al., \it LMS Lecture Notes Series \bf 459, \rm 393--427 (2020), \rm the authors proved an analogue of Kronecker's limit formula associated to any divisor $\mathcal D$ which is smooth in codimension one on any smooth Kähler manifold $X$. In the present article, we apply the aforementioned Kronecker limit formula in the case when $X$ is complex projective space $\CC\PP^n$ for $n \geq 2$ and $\mathcal D$ is a hyperplane, meaning the divisor of a linear form $P_D({z})$ for ${z} = (\mathcal{Z}_{j}) \in \CC\PP^n$. Our main result is an explicit evaluation of the Mahler measure of $P_{D}$ as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the $L^{2}$-norm of the vector of coefficients of $P_{D}$.

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arXiv2021

Kronecker limit functions and an extension of the Rohrlich-Jensen formula

In 1984 Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert$ of a $\text{\rm PSL}(2,\ZZ)$ modular form $f$ in terms of the Dedekind Delta function evaluated at the divisor of $f$. Recently, Bringmann-Kane re-interpreted the Rohrlich-Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert$ and extended the result to compute a regularized inner product of $\log \Vert f \Vert$ with what amounts to powers of the Hauptmoduli of $\text{\rm PSL}(2,\ZZ)$. In the present article, we revisit the Rohrlich-Jensen formula and prove that it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur-Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group $Γ$ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when $Γ$ is the full modular group $\text{\rm PSL}(2,\ZZ)$, thus reproving the theorems from \cite{BK19}; and second when $Γ$ is an Atkin-Lehner group $Γ_{0}(N)^+$, where explicit computations are given for certain genus zero, one and two levels.

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arXiv2020

Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encode the spectrum of $Δ$ in such a way that they can be used to define the regularized determinant of $Δ-z(1-z)I$. The resulting formula for the regularized determinant of $Δ-z(1-z)I$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, which could not be seen in previous works, due to a different definition of the regularized determinant.

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arXiv2018

Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps

Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In the case $f$ admits a Dirichlet series expansion in some right half-plane, we derive the meromorphic continuation in $s$ of $\Z_{f}(s,z)$ as an integral transform of $f'/f$. We apply these results to obtain superzeta product evaluations of Selberg zeta function associated to finite volume hyperbolic manifolds with cusps.

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arXiv2016

An evaluation of the central value of the automorphic scattering determinant

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $ϕ(s)$ denote the automorphic scattering determinant. From the known functional equation $ϕ(s)ϕ(1-s)=1$ one concludes that $ϕ(1/2)^{2} = 1$. However, except for the relatively few instances when $ϕ(s)$ is explicitly computable, one does not know $ϕ(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $ϕ(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $ϕ(s)$. Then $ϕ(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.

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arXiv2016

The determinant of the Lax-Phillips scattering operator

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $ζ^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $ζ^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.

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arXiv2015

Certain aspects of holomorphic function theory on some genus zero arithmetic groups

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weight four and six; the smallest weight cusp form Delta has weight twelve and can be written as a polynomial in E4 and E6; and the Hauptmodul j can be written as a multiple of E4 cubed divided by Delta. The goal of the present article is to seek generalizations of these results to some other genus zero arithmetic groups, namely those generated by Atkin-Lehner involutions of level N with square-free level N.

math.NT↗

arXiv2015

Applications of Kronecker's limit formula for elliptic Eisenstein series

We develop two applications of the Kronecker's limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil's reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic points for a few small level moonshine groups.

math.NT↗

arXiv2015

On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave operator.

math.NT↗

arXiv2013

On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness for the number of zeros in the half plane to the left of the critical line; an asymptotic expansion for the counting function measuring the vertical distribution of zeros; and an asymptotic expansion for the counting function measuring the horizontal distance of zeros from the critical line. In the present article, we study the more complicated setting of distribution of zeros of the derivative of the Selberg zeta function associated to a non-compact, finite volume hyperbolic Riemann surface. There are numerous difficulties which exist in the non-compact case that are not present in the compact setting, beginning with the fact that in the non-compact case the Selberg zeta function does not satisfy the analogue of the Riemann hypothesis. To be more specific, we actually study the zeros of the derivative of ZH, where Z is the Selberg zeta function and H is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite-volume hyperbolic Riemann surface. Our main results address finiteness of zeros in the half plane to the left of the critical line, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros.

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