arXiv subjects
Publications and source records attributed to Lejla Smajlović.
We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $Γ_0(N)$, and the non-congruence arithmetic groups $Γ_0(N)^+$.
Let $N$ be a positive square-free integer such that the discrete group $Γ_{0}(N)^{+}$ has genus one. In a previous article, we constructed canonical generators $x_{N}$ and $y_{N}$ of the holomorphic function field associated to $Γ_{0}(N)^{+}$ as well as an algebraic equation $P_{N}(x_{N},y_{N}) = 0$ with integer coefficients satisfied by these generators. In the present paper, we study the singular moduli problem corresponding to $x_{N}$ and $y_{N}$, by which we mean the arithmetic nature of the numbers $x_{N}(τ)$ and $y_{N}(τ)$ for any CM point $τ$ in the upper half plane $\mathbb{H}$. If $τ$ is any CM point which is not equivalent to an elliptic point of $Γ_{0}(N)^{+}$, we prove that the complex numbers $x_{N}(τ)$ and $y_{N}(τ)$ are algebraic integers. Going further, we characterize the algebraic nature of $x_{N}(τ)$ as the generator of a certain ring class field of $\mathbb{Q}(τ)$ of prescribed order and discriminant depending on properties of $τ$ and level $N$. The theoretical considerations are supplemented by computational examples. As a result, several explicit evaluations are given for various $N$ and $τ$, and further arithmetic consequences of our analysis are presented. In one example, we explicitly construct a set of minimal polynomials for the Hilbert class field of $\mathbb{Q}(\sqrt{-74})$ whose coefficients are less than $2.2\times 10^{4}$, whereas the minimal polynomials obtained from the Hauptmodul of $\textrm{PSL}(2,\mathbb{Z})$ has coefficients as large as $6.6\times 10^{73}$.
Let $N$ be a square-free integer such that the arithmetic group $Γ_0(N)^+$ has genus zero; there are $44$ such groups. Let $j_N$ denote the associated Hauptmodul normalized to have residue equal to one and constant term equal to zero in its $q$-expansion. In this article we prove that the Hauptmodul at any elliptic point of the surface associated to $Γ_0(N)^+$ is an algebraic integer. Moreover, for each such $N$ and elliptic point $e$, we show how to explicitly evaluate $j_{N}(e)$ and provide the list of generating polynomials (with small coefficients) of the class fields or their subfields corresponding to the orders over the imaginary quadratic extension of rationals stemming from the elliptic points under consideration.
In this paper we study, both analytically and numerically, questions involving the distribution of eigenvalues of Maass forms on the moonshine groups $Γ_0(N)^+$, where $N>1$ is a square-free integer. After we prove that $Γ_0(N)^+$ has one cusp, we compute the constant term of the associated non-holomorphic Eisenstein series. We then derive an "average" Weyl's law for the distribution of eigenvalues of Maass forms, from which we prove the "classical" Weyl's law as a special case. The groups corresponding to $N=5$ and $N=6$ have the same signature; however, our analysis shows that, asymptotically, there are infinitely more cusp forms for $Γ_0(5)^+$ than for $Γ_0(6)^+$. We view this result as being consistent with the Phillips-Sarnak philosophy since we have shown, unconditionally, the existence of two groups which have different Weyl's laws. In addition, we employ Hejhal's algorithm, together with recently developed refinements from [31], and numerically determine the first $3557$ of $Γ_0(5)^+$ and the first $12474$ eigenvalues of $Γ_0(6)^+$. With this information, we empirically verify some conjectured distributional properties of the eigenvalues.
Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of the Dedekind eta function. This eta function is a weight $1/2$ modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $Γ$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $S:Γ\to {\mathbb R}$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $Γ$ to ${\mathbb C}$ produces higher-order Eisenstein series. These series share many of the properties of $E(z,s)$ though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $S^*:Γ\to {\mathbb R}$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $Γ_0(N)$ are rational.
In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $λ_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(λ_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $GL_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-functions.
We define generalized Li coefficients, called $τ-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$, derive an arithmetic formula for the computation of $τ-$Li coefficients and conduct numerical investigation of $τ-$Li coefficients for a certain product of shifts of the Riemann zeta function.
For any square-free integer $N$ such that the "moonshine group" $Γ_0(N)^+$ has genus zero, the Monstrous Moonshine Conjectures relate the Hauptmoduli of $Γ_0(N)^+$ to certain McKay-Thompson series associated to the representation theory of the Fischer-Griess monster group. In particular, the Hauptmoduli admits a $q$-expansion which has integer coefficients. In this article, we study the holomorphic function theory associated to higher genus moonshine groups $Γ_0(N)^+$. For all moonshine groups of genus up to and including three, we prove that the corresponding function field admits two generators whose $q$-expansions have integer coefficients, has lead coefficient equal to one, and has minimal order of pole at infinity. As corollary, we derive a polynomial relation which defines the underlying projective curve, and we deduce whether $i\infty$ is a Weierstrass point. Our method of proof is based on modular forms and includes extensive computer assistance, which, at times, applied Gauss elimination to matrices with thousands of entries, each one of which was a rational number whose numerator and denominator were thousands of digits in length.