Difference of a Hauptmodul for $Γ_{0}(N)$

In this short note, by modifying the results and proofs given in the author's recent work, we simply show that the difference of a Hauptmodul for any genus zero group $\Gamma_{0}(N)$ is a Borcherds lift. This work extends Scheithauer's results in a different direction. As a consequence, we derive Monster denominator formula like product expansion for these Hilbert modular forms.

Comments: arXiv admin note: substantial text overlap with arXiv:1701.08433

Similar Publications

In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincar\'e series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms. Read More


Une conjecture de Colliot-Th\'el\`ene \'etablit que l'obstruction de Brauer-Manin est la seule obstruction au principe de Hasse et \`a l'approximation faible pour les espaces homog\`enes des groupes lin\'eaires. Nous montrons que la question peut \^etre r\'eduite au cas des espaces homog\`enes de $\mathrm{SL}_{n,k}$ \`a stabilisateurs finis en suivant les travaux du deuxi\`eme auteur sur l'approximation faible. A conjecture of Colliot-Th\'el\`ene states that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and to weak approximation for homogeneous spaces of linear groups. Read More


We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence and the case where all variables are the same. Next, under an assumption on variables, some determinant formulas coming from theory of Schur functions such as the Jacobi-Trudi, Giambelli and dual Cauchy formula are established with the help of Macdonald's ninth variation of Schur functions. Read More


Given $k\in\mathbb N$, we study the vanishing of the Dirichlet series $$D_k(s,f):=\sum_{n\geq1} d_k(n)f(n)n^{-s}$$ at the point $s=1$, where $f$ is a periodic function modulo a prime $p$. We show that if $(k,p-1)=1$ or $(k,p-1)=2$ and $p\equiv 3\mod 4$, then there are no odd rational-valued functions $f\not\equiv 0$ such that $D_k(1,f)=0$, whereas in all other cases there are examples of odd functions $f$ such that $D_k(1,f)=0$. As a consequence, we obtain, for example, that the set of values $L(1,\chi)^2$, where $\chi$ ranges over odd characters mod $p$, are linearly independent over $\mathbb Q$. Read More


In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining equation (`cluster pictures'). We construct explicit combinatorial one-to-one correspondences between the three, which furthermore respect automorphisms and allow to keep track of the monodromy pairing and the Tamagawa group of the Jacobian. We introduce a classification scheme and a naming convention for semistable types of hyperelliptic curves and types with a Frobenius action. Read More


The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi presented an algorithm to enumerate all local optima in a finite computation, which has been implemented in up to d = 8 dimensions. Read More


We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincar\'e series with evaluations of Niebur-Poincar\'e series as Fourier coefficients. This allows us to study twisted traces of singular moduli in an integral weight setting. Read More


The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(\alpha^n x)^{}_{n\in\mathbb{N}}$, where $\alpha$ is a fixed real number with $| \alpha | > 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every $x\in\mathbb{R}$. Read More


We prove that there exist hypersurfaces that contain a given closed subscheme $Z$ of the projective space over a finite field and intersect a given smooth scheme $X$ off of $Z$ smoothly, if the intersection $V = Z \cap X$ is smooth. Furthermore, we can give a bound on the dimension of the singular locus of the hypersurface section and prescribe finitely many local conditions on the hypersurface. This is an analogue of a Bertini theorem of Bloch over finite fields and is proved using Poonen's closed point sieve. Read More


We call a positive real number $\lambda$ admissible if it belongs to the Lagrange spectrum and there exists an irrational number $\alpha$ such that $\mu(\alpha)=\lambda$. Here $\mu(\alpha)$ denotes the Lagrange constant of $\alpha$ - maximal value of real numbers $c$ such that $\forall \varepsilon>0$ the inequality $|\alpha-\frac{p}{q}|<\frac{1}{cq^2}$. In this paper we establish a necessary and sufficient condition of admissibility of the Lagrange spectrum element and construct an infinite series of not admissible elements. Read More