We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_1,...,p_s\in X\left(K\right)$ and $X^o:=X̅\setminus D,p_1,...,p_s$ (the $p_j$’s may be in the support of D). Using tools from Nevanlinna theory and formal geometry, we give the definition of E-section of arithmetic type of the vector bundle E with respect to the points $p_j$; this is the natural generalization of the notion of E-function defined in Siegel-Shidlovskiĭ theory. We prove...

Let E be an elliptic curve over a field K of characteristic not equal to 2 and let N > 1 be an integer prime to char(K). The purpose of this paper is to construct the (two-dimensional) Hurwitz moduli space H (E/K,N,2) which classifies genus 2 covers of E of degree N and to show that it is closely related to the modular curve X(N) which parametrizes elliptic curves with level-N-structure. More precisely, we introduce the notion of a normalized genus 2 cover of E/K and show that the corresponding...

The article presents some possibilities to create non-convex star polygons from regular polygons. The text includes exercises about the construction of the star polygons and exercises inciting to study their attributes and their using in the everyday life of the pupil. The subject of the star polygons deepens the basic curriculum in plane geometry in RVP and it is a suitable motivation complement in teaching mathematics at the second stage of elementary school as well as at grammar school.

Let $f$ be a Hecke–Maass cusp form of eigenvalue $\lambda$ and square-free level $N$. Normalize the hyperbolic measure such that $\mathrm{vol}\left(Y_0\left(N\right)\right)=1$ and the form $f$ such that ${\parallel f\parallel }_2=1$. It is shown that ${\parallel f\parallel }_{\infty }{\ll }_{ϵ}{\lambda }^{\frac{5}{24}+ϵ}{N}^{\frac{1}{3}+ϵ}$ for all $ϵ>0$. This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

Let ${ℳ}_0^{ℝ}$ be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space $H^4$ and form the quotient by an arithmetic group to obtain an orbifold isomorphic to a component of the moduli space. There are five components. For each we describe the corresponding lattices in $\mathrm{PO}(4,1)$. We also derive several new and several old results on the topology of ${ℳ}_0^{ℝ}$....

Etant donnés $A_1,\cdots ,A_s$ ($s\ge 3$) des $S{L}_2\left(ℂ\right)$-modules non triviaux de dimensions respectives $n_1+1\ge \cdots \ge n_s+1$ (avec $n_1=n_2+\cdots +n_s$) et $\phi \in ℒ(A_2\otimes \cdots \otimes A_s,A_1^{*})$ un $S{L}_2\left(ℂ\right)$-homomorphisme, nous montrons que l’hyperdéterminant de $\phi$ est nul sauf si les modules $A_i$ sont irréductibles et si l’homomorphisme est la multiplication des polynômes homogènes à deux variables.

Applying the “exact WKB method” (cf. Delabaere-Dillinger-Pham) to the stationary one-dimensional Schrödinger equation with polynomial potential, one is led to a multivalued complex action-integral function. This function is a (hyper)elliptic integral; the sheet structure of its Riemann surface above the plane of its values has interesting properties: the projection of its branch-points is in general a dense subset of the plane, and there is a group of symmetries acting on the surface. The distribution...