arXiv subjects

Devendra Tiwari

Publications and source records attributed to Devendra Tiwari.

8 recordsLinked to original sources

arXiv2026

Shimura curves of discriminant 14 and 15 and associated Heun Functions

The Shimura curve of discriminant $D$ for $D=14, 15$ is uniformized by a subgroup of an arithmetic quadrilateral Fuchsian group $(2, 2, 2, q)$, where $q=4, 6$. We relate the generator of the ring of quaternionic modular forms on this Shimura curve to explicit Heun functions for the quadrilateral group. We also discuss how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms on $X^{D}(1) / W_{D}$, where $W_D$ is the full group of Atkin-Lehner involutions. This leads us to completely describe the rational exceptional sets of the associated Heun functions, and the algebraic values attained by the Heun function on these points, for example $${\rm He}\left( 81, \frac{1}{2}; \frac{1}{3}, \frac{1}{6}, \frac{1}{2}, \frac{1}{2};-\frac{729}{112}\right)= \left( \frac{2^2 \cdot 3^3 \cdot 5^3}{7^5} \right)^{\frac{1}{6}}$$.

math.NT↗

arXiv2026

Modular elliptic curves and hyperbolic uniformization

In an article published a few years before the modularity of elliptic curves over $\Q$ was proved, Mazur \cite{maz} looked at modularity as a purely complex analytic phenomenon, defining a notion of an elliptic curve over $\Q$ having a hyperbolic uniformisation of arithmetic type. Such an elliptic curve (of conductor $N$, say) is necessarily geometrically modular, i.e. a quotient of the jacobian of the modular curve $X_0(N)$, by a morphism defined over $\Q$. We extend these ideas to elliptic curves over totally real fields of odd degree, using Shimura curves for quaternion algebras split at all finite places and one real place. In particular, we prove that the existence of a hyperbolic uniformisation of arithmetic type would imply geometric modularity.

math.NT↗

arXiv2022

High temperature equilibrium of 3D and 2D chalcogenide perovskites

Chalcogenide perovskites have been recently under the researchers spotlight as novel absorber materials for photovoltaic applications. BaZrS$_3$, the most investigated compound of this family, shows a high absorption coefficient, a bandgap of around 1.8 eV, and excellent environmental and thermal stability. In addition to the 3D perovskite BaZrS$_3$, the Ba-Zr-S compositional space contains various 2-D Ruddlesden-Popper phases Ba$_{x+1}$Zr$_x$S$_{3x+1}$ (with $x=$ 1, 2, 3) which have recently been reported. In this work it will be shown that at high temperature the Gibbs free energies of 3D and 2D perovskites are very close, suggesting that 2D phases can be easily formed at high temperatures. The analysis of the product of the BaS and ZrS$_2$ solid-state reaction, in different stoichiometric conditions, present a mixture of BaZrS$_3$ and Ba$_4$Zr$_3$S$_{10}$. To carefully resolve the composition, XRD, SEM and EDS analysis were complemented with Raman spectroscopy. For this purpose, the phonon modes, and the consequent Raman spectra, were calculated for the 3D and 2D chalcogenide perovskites, as well as for the binary precursors. This thorough characterization demonstrates the thermodynamic limitations and experimental difficulties in forming phase-pure chalcogenide perovskites through solid state synthesis, and the importance of using multiple techniques to soundly resolve the composition of these chalcogenide materials.

cond-mat.mtrl-sci↗

arXiv2021

Hecke triangle Groups and Dessin d'enfant

In this work we will construct bipartite graphs, famously known as Dessin d'enfant, corresponding to finite index subgroups of Hecke triangle groups $(2, q, \infty )$. Then using a results of \cite{ll} we shall show the correspondences among the special polygons, the bi-partite graph, and the tree diagram for a finite index subgroup of the Hecke triangle groups $(2, q, \infty )$.

math.CO↗

arXiv2020

On Free Group Generated by Two Heisenberg Translations

In this paper, we will discuss the groups generated by two Heisenberg translations of ${\rm PSp}(2,1)$ and determine when they are free. We improve a result given in \cite{xwy} by Xie, Wang, Jiang in Canad. Math. Bull. $56(2013), 881-889.$ and from that derive result for the ${\rm PSp}(2,1)$ case.

math.GR↗

arXiv2018

Discreteness Of Hyperbolic Isometries by Test Maps

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is called \emph{Zariski dense} if it does not fix a point on the closure of the $\mathbb F$-hyperbolic space, and neither it preserves a totally geodesic subspace of it. We prove that a Zariski dense subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is discrete if for every loxodromic element $g \in G$, the two generator subgroup $\langle f, g \rangle$ is discrete, where $f \in {\rm U}(n,1; \mathbb F)$ is a test map not necessarily from $G$.

math.GT↗

arXiv2018

On discreteness of subgroups of quaternionic hyperbolic isometries

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_{\mathbb H}}^n$ and neither it preserves a totally geodesic subspace of ${{{\bf H}}_{\mathbb H}}^n$. We prove that a Zariski dense subgroup $G$ of ${\rm{ Sp}}(n,1)$ is discrete if for every loxodromic element $g \in G$ the two generator subgroup $\langle f, g f g^{-1} \rangle$ is discrete, where the generator $f \in {\rm{Sp}}(n,1)$ is certain fixed element not necessarily from $G$.

math.GT↗

arXiv2018

On generalized J\o{}rgensen inequality in ${\rm SL}(2, \mathbb C)$

Wang, Jiang and Cao have obtained a generalized version of the J\o{}rgensen inequality in Proc. Indian Acad. Sci. Math. Sci., 123(2):245--251, 2013, for two generator subgroups of ${\rm SL}(2, \mathbb C)$ where one of the generators is loxodromic. We prove that their inequality is strict.

math.GR↗