We give a detailed presentation of $\ell$-isogeny graphs associated with ordinary elliptic curves defined over $\mathbb{F}_p$. We then focus on the following inverse problem: given an abstract volcano $V$, do there always exist primes $\ell, p \in \m...
The two brightest elliptical galaxies in the Eridanus A group, NGC 1400 and NGC 1407, have been observed in both the Washington T_1 and Kron-Cousins I filters to obtain photometry of their globular cluster systems (GCSs). This group of galaxies is...
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use...
In this text we expound recent results by Idrisse Khemar on the construction of various geometric completely integrable systems generalizing the structure of Hamiltonian stationary Lagrangian surfaces (HSLS) discovered by F. Hélein and P. Romon. W...
In this proceeding, we briefly describe the viscous hydrodynamics + hadron cascade hybrid model VISHNU for relativistic heavy ion collisions and report the current status on extracting the QGP viscosity from elliptic flow data....
Strange duality is shown to hold over generic $K3$ surfaces in a large number of cases. The isomorphism for elliptic $K3$ surfaces is established first via Fourier-Mukai techniques. Applications to Brill-Noether theory for sheaves on $K3$s are also o...
This is an English translation of Felix Klein's paper "Ueber die Transformation elfter Ordnung der elliptischen Functionen" from 1879....
An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^{\infty}])/K(μ_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism classes...
Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\ell$-adic Galois representations for some rational prime $\ell$. At the ICMS Workshop held in November 2024, we presented evidence for two finitene...
Some new classes of exact solutions (so-called functionally-invariant solutions) of the elliptic and hyperbolic complex Monge-Amp$\grave{e}$re equations and of the second heavenly equation, mixed heavenly equation, asymmetric heavenly equation, evolu...
This document focuses on developing a key expansion method based on the internet X.509 public key infrastructure and elliptic curve cryptography, which is applied in the context of anonymous voting. The method enables end entities to maintain anonymi...
In 2016 Izadi and Nabardi (b) showed (4-2-4) has infinitely many integer solutions. They used a specific congruent number elliptic curve.In 2019 Janfada and Nabardi,item C, showed that a necessary condition for n to have an integral solution for the...
In this paper we obtain a parametric solution of the hitherto unsolved diophantine equation $(x_1^5+x_2^5)(x_3^5+x_4^5)=(y_1^5+y_2^5)(y_3^5+y_4^5)$. Further, we show, using elliptic curves, that there exist infinitely many parametric solutions of the...
According to the flow narrative commonly applied to high-energy nuclear collisions a cylindrical-quadrupole component of 1D azimuth angular correlations is conventionally denoted by quantity $v_2$ and interpreted to represent elliptic flow. Jet angul...
A fundamental problem in statistics is estimating the shape matrix of an Elliptical distribution. This generalizes the familiar problem of Gaussian covariance estimation, for which the sample covariance achieves optimal estimation error. For Elliptic...
Let X0(N) denote the modular curve classifying elliptic curves with a cyclic N-isogeny, A0(N) its group of algebraic autmorphisms and B0(N) the subgroup of automorphisms coming from matrices acting on the upper halfplane. In a well-known paper, Kenku...
SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a secret isogeny...
We consider algebraic affine and projective curves of Edwards \cite{E, SkOdProj} over a finite field $\text{F}_{p^n}$. Most cryptosystems of the modern cryptography \cite{SkBlock} can be naturally transform into elliptic curves \cite{Kob}. We researc...
We study the geometric engineering of gauge theories with gauge group Spin(4) and SO(4) using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual...
The asymptotic efficiency of the spatial sign covariance matrix (SSCM) relative to affine equivariant estimates of scatter is studied in detail. In particular, the SSCM is shown to be asymptoticaly inadmissible, i.e. the asymptotic variance-covarianc...