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Artur Elezi

Publications and source records attributed to Artur Elezi.

9 recordsLinked to original sources
arXiv2026

Equilibrium Laws for Julia's Zero and the Hyperbolic Zero of Binary Forms

For a real binary form with no real roots, Julia's zero and the hyperbolic zero are two SL(2,R)-equivariant points in the upper half-plane. We show that they admit closely related equilibrium characterizations, with radial weights given respectively by the hyperbolic tangent and hyperbolic sine of the distances to the roots. This common framework gives geometric criteria for coincidence of the two zero maps. They always agree for binary quartics; for binary sextics they agree exactly when the three upper-half-plane roots form an equilateral hyperbolic triangle, or are collinear with one root the hyperbolic midpoint of the other two. We also obtain a corresponding result for collinear binary octics. The two equilibrium laws further reveal a sharp difference in the influence of distant roots. A strict majority of roots confined to a compact set keeps Julia's zero in a compact set, and the threshold one-half is optimal. In contrast, an escaping minority can force the hyperbolic zero to escape at linear scale. We also illustrate the computational advantages of the explicit formula for the hyperbolic zero.

math.MG↗
arXiv2019

Isogenous components of Jacobian surfaces

Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, ι)$ its Jacobian, where $ι$ is the principal polarization of $\mbox{Jac} (\mathcal X)$ attached to $\mathcal X$. Assume that $\mbox{Jac} (\mathcal X)$ is $(n, n)$- geometrically reducible with $E_1$ and $E_2$ its elliptic components. We prove that there are only finitely many curves $\mathcal X$ (up to isomorphism) defined over $K$ such that $E_1$ and $E_2$ are $N$-isogenous for $n=2$ and $N=2,3, 5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X )\cong V_4$ or $n = 2$, $N = 3,5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X ) \cong D_4$. The same holds if $n=3$ and $N=5$. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above $\mbox{Jac} \mathcal X$ and show how such results in positive characteristic $p>2$ suggest nice applications in cryptography.

math.AG↗
arXiv2017

Reduction of binary forms via the hyperbolic center of mass

In this paper we provide an alternative reduction theory for real, binary forms with no real roots. Our approach is completely geometric, making use of the notion of hyperbolic center of mass in the upper half-plane. It appears that our model compares favorably with existing reduction theories, at least in certain aspects related to the field of definition. Various tools and features of hyperbolic geometry that are interesting in themselves, but also relevant for our and various other reduction theories papers (\cite{julia} and \cite{SC}), are also treated in detail and in a self-contained way here.

math.MG↗
arXiv2006

A mirror conjecture for projective bundles

We propose, motivate and give evidence for a relation between the $\mathcal D$-modules of the quantum cohomology of a smooth complex projective manifold $X$ and a projective bundle $\PP(\oplus L_i)$ over $X$.

math.AG↗
arXiv2006

Mirror symmetry and quantum cohomology of projective bundles

In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete intersection in a toric variety. We also use the conjecture to show that the relations of the small quantum cohomology ring of $X$ that come from differential operators lift to the projective bundle. The basic cohomology relation of the projective bundle deforms to a relation in the small quantum cohomology.

math.AG↗
arXiv2005

Mirror symmetry for concavex vector bundles on projective spaces

Let $X\subset Y$ be smooth, projective manifolds. Assume that $X$ is the zero locus of a generic section of a direct sum $V+$ of positive line bundles on $\PP^n$. Furthermore assume that the normal bundle $N_{X/Y}$ is a direct sum $V-$ of negative line bundles. We show that a $V:=V+\oplus V-$-twisted Gromov-Witten theory of $\PP^n$ restricts to the Gromov-Witten theory of $X$ inherited form $Y$. The later one can be computed via a Mirror Theorem which we prove in this paper.

math.AG↗
arXiv2005

Virtual Class of Zero Loci and Mirror Theorems

Let $Y$ be the zero loci of a regular section of a convex vector bundle $E$ over $X$. We provide a new proof of a conjecture of Cox, Katz and Lee for the virtual class of the genus zero moduli of stable maps to $Y$. This in turn yields the expected relationship between Gromov-Witten theories of $Y$ and $X$ which together with Mirror Theorems allows for the calculation of enumerative invariants of $Y$ inside of $X$.

math.AG↗