Toric Elliptic Pairs with Picard Number Three
Aditya Khurmi

TL;DR
This paper classifies certain elliptic pairs on rational surfaces with Picard number three, showing that Lang-Trotter quadrilaterals do not exist and analyzing the structure of these pairs.
Contribution
It proves the non-existence of Lang-Trotter quadrilaterals among elliptic pairs with Picard number three and studies the Zariski decomposition of related divisors.
Findings
Lang-Trotter quadrilaterals do not exist in this setting.
Classification of elliptic pairs with Picard number three.
Analysis of the Zariski decomposition of $K_X+C$.
Abstract
An elliptic pair is a generalization of a rational elliptic fibration with fiber introduced in \cite{jenia_blowup}. Here, is a projective rational surface with log terminal singularities, and is an irreducible curve contained in the smooth locus of with and These naturally arise as blowups of projective toric surfaces, whose Newton polygon is elliptic. The order of in gives a quantitative way to check if is an elliptic fibration, which is equivalent to finiteness of the order. We call a Lang-Trotter polygon when this order is infinite, in which case is non-polyhedral. The paper \cite{lizzie} shows there are exactly elliptic triangles up to none of…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Differential Equations and Dynamical Systems · Coding theory and cryptography
