On the arithmetic of simple singularities of type E

TL;DR

This paper investigates the distribution of integral points on algebraic curves associated with exceptional Dynkin diagrams E6, E7, and E8, revealing many locally solvable but globally insoluble cases.

Contribution

It applies arithmetic invariant theory to analyze integral points on non-hyperelliptic genus 3 or 4 curves linked to E-type Dynkin diagrams, highlighting their local-global failure.

Findings

A positive proportion of curves have points everywhere locally but not globally.

The curves studied are non-hyperelliptic of genus 3 or 4.

The study advances understanding of local-global principles for these algebraic curves.

Abstract

An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams $E_6, E_7$, $E_8$. These curves are non-hyperelliptic of genus 3 or 4. We prove that a positive proportion of each family consists of curves with integral points everywhere locally but no integral points globally.