Families of curves in Vinberg representations

TL;DR

This paper develops a Lie-theoretic framework to construct and classify families of algebraic curves linked to nilpotent elements in graded Lie algebras, extending previous work and providing new interpretations of orbit parametrizations.

Contribution

It generalizes Thorne's orbit parametrization approach to broader cases, including non-simply laced Lie algebras and higher moduli, and offers a Lie-theoretic proof for 5-Selmer element parametrization.

Findings

Classified families of curves from subregular nilpotents in stable gradings.

Connected orbit parametrizations to algebraic curves in existing literature.

Provided a Lie-theoretic proof for the 5-Selmer elements of elliptic curves.

Abstract

Inspired by orbit parametrizations in arithmetic statistics, we explain how to construct families of curves associated to certain nilpotent elements in $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras, generalizing work of Thorne to the $m\geq 3$ case and the non-simply laced case. We classify such families arising from subregular nilpotents in stable gradings and interpret almost all orbit parametrizations associated with algebraic curves appearing in the literature in this framework. As an extended example, we give a Lie-theoretic proof of the integral orbit parametrization of $5$-Selmer elements of elliptic curves over $\mathbb{Q}$, using a $\mathbb{Z}/5\mathbb{Z}$-grading on a Lie algebra of type $E_8$.