Families of curves in Vinberg representations

2025/08/13 by Laga, Jef, Romano, Beth · 1 citation
#11G30 (Primary) 17B08 #14H45 (Secondary) #17B70 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2508.09607

Abstract

Inspired by orbit parametrizations in arithmetic statistics, we explain how to construct families of curves associated to certain nilpotent elements in ℤ/mℤ-graded Lie algebras, generalizing work of Thorne to the m≥ 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 ℚ, using a ℤ/5ℤ-grading on a Lie algebra of type E₈.

Citations

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Cited by

• Hyperplane arrangements and Vinberg's θ-groups

Related

• Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
• The density of ADE families of curves having squarefree discriminant
• Series of nilpotent orbits
• Reduction theory for stably graded Lie algebras
• Geometry of orbit closures for the representations associated to gradings of Lie algebras of types E₇