The density of ADE families of curves having squarefree discriminant

2023/06/09 by Oller, Martí
#11G30 #11N35 #17B70 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.05961

Abstract

We determine the density of curves having squarefree discriminant in some families of curves that arise from Vinberg representations, showing that the global density is the product of the local densities. We do so using the framework of Thorne and Laga's PhD theses and Bhargava's orbit-counting techniques. This paper generalises a previous result by Bhargava, Shankar and Wang. As an application, we give an asymptotic for the number of reducible orbits of our representations.

Related

• Geometry-of-numbers over number fields and the density of ADE families of curves having squarefree discriminant
• Families of curves in Vinberg representations
• Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
• The geometric sieve and the density of squarefree values of invariant polynomials
• Local and global densities for Weierstrass models of elliptic curves