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.
Nous déterminons la densité des courbes à discriminant sans facteur carré dans certaines familles de courbes qui proviennent des représentations de Vinberg, en montrant que la densité globale est le produit des densités locales. Pour ce faire, nous travaillons dans le contexte des thèses de doctorat de Thorne et Laga et utilisons les techniques de comptage d’orbites de Bhargava. Cet article généralise un résultat précédent de Bhargava, Shankar et Wang.
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Keywords: Arithmetic statistics, geometry-of-numbers, squarefree sieve, Vinberg representations, graded Lie algebras
Martí Oller 1
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@article{JTNB_2025__37_3_989_0,
author = {Mart{\'\i} Oller},
title = {The density of {ADE} families of curves having squarefree discriminant},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {989--1029},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {3},
doi = {10.5802/jtnb.1349},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1349/}
}
TY - JOUR AU - Martí Oller TI - The density of ADE families of curves having squarefree discriminant JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 989 EP - 1029 VL - 37 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1349/ DO - 10.5802/jtnb.1349 LA - en ID - JTNB_2025__37_3_989_0 ER -
%0 Journal Article %A Martí Oller %T The density of ADE families of curves having squarefree discriminant %J Journal de théorie des nombres de Bordeaux %D 2025 %P 989-1029 %V 37 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1349/ %R 10.5802/jtnb.1349 %G en %F JTNB_2025__37_3_989_0
Martí Oller. The density of ADE families of curves having squarefree discriminant. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 989-1029. doi: 10.5802/jtnb.1349
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