Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
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arXiv
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| Natura: | Preprint |
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2022
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| _version_ | 1866914849438564352 |
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| author | Laga, Jef |
| author_facet | Laga, Jef |
| contents | A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Bialynicki-Birula decomposition, and a general construction of integral orbit representatives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_02048 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics Laga, Jef Number Theory Algebraic Geometry Representation Theory 11G30, 11E72, 17B70 A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a strong link between the arithmetic of these curves and the rational orbits of these representations. In this paper, we complete Thorne's picture and show that $2$-Selmer elements of the Jacobians of the smooth curves in each family can be parametrised by integral orbits of the corresponding representation. Using geometry-of-numbers techniques, we deduce statistical results on the arithmetic of these curves. We prove these results in a uniform manner. This recovers and generalises results of Bhargava, Gross, Ho, Shankar, Shankar and Wang. The main innovations are: an analysis of torsors on affine spaces using results of Colliot-Thélène and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Bialynicki-Birula decomposition, and a general construction of integral orbit representatives. |
| title | Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics |
| topic | Number Theory Algebraic Geometry Representation Theory 11G30, 11E72, 17B70 |
| url | https://arxiv.org/abs/2204.02048 |