Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
cam.depositDate | 2022-05-25 | |
cam.restriction | thesis_access_open | |
cam.supervisor | Thorne, Jack | |
dc.contributor.author | Laga, Jef | |
dc.contributor.orcid | Laga, Jef [0000-0003-3950-6490] | |
dc.date.accessioned | 2022-05-26T10:52:33Z | |
dc.date.available | 2022-05-26T10:52:33Z | |
dc.date.submitted | 2021-11-30 | |
dc.date.updated | 2022-05-25T16:18:25Z | |
dc.description.abstract | A simply laced Dynkin diagram gives rise to a family of curves over 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 thesis, 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--Thelene and the Grothendieck--Serre conjecture, a study of geometric properties of compactified Jacobians using the Białynicki-Birula decomposition, and a general construction of integral orbit representatives. | |
dc.description.sponsorship | ERC (714405) | |
dc.identifier.doi | 10.17863/CAM.84919 | |
dc.identifier.uri | https://www.repository.cam.ac.uk/handle/1810/337504 | |
dc.language.iso | eng | |
dc.publisher.institution | University of Cambridge | |
dc.rights | All Rights Reserved | |
dc.rights.uri | https://www.rioxx.net/licenses/all-rights-reserved/ | |
dc.subject | Algebraic curves | |
dc.subject | Arithmetic statistics | |
dc.subject | Lie algebras | |
dc.subject | Rational points | |
dc.subject | Geometry of numbers | |
dc.title | Graded Lie Algebras, Compactified Jacobians and Arithmetic Statistics | |
dc.type | Thesis | |
dc.type.qualificationlevel | Doctoral | |
dc.type.qualificationname | Doctor of Philosophy (PhD) | |
pubs.funder-project-id | EPSRC (2114472) | |
pubs.licence-display-name | Apollo Repository Deposit Licence Agreement | |
pubs.licence-identifier | apollo-deposit-licence-2-1 | |
rioxxterms.licenseref.uri | https://www.rioxx.net/licenses/all-rights-reserved/ | |
rioxxterms.type | Thesis |
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