dc.contributor.author Laga, Jef dc.date.accessioned 2022-05-26T10:52:33Z dc.date.available 2022-05-26T10:52:33Z dc.date.submitted 2021-11-30 dc.identifier.uri https://www.repository.cam.ac.uk/handle/1810/337504 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.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) dc.publisher.institution University of Cambridge dc.date.updated 2022-05-25T16:18:25Z dc.identifier.doi 10.17863/CAM.84919 rioxxterms.licenseref.uri https://www.rioxx.net/licenses/all-rights-reserved/ dc.contributor.orcid Laga, Jef [0000-0003-3950-6490] rioxxterms.type Thesis pubs.funder-project-id EPSRC (2114472) cam.supervisor Thorne, Jack cam.depositDate 2022-05-25 pubs.licence-identifier apollo-deposit-licence-2-1 pubs.licence-display-name Apollo Repository Deposit Licence Agreement
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