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dc.contributor.authorLaga, Jef
dc.date.accessioned2022-05-26T10:52:33Z
dc.date.available2022-05-26T10:52:33Z
dc.date.submitted2021-11-30
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/337504
dc.description.abstractA 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.sponsorshipERC (714405)
dc.rightsAll Rights Reserved
dc.rights.urihttps://www.rioxx.net/licenses/all-rights-reserved/
dc.subjectAlgebraic curves
dc.subjectArithmetic statistics
dc.subjectLie algebras
dc.subjectRational points
dc.subjectGeometry of numbers
dc.titleGraded Lie Algebras, Compactified Jacobians and Arithmetic Statistics
dc.typeThesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnameDoctor of Philosophy (PhD)
dc.publisher.institutionUniversity of Cambridge
dc.date.updated2022-05-25T16:18:25Z
dc.identifier.doi10.17863/CAM.84919
rioxxterms.licenseref.urihttps://www.rioxx.net/licenses/all-rights-reserved/
dc.contributor.orcidLaga, Jef [0000-0003-3950-6490]
rioxxterms.typeThesis
pubs.funder-project-idEPSRC (2114472)
cam.supervisorThorne, Jack
cam.depositDate2022-05-25
pubs.licence-identifierapollo-deposit-licence-2-1
pubs.licence-display-nameApollo Repository Deposit Licence Agreement


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