Vinberg Representations and 2-Descent on Jacobians of Curves
dc.contributor.author | Liang, Jason | |
dc.date.accessioned | 2022-09-06T16:03:07Z | |
dc.date.available | 2022-09-06T16:03:07Z | |
dc.date.issued | 2022 | |
dc.date.submitted | 2022 | |
dc.identifier.uri | https://hdl.handle.net/2027.42/174278 | |
dc.description.abstract | This dissertation applies Vinberg theory to the problem of constructing 2-descent maps on the Jacobians of hyperelliptic curves. In the first part, we construct, given a hyperelliptic curve C with certain marked points and tangent vectors, a smooth, complete surface S. Using the Picard group of S, we obtain a 2-graded simple adjoint Lie group H of Dynkin type An, where n depends on the genus of C and the nature of the marked points. Assuming that H is split, we show that there exists an injective 2-descent map from the Jacobian of C into the orbit spaces of the local and global Vinberg representations associated to H. The approach is based on previous work of Thorne which addresses the cases where H is of Dynkin type E6 or E7. In the second part, we construct, given a polynomial f(x) of odd degree n and satisfying certain properties, an explicit map from J(C) into the orbit space of the Dynkin representation (G,V) of type Dn, where C is the hyperelliptic curve with Weierstrass points at the roots of f(x). Our map extends a previous construction of Thorne which gives an explicit descent map for the degree 2 Vinberg representation of Dynkin type An. In both parts, we work over an arbitrary field K of characteristic 0. | |
dc.language.iso | en_US | |
dc.subject | representation theory | |
dc.subject | number theory | |
dc.subject | Vinberg theory | |
dc.title | Vinberg Representations and 2-Descent on Jacobians of Curves | |
dc.type | Thesis | |
dc.description.thesisdegreename | PhD | en_US |
dc.description.thesisdegreediscipline | Mathematics | |
dc.description.thesisdegreegrantor | University of Michigan, Horace H. Rackham School of Graduate Studies | |
dc.contributor.committeemember | Ho, Wei | |
dc.contributor.committeemember | Booth, Victoria | |
dc.contributor.committeemember | Liu, Yuan | |
dc.contributor.committeemember | Snowden, Andrew | |
dc.subject.hlbsecondlevel | Mathematics | |
dc.subject.hlbtoplevel | Science | |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/174278/1/liangji_1.pdf | |
dc.identifier.doi | https://dx.doi.org/10.7302/6009 | |
dc.identifier.orcid | 0000-0002-5920-8522 | |
dc.identifier.name-orcid | Liang, Jiayu; 0000-0002-5920-8522 | en_US |
dc.working.doi | 10.7302/6009 | en |
dc.owningcollname | Dissertations and Theses (Ph.D. and Master's) |
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