Saito-Kurokawa lifts, L-values for GL2, and congruences between Siegel modular forms.
dc.contributor.author | Brown, Jim L. | |
dc.contributor.advisor | Skinner, Christopher M. | |
dc.date.accessioned | 2016-08-30T15:53:50Z | |
dc.date.available | 2016-08-30T15:53:50Z | |
dc.date.issued | 2005 | |
dc.identifier.uri | http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3192587 | |
dc.identifier.uri | https://hdl.handle.net/2027.42/125307 | |
dc.description.abstract | Let <italic>f</italic> be a newform of weight 2<italic>k </italic> - 2 and level 1. There is a conjecture of Bloch and Kato that states that the special values of the <italic>L</italic>-function attached to <italic> f</italic> should be the size of the Selmer group attached to <italic>f</italic>. In this thesis we provide evidence for this conjecture by demonstrating an implication that if o | <italic>L</italic><sub> alg</sub>(<italic>k</italic>, <italic>f</italic>), then o must divide the order of the Selmer group as well where o is the uniformizer of a finite extension of <blkbd>Qp</blkbd> . The method we employ is the following. Let <italic>F<sub>f</sub></italic> be the Saito-Kurokawa lift of <italic>f</italic>. Given a prime <italic>p </italic> so that o | <italic>L</italic><sub>alg </sub>(<italic>k</italic>, <italic>f</italic>), we construct a cuspidal Siegel eigenform <italic>G</italic> that is not a CAP form so that Hecke eigenvalues of <italic>G</italic> are congruent to the Hecke eigenvalues of <italic>F<sub> f</sub></italic> modulo o. Associated to a Siegel eigenform we have a 4-dimensional Galois representation. Once we have such an eigenform <italic>G</italic>, we use the explicit nature of the Saito-Kurokawa correspondence to make deductions about the Galois representation associated to <italic>G</italic>. These deductions allow us to construct a non-zero element of <italic>p</italic>-power order in the Selmer group, thereby providing the divisibility we seek. | |
dc.format.extent | 122 p. | |
dc.language | English | |
dc.language.iso | EN | |
dc.subject | Congruences | |
dc.subject | Gl2 | |
dc.subject | L-values | |
dc.subject | Saito-kurokawa Lifts | |
dc.subject | Siegel Modular Forms | |
dc.title | Saito-Kurokawa lifts, L-values for GL2, and congruences between Siegel modular forms. | |
dc.type | Thesis | |
dc.description.thesisdegreename | PhD | en_US |
dc.description.thesisdegreediscipline | Mathematics | |
dc.description.thesisdegreediscipline | Pure Sciences | |
dc.description.thesisdegreegrantor | University of Michigan, Horace H. Rackham School of Graduate Studies | |
dc.description.bitstreamurl | http://deepblue.lib.umich.edu/bitstream/2027.42/125307/2/3192587.pdf | |
dc.owningcollname | Dissertations and Theses (Ph.D. and Master's) |
Files in this item
Remediation of Harmful Language
The University of Michigan Library aims to describe library materials in a way that respects the people and communities who create, use, and are represented in our collections. Report harmful or offensive language in catalog records, finding aids, or elsewhere in our collections anonymously through our metadata feedback form. More information at Remediation of Harmful Language.
Accessibility
If you are unable to use this file in its current format, please select the Contact Us link and we can modify it to make it more accessible to you.