p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes

p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes

Let E be a CM elliptic curve over the rationals and p a good, ordinary prime. We show the implication: corank<sub>(Z<sub>p</sub>)</sub>Sel<sub>(p<sup>∞</sup>)</sub>(E/<sub>Q</sub>) = 1 ⟹ ord<sub>(s=1)</sub>L(s,E/<sub>Q<...

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Main Author: Yu, Qiyao (Vivian)
Format: Others
Language:en
Published: 2021
Online Access:https://thesis.library.caltech.edu/14225/3/Caltech_Senior_Thesis.pdf
Yu, Qiyao (Vivian) (2021) p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes. Senior thesis (Major), California Institute of Technology. doi:10.7907/ccbk-ht28. https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 <https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935>
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spelling ndltd-CALTECH-oai-thesis.library.caltech.edu-142252021-11-04T05:01:39Z https://thesis.library.caltech.edu/14225/ p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes Yu, Qiyao (Vivian) Let E be a CM elliptic curve over the rationals and p a good, ordinary prime. We show the implication: corank<sub>(Z<sub>p</sub>)</sub>Sel<sub>(p<sup>∞</sup>)</sub>(E/<sub>Q</sub>) = 1 ⟹ ord<sub>(s=1)</sub>L(s,E/<sub>Q</sub>) = 1 for the p<sup>∞</sup>-Selmer group Sel<sub>(p<sup>∞</sup>)</sub>(E/<sub>Q</sub>) and the complex L-function L(s,E/<sub>Q</sub>Q) in the case of p = 2, 3. 2021 Thesis NonPeerReviewed application/pdf en other https://thesis.library.caltech.edu/14225/3/Caltech_Senior_Thesis.pdf Yu, Qiyao (Vivian) (2021) p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes. Senior thesis (Major), California Institute of Technology. doi:10.7907/ccbk-ht28. https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 <https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935> https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 CaltechTHESIS:06022021-000654935 10.7907/ccbk-ht28
collection NDLTD
language en
format Others
sources NDLTD
description Let E be a CM elliptic curve over the rationals and p a good, ordinary prime. We show the implication: corank<sub>(Z<sub>p</sub>)</sub>Sel<sub>(p<sup>∞</sup>)</sub>(E/<sub>Q</sub>) = 1 ⟹ ord<sub>(s=1)</sub>L(s,E/<sub>Q</sub>) = 1 for the p<sup>∞</sup>-Selmer group Sel<sub>(p<sup>∞</sup>)</sub>(E/<sub>Q</sub>) and the complex L-function L(s,E/<sub>Q</sub>Q) in the case of p = 2, 3.
author Yu, Qiyao (Vivian)
spellingShingle Yu, Qiyao (Vivian)
p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
author_facet Yu, Qiyao (Vivian)
author_sort Yu, Qiyao (Vivian)
title p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
title_short p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
title_full p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
title_fullStr p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
title_full_unstemmed p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes
title_sort p-converse to a theorem of gross-zagier, kolyvagin, and rubin for small primes
publishDate 2021
url https://thesis.library.caltech.edu/14225/3/Caltech_Senior_Thesis.pdf
Yu, Qiyao (Vivian) (2021) p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes. Senior thesis (Major), California Institute of Technology. doi:10.7907/ccbk-ht28. https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935 <https://resolver.caltech.edu/CaltechTHESIS:06022021-000654935>
work_keys_str_mv AT yuqiyaovivian pconversetoatheoremofgrosszagierkolyvaginandrubinforsmallprimes
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