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Title: | A Birch and Swinnerton-Dyer formula for high-weight modular forms |

Authors: | Thackeray, Henry Robert |

Advisors: | Skinner, Christopher |

Contributors: | Mathematics Department |

Keywords: | Birch and Swinnerton-Dyer BSD Heegner cycle modular form number theory Shafarevich-Tate |

Subjects: | Mathematics |

Issue Date: | 2020 |

Publisher: | Princeton, NJ : Princeton University |

Abstract: | The Birch and Swinnerton-Dyer conjecture -- which is one of the seven million-dollar Clay Mathematics Institute Millennium Prize Problems -- and its generalizations to modular forms, motives, etc. are a significant focus of current number theory research. A 2017 article of Jetchev, Skinner and Wan proved a Birch and Swinnerton-Dyer formula at a prime $p$ for certain rational elliptic curves of rank 1. I generalize and adapt that article's arguments to prove an analogous formula for certain modular forms. For newforms $f$ of even weight higher than 2 with Galois representation $V_{f}$ containing a Galois-stable lattice $T_{f}$, let $W_{f} = V_{f}/T_{f}$ and let $K$ be an imaginary quadratic field in which the prime $p$ splits as $v_{0}\overline{v}_{0}$. My main result is that under some conditions, the $p$-index of the size of the Shafarevich-Tate group $\Sha(K,W_{f})$ is twice the $p$-index of a logarithm of the Abel-Jacobi map of a Heegner cycle defined by Bertolini, Darmon and Prasanna. Significant original adaptations I make to the 2017 arguments are (1) a generalized version of a previous calculation of the size of the cokernel of the localization-modulo-torsion map $H^{1}_{f}(K,T_{f}) \rightarrow H^{1}_{f}(K_{v_{0}},T_{f})/$tor, and (2) a comparison of different Heegner cycles. |

URI: | http://arks.princeton.edu/ark:/88435/dsp01d504rp250 |

Alternate format: | The Mudd Manuscript Library retains one bound copy of each dissertation. Search for these copies in the library's main catalog: catalog.princeton.edu |

Type of Material: | Academic dissertations (Ph.D.) |

Language: | en |

Appears in Collections: | Mathematics |

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Thackeray_princeton_0181D_13289.pdf | 660.31 kB | Adobe PDF | View/Download |

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