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`http://arks.princeton.edu/ark:/88435/dsp019p290c64f`

Title: | A Dual Description of Integral Binary Cubic Forms and the Ohno-Nakagawa Identities |

Authors: | Marinescu, Monica |

Advisors: | Bhargava, Manjul |

Contributors: | Skinner, Christopher |

Department: | Mathematics |

Class Year: | 2015 |

Abstract: | Given a nonzero integer D, we analyze the action of SL2(Z) on the space of integral binary cubic forms of discriminant D, and on the space of integer-matrix binary cubic forms of reduced discriminant D. Ohno conjectured that the ratio of SL2(Z)-equivalence classes is 1-to-1, when D is negative, and 1-to-3, when D is positive. The current proof, due to Nakagawa, offers an analytical perspective by studying these class numbers as coefficients in four Dirichlet series; using a result of Datskovsky and Wright, Nakagawa relates these series to certain products of Dedekind zeta functions of cubic fields and the Riemann zeta function. Our goal is to present a proof that does not characterize these class numbers in terms of the Dirichlet series, but only in terms of quadratic and cubic orders. In this respect, we construct a correspondence between the SL2(Z)-classes of integral binary cubic forms and isomorphism classes of oriented cubic rings of discriminant D, and a correspondences between SL2(Z)-classes of integer-matrix binary cubic forms and 3-torsion ideal classes in the unique quadratic order of discriminant D. |

Extent: | 45 pages |

URI: | http://arks.princeton.edu/ark:/88435/dsp019p290c64f |

Type of Material: | Princeton University Senior Theses |

Language: | en_US |

Appears in Collections: | Mathematics, 1934-2020 |

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PUTheses2015-Marinescu_Monica.pdf | 516.38 kB | Adobe PDF | Request a copy |

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