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## The cubic Pell equation L-function

Equations of the form ππ₯Β³ + ππ¦Β³ = 1, where the constants π and π are integers of some number field such that ππ₯Β³ + ππ¦Β³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer π, we consider the family of equations of the form ππ₯Β³ β πππ¦Β³ = 1 where π and π are squarefree.

We define an πΏ-function associated to π whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (τ°β τ°-), and we study that πΏ-functionβs analytic properties by using a method generalizing the approach used by Takhtajan and Vinogradov to derive a trace formula using the quadratic theta function for Q. We construct its meromorphic continuation and determine the locations and orders of its poles. Specifically, the poles occur at the eigenvalues of the Laplacian for the Maass forms π’_π , π = 1, 2, 3, Β· Β· Β· in the discrete spectrum, with a double pole at π = Β½ and possible simple poles at π =π _π,1βπ _π,whereππ =2π _π(2β2π _π)istheLaplaceeigenvalueofπ’π andππ β 1.

Identifer | oai:union.ndltd.org:columbia.edu/oai:academiccommons.columbia.edu:10.7916/w998-0e28 |

Date | January 2022 |

Creators | Hinkle, Gerhardt Nicholaus Farley |

Source Sets | Columbia University |

Language | English |

Detected Language | English |

Type | Theses |

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