## The cubic Pell equation L-function

Equations of the form ππ₯Β³ + ππ¦Β³ = 1, where the constants π and π are integers of some number field such that ππ₯Β³ + ππ¦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 ππ₯Β³ β πππ¦Β³ = 1 where π and π are squarefree.

We define an πΏ-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 πΏ-functionβs 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,1βπ _π,whereππ =2π _π(2β2π _π)istheLaplaceeigenvalueofπ’π andππ 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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