The cubic Pell equation L-function

Equations of the form 𝑎𝑥³ + 𝑏𝑦³ = 1, where the constants 𝑎 and 𝑏 are integers of some number field such that 𝑎𝑥³ + 𝑏𝑦³ is irreducible, are a particularly significant class of cubic Thue equations that notably includes the cubic Pell equation. For a positive cubefree rational integer 𝑑, we consider the family of equations of the form 𝑚𝑥³ − 𝑑𝑛𝑦³ = 1 where 𝑚 and 𝑛 are squarefree. We define an 𝐿-function associated to 𝑑 whose nonvanishing coefficients correspond to the nontrivial solutions of those equations. That definition uses expressions related to the cubic theta function Q (􏰇√ 􏰈-), and we study that 𝐿-function’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 𝑢_𝑗 , 𝑗 = 1, 2, 3, · · · in the discrete spectrum, with a double pole at 𝑠 = ½ and possible simple poles at 𝑠=𝑠_𝑗,1−𝑠_𝑗,where𝜆𝑗 =2𝑠_𝑗(2−2𝑠_𝑗)istheLaplaceeigenvalueof𝑢𝑗 and𝜆𝑗 ≠1.