“The ζ3-Pell Equation” Abstract
The negative Pell equation asks whether, for a squarefree integer D, do there exist integers x and y such that x2 − Dy2 = −1. This question is equivalent to whether there is a unit u ∈ Q(√D) with norm −1. It is easy to see that D must be positive and have no prime factors that are 3 mod 4. A question of arithmetic statistics is ”how many positive squarefree D with no prime factors that are 3 mod 4 have a solution to the negative Pell equation?” with the major result being due to Fouvry and Kluners showing that between 41.8% and 66.7%. In this talk, I will generalize both the problem and result of Fouvry and Kluners to the setting of cyclic cubic extensions of Q(ζ3), showing that between 63.8% and 75% of such fields that could have a unit whose norm is ζ3 actually do. This is joint work with Stanley Xiao.