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Number Theory SeminarExport this event to calendar

Tuesday, February 26, 2019 — 12:30 PM EST

Erick Knight University of Toronto

“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.

MC 5417

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