## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.