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Metacommutation as a Group Action on the Projective Line Over $ \mathbb{F}_p$

Authors: Adam Forsyth, Jacob Gurev and Shakthi Shrima
Journal: Proc. Amer. Math. Soc. 144 (2016), 4583-4590
MSC (2010): Primary 11R52; Secondary 11R27
Published electronically: July 21, 2016
MathSciNet review: 3544510
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Abstract: Cohn and Kumar showed the quadratic character of $ q$ modulo $ p$ gives the sign of the permutation of Hurwitz primes of norm $ p$ induced by the Hurwitz primes of norm $ q$ under metacommutation. We demonstrate that these permutations are equivalent to those induced by the right standard action of $ \operatorname {PGL}_2 (\mathbb{F}_p)$ on $ \mathbb{P}^1 (\mathbb{F}_p)$. This equivalence provides simpler proofs of the results of Cohn and Kumar and characterizes the cycle structure of the aforementioned permutations. Our methods are general enough to extend to all orders over the quaternions with a division algorithm for primes of a given norm $ p$.

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Adam Forsyth
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Jacob Gurev
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Shakthi Shrima
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08540

Received by editor(s): March 18, 2015
Received by editor(s) in revised form: December 24, 2015
Published electronically: July 21, 2016
Communicated by: Ken Ono
Article copyright: © Copyright 2016 American Mathematical Society

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