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Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials


Authors: John Cullinan and Farshid Hajir
Journal: Proc. Amer. Math. Soc. 145 (2017), 2291-2304
MSC (2010): Primary 11R32, 11R09, 33C45
DOI: https://doi.org/10.1090/proc/13212
Published electronically: February 15, 2017
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Abstract: The supersingular polynomial $ \mathfrak{S}_\ell (x) \in \mathbf {F}_\ell [x]$ has many well-
studied lifts to $ \mathbf {Q}[x]$. Among these is one introduced by Kaneko and Zagier, which, when interpreted as a specialized Jacobi polynomial, is seen to coincide with a lift discovered by Brillhart and Morton a few years later. The algebraic properties of this family of lifts of $ \mathfrak{S}_\ell (x)$ are not well-understood. We focus on a conjecture of Mahlburg and Ono regarding the maximality of their Galois groups (when shorn of their trivial linear factors) and also establish their irreducibility in some previously unknown cases.


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Additional Information

John Cullinan
Affiliation: Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504
Email: cullinan@bard.edu

Farshid Hajir
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01002
Email: hajir@math.umass.edu

DOI: https://doi.org/10.1090/proc/13212
Keywords: Kaneko-Zagier polynomials, Galois group, Hardy-Littlewood conjectures
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: March 21, 2016
Published electronically: February 15, 2017
Communicated by: Ken Ono
Article copyright: © Copyright 2017 American Mathematical Society