Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials
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- by John Cullinan and Farshid Hajir PDF
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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.References
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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
- MR Author ID: 337116
- Email: hajir@math.umass.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2291-2304
- MSC (2010): Primary 11R32, 11R09, 33C45
- DOI: https://doi.org/10.1090/proc/13212
- MathSciNet review: 3626489