Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials

Cullinan, John; Hajir, Farshid

doi:10.1090/proc/13212

Algebraic properties of Kaneko-Zagier lifts of supersingular polynomials
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Proc. Amer. Math. Soc. 145 (2017), 2291-2304 Request permission

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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