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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
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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  • [1] John Brillhart and Patrick Morton, Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial, J. Number Theory 106 (2004), no. 1, 79-111. MR 2049594,
  • [2] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478,
  • [3] Michael R. Bush and Farshid Hajir, An irreducibility lemma, J. Ramanujan Math. Soc. 23 (2008), no. 1, 33-41. MR 2410519
  • [4] John Cullinan and Farshid Hajir, On the Galois groups of Legendre polynomials, Indag. Math. (N.S.) 25 (2014), no. 3, 534-552. MR 3188847,
  • [5] John Cullinan and Farshid Hajir, Primes of prescribed congruence class in short intervals, Integers 12 (2012), Paper No. A56, 4. MR 3083429
  • [6] G. Dumas, Sur quelques cas d'irréductibilité des polynomes à coefficients rationnels, J. de Math. Pures et Appl. 2 (1906), 191-258.
  • [7] Farshid Hajir, On the Galois group of generalized Laguerre polynomials, J. Théor. Nombres Bordeaux 17 (2005), no. 2, 517-525 (English, with English and French summaries). MR 2211305
  • [8] G. H. Hardy and J. E. Littlewood, Some problems of `Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), no. 1, 1-70. MR 1555183,
  • [9] M. Kaneko and D. Zagier, Supersingular $ j$-invariants, hypergeometric series, and Atkin's orthogonal polynomials, Computational perspectives on number theory (Chicago, IL, 1995) AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 97-126. MR 1486833
  • [10] Karl Mahlburg and Ken Ono, Arithmetic of certain hypergeometric modular forms, Acta Arith. 113 (2004), no. 1, 39-55. MR 2046967,
  • [11] PARI/GP, version 2.5.0, Bordeaux, 2011,
  • [12] Olivier Ramaré and Robert Rumely, Primes in arithmetic progressions, Math. Comp. 65 (1996), no. 213, 397-425. MR 1320898,
  • [13] Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
  • [14] Helmut Wielandt, Finite permutation groups, translated from the German by R. Bercov, Academic Press, New York-London, 1964. MR 0183775

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Additional Information

John Cullinan
Affiliation: Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504

Farshid Hajir
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01002

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

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