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Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities

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Abstract

Recently, Ono et al. answered problems of Manin by defining zeta-polynomials \(Z_f(s)\) for even weight newforms \(f\in S_k(\varGamma _0(N)\); these polynomials can be defined by applying the “Rodriguez-Villegas transform” to the period polynomial of f. It is known that these zeta-polynomials satisfy a functional equation \(Z_f(s) = \pm \, Z_f(1-s)\) and they have a conjectural arithmetic-geometric interpretation. Here, we give analogous results for a slightly larger class of polynomials which are also defined using the Rodriguez–Villegas transform.

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References

  1. Brenti, F.: Hilbert polynomials in combinatorics. J. Algebraic Combin. 7(2), 127–156 (1998). https://doi.org/10.1023/A:1008656320759

    Article  MathSciNet  MATH  Google Scholar 

  2. Choie, Y., Park, Y.K., Zagier, D.B.: Periods of modular forms on \(\gamma _0(n)\) and products of jacobi theta functions. J. Eur. Math. Soc. 21(5), 1379–1420 (2019). https://doi.org/10.4171/JEMS/864

    Article  MathSciNet  MATH  Google Scholar 

  3. Conrey, J.B., Farmer, D.W., Imamoglu, O.: The nontrivial zeros of period polynomials of modular forms Lie on the unit circle. Int. Math. Res. Not. IMRN 20, 4758–4771 (2013). https://doi.org/10.1093/imrn/rns183

    Article  MathSciNet  MATH  Google Scholar 

  4. El-Guindy, A., Raji, W.: Unimodularity of zeros of period polynomials of Hecke eigenforms. Bull. Lond. Math. Soc. 46(3), 528–536 (2014). https://doi.org/10.1112/blms/bdu007

    Article  MathSciNet  MATH  Google Scholar 

  5. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/

  6. Jin, S., Ma, W., Ono, K., Soundararajan, K.: Riemann hypothesis for period polynomials of modular forms. Proc. Natl. Acad. Sci. USA 113(10), 2603–2608 (2016). https://doi.org/10.1073/pnas.1600569113

    Article  MathSciNet  MATH  Google Scholar 

  7. Kohnen, W., Zagier, D.: Modular forms with rational periods. In: Modular forms (Durham, 1983), Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., pp. 197–249. Horwood, Chichester (1984)

  8. Manin, Y.I.: Local zeta factors and geometries under \({\rm Spec}\,{\mathbf{Z}}\).Izv. Ross. Akad. Nauk Ser. Mat.80(4), 123–130 (2016). https://doi.org/10.4213/im8392. [Reprinted in Izv. Math. 80(4), 751–758 (2016)]

    Article  MathSciNet  Google Scholar 

  9. Ono, K., Rolen, L., Sprung, F.: Zeta-polynomials for modular form periods. Adv. Math. 306, 328–343 (2017). https://doi.org/10.1016/j.aim.2016.10.004

    Article  MathSciNet  MATH  Google Scholar 

  10. Rodriguez-Villegas, F.: On the zeros of certain polynomials. Proc. Am. Math. Soc. 130(8), 2251–2254 (2002). https://doi.org/10.1090/S0002-9939-02-06454-7

    Article  MathSciNet  MATH  Google Scholar 

  11. The PARI Group, Univ. Bordeaux: PARI/GP version 2.11.1 (2018). Available from http://pari.math.u-bordeaux.fr/

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Acknowledgements

The author thanks Nick Andersen, Maddie Locus Dawsey, Michael Griffin, Tim Huber, Larry Rolen, and Armin Straub for their helpful discussions and correspondence.

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Correspondence to Marie Jameson.

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Jameson, M. Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities. Res Math Sci 6, 27 (2019). https://doi.org/10.1007/s40687-019-0190-4

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