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On the sparsity of positive-definite automorphic forms within a family

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Abstract

Baker and Montgomery have proved that almost all Fekete polynomials with respect to a certain ordering have at least one zero on the interval (0, 1). It is also known that a Fekete polynomial has no zeros on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. Generalizing their result, we formulate an axiomatic result about sets of automorphic forms π satisfying certain averages when suitably ordered to ensure that almost all p’s are not positive-definite within such sets. We then apply the result to various families, including the family of holomorphic cusp forms, the family of Hilbert class characters of imaginary quadratic fields, and the family of elliptic curves. In the appendix, we apply the result to general families of automorphic forms defined by Sarnak, Shin, and Templer.

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References

  1. J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, American Mathematical Society, Providence, RI, 2013.

    Book  MATH  Google Scholar 

  2. E. Bachmat, Mathematical Adventures in Performance Analysis, Birkhäuser/Springer, 2014.

    Book  MATH  Google Scholar 

  3. R C. Baker and H. L. Montgomery, Oscillations of quadratic L-functions, in Analytic Number Theory, Birkhäuser Boston, Boston, MA, 1990, pp. 23–40.

    Chapter  Google Scholar 

  4. N. Bergeron, J. Millson, and C. Moeglin, Hodge type theorems for arithmetic manifolds associated to orthogonal groups,arXiv:1110.3049v3[math. NT].

  5. B. J. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc., 43 (1968), 57–60.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math., 213(1965), 79–108.

    MathSciNet  MATH  Google Scholar 

  7. E. Bombieri, A variational approach to the explicit formula, Comm. Pure Appl. Math. 56 (2003), 1151–1164.

  8. A. Borel, Automorphic L-functions, in Automorphic forms, Representations and Lfunctions, American Mathematical Society, Providence, RI, 1979, pp. 27–61.

    Book  Google Scholar 

  9. C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc., 14 (2001), 843–939.

    Article  MathSciNet  MATH  Google Scholar 

  10. P. Cartier, Representations of p-adic groups: A survey, in Automorphic Forms, Representations and L-functions, American Mathematical Society, Providence, RI, 1979, pp. 111–155.

    Google Scholar 

  11. S. Chowla, Note on Dirichlet’s L-functions, Acta. Arith., 1 (1935), 113–114.

    MathSciNet  MATH  Google Scholar 

  12. R. Courant, Ein allgemeiner satz zur theorie der eigenfunktionen selbstadjungierter differentialausdr ücke, Nachr. Ges. Wiss. Göttingen (1923), 81–84.

    Google Scholar 

  13. B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials, Ann. Inst. Fourier (Grenoble), 50 (2000), 865–889.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Hamer, A formula for the traces of the Hecke operators on certain spaces of newforms, Arch. Math. (Basel) 70 (1998), 204–210.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Heilbronn, On real characters, Acta Arith., 2 (1937), 212–213.

    MATH  Google Scholar 

  16. H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms II, Amer. J. Math., 103 (1981), 777–815.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Karlin, Total Positivity, Vol. I, Stanford University Press, Stanford, CA, 1968.

    MATH  Google Scholar 

  18. S. Kato, Spherical functions and a q-analogue of Kostant’s weight multiplicity formula, Invent. Math., 66 (1982), 461–468.

    Article  MathSciNet  MATH  Google Scholar 

  19. N. M. Katz, Exponential sums over finite fields and differential equations over the complex numbers: some interactions, Bull. Amer. Math. Soc. (N. S.), 23 (1990), 269–309.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. A. Kenku, On the number of Q-isomorphism classes of elliptic curves in each Qisogeny class, J. Number Theory, 15 (1982), 199–202.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. H. Kim and F. Shahidi, Functorial products for GL2 × GL3 and functorial symmetric cube for GL2, C. R. Acad. Sci. Paris Sér. I Math., 331 (2000), 599–604.

    Article  MathSciNet  MATH  Google Scholar 

  22. N. N. Lebedev, Special Functions and their Applications, revised English edition, Prentice-Hall Inc., Englewood Cliffs, NJ, 1965.

    Google Scholar 

  23. W. Luo, Z. Rudnick, and P. Sarnak, On the generalized Ramanujan conjecture for GL(n), in Automorphic Forms, Automorphic Representations, and Arithmetic, Amer. Math. Soc., Providence, RI, 1999, pp. 301–310.

    Chapter  Google Scholar 

  24. G. Martin, Dimensions of the spaces of cusp forms and newforms on 0(N) and 1(N), J. Number Theory, 112 (2005), 298–331.

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Mertens, Ein beitrag zur analytischen zahlentheorie. Ueber die vertheilung der primzahlen, J. Reine Angew. Math., 78 (1874), 44–63.

    MathSciNet  Google Scholar 

  26. C. P. Mok, C. P. Mok, Endoscopic classification of representation of quasi-split arbitrary groups, Mem. Amer. Math. Soc., 235 (2015), no. 108.

    MathSciNet  MATH  Google Scholar 

  27. H. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical theory, Cambridge University Press, Cambridge, 2007.

    MATH  Google Scholar 

  28. G. Polya, Verschiedene Bemerkungen zur Zahlentheorie, Jahresber. Dtsch. Math. Ver., 28 (1919), 31–40.

    MATH  Google Scholar 

  29. P. Sarnak, Class numbers of indefinite binary quadratic forms. II, J. Number Theory, 21 (1985), 333–346.

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Sarnak, Letter to Eitan Bachmat on positive definite L-functions, publications.ias.edu/sarnak/paper/511.

  31. P. Sarnak, S. W. Shin, and N. Templier, Families of L-functions and their symmetry, Proceedings of the Simons Symposion on Families of Automorphic Forms and the Trace Formula, January 26 February 1, 2014.

    Google Scholar 

  32. C. L. Siegel, Uber die Classenzahl quadratischer Zahlkörper, Acta. Arith., 1 (1935), 83–86.

    MATH  Google Scholar 

  33. A. Selberg, On the Estimation of Fourier Coefficients of Modular Forms, American Mathematical Society, Providence, RI, 1965.

  34. J.-P. Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke Tp, J. Amer. Math. Soc. 10 (1997), 75–102.

    Article  MathSciNet  Google Scholar 

  35. S. W. Shin and N. Templier, Sato-Tate theorem for families and low-lying zeros of automorphic L-functions, Invent. Math., 203 (2016), 1–177.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Science, KAIST, Daejeon, 305-701, South Korea

    Junehyuk Jung

  2. School of Mathematics, Institute of Advanced Study, Princeton, NJ, 08540, USA

    Junehyuk Jung

  3. University of California, Berkeley 901 Evans Hall, Berkeley, CA, 94720, USA

    Sug Woo Shin

  4. Korea Institute for Advanced Study, 85 Hoegiro Dongdaemun-Gu, Seoul, 130-722, Republic of Korea

    Sug Woo Shin

Authors
  1. Junehyuk Jung
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  2. Sug Woo Shin
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Correspondence to Junehyuk Jung.

Additional information

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(No. 2013042157). The author was also partially supported by NSF grant DMS-1128155 and by TJ Park Post-doc Fellowship funded by POSCO TJ Park Foundation.

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Jung, J., Shin, S.W. On the sparsity of positive-definite automorphic forms within a family. JAMA 129, 105–138 (2016). https://doi.org/10.1007/s11854-016-0017-9

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  • DOI: https://doi.org/10.1007/s11854-016-0017-9

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