Statistics for products of traces of high powers of the frobenius class of hyperelliptic curves

Article, Preprint English OPEN
Roditty-Gershon, Edva (2011)
  • Publisher: Elsevier BV
  • Journal: Journal of Number Theory, volume 132, issue 3, pages 467-484 (issn: 0022-314X)
  • Related identifiers: doi: 10.1016/j.jnt.2011.09.008
  • Subject: Algebra and Number Theory | Mathematics - Number Theory
    arxiv: Mathematics::Algebraic Geometry

We study the averages of products of traces of high powers of the Frobenius class of hyperelliptic curves of genus g over a fixed finite field. We show that for increasing genus g, the limiting expectation of these products equals to the expectation when the curve varies over the unitary symplectic group USp(2g). We also consider the scaling limit of linear statistics for eigenphases of the Frobenius class of hyperelliptic curves, and show that their first few moments are Gaussian.
  • References (9)

    [1] A. Bucur; C. David; B. Feigon and M. Lal´in, Statistics for traces of cyclic trigonal curves over finite fields, Int. Math. Res. Not. IMRN 2010, no. 5, 932-967.

    [2] P. Diaconis and S. N. Evans, Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), 2615-2633.

    [3] P. Diaconis and M. Shahshahani, On the eigenvalues of random matrices, Studies in Applied Probability, J. Appl. Probab. 31A (1994),49-62.

    [4] C. P. Hughes and Z. Rudnick, Mock-Gaussian behavior for linear statistics of classical compact groups, Journal of Physics A: Mathematical and General J. Phys. A: Math. Gen. 36 (2003), 2919-2932.

    [5] N. M. Katz and P. Sarnak , Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, RI, 1999.

    [6] P. Kurlberg and Z. Rudnick, The fluctuations in the number of points on a hyperelliptic curve over a finite field, J. Number Theory 129 (2009), 580-587.

    [7] M. Rosen, Number Theory in Function Fields, Grad. Texts in Math. 210, Springer, New York, 2002.

    [8] Z. Rudnick Traces of high powers of the Frobenius class in the hyperelliptic ensemble , Acta Arith. 143 (2010), 81-99

    [9] A. Weil, Sur les courbes alge´briques et les varie´te´s qui s'en de´duisent, Hermann, Paris, 1948. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail address:

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