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Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field

Published online by Cambridge University Press:  11 December 2009

Dmitry Faifman
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: [email protected])
Zeév Rudnick
Affiliation:
Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (email: [email protected])
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Abstract

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We study the fluctuations in the distribution of zeros of zeta functions of a family of hyperelliptic curves defined over a fixed finite field, in the limit of large genus. According to the Riemann hypothesis for curves, the zeros all lie on a circle. Their angles are uniformly distributed, so for a curve of genus g a fixed interval ℐ will contain asymptotically 2g∣ℐ∣ angles as the genus grows. We show that for the variance of number of angles in ℐ is asymptotically (2/π2)log (2g∣ℐ∣) and prove a central limit theorem: the normalized fluctuations are Gaussian. These results continue to hold for shrinking intervals as long as the expected number of angles 2g∣ℐ∣ tends to infinity.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Artin, E., Quadratische Körper in Geibiet der Höheren Kongruzzen I and II, Math. Z. 19 (1924), 153296.CrossRefGoogle Scholar
[2]Baker, T. H. and Forrester, P. J., Finite N fluctuation formulas for random matrices, J. Stat. Phys. 88 (1997), 13711385.CrossRefGoogle Scholar
[3]Costin, O. and Lebowitz, J., Gaussian fluctuation in random matrices, Phys. Rev. Lett. 75 (1995), 6972.CrossRefGoogle ScholarPubMed
[4]Diaconis, P. and Evans, S., Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), 26152633.CrossRefGoogle Scholar
[5]Faifman, D., Counting zeros of L-functions over the rational function field, MSc thesis, Tel Aviv University (2008).Google Scholar
[6]Hughes, C. P., Keating, J. P. and O’Connell, N., On the characteristic polynomial of a random unitary matrix, Comm. Math. Phys. 220 (2001), 429451.CrossRefGoogle Scholar
[7]Johansson, K., On random matrices from classical compact groups, Ann. of Math. (2) 145 (1997), 519545.CrossRefGoogle Scholar
[8]Katz, N. M. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy, American Mathematical Society Colloquium Publications, vol. 45 (American Mathematical Society, Providence, RI, 1999).Google Scholar
[9]Katz, N. M. and Sarnak, P., Zeroes of zeta functions and symmetry, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 126.CrossRefGoogle Scholar
[10]Keating, J. P. and Snaith, N., Random matrix theory and ζ(1/2+it), Comm. Math. Phys. 214 (2000), 5789.CrossRefGoogle Scholar
[11]Montgomery, H. L., Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
[12]Politzer, H. D., Random-matrix description of the distribution of mesoscopic conductance, Phys. Rev. B 40 (1989), 1191711919.CrossRefGoogle ScholarPubMed
[13]Rosen, M., Number theory in function fields, Graduate Texts in Mathematics, vol. 210 (Springer, New York, 2002).CrossRefGoogle Scholar
[14]Selberg, A., On the remainder in the formula for N(T), the number of zeros of ζ(s) in the strip 0<t<T, Avh. Nor. Vidensk. -Akad. Oslo I 1944 (1944), 127.Google Scholar
[15]Selberg, A., Contributions to the theory of Dirichlet’s L-functions, Skr. Nor. Vidensk. -Akad. Oslo I 1946 (1946), 162.Google Scholar
[16]Selberg, A., Contributions to the theory of the Riemann zeta-function, Arch. Math. Naturvid. 48 (1946), 89155.Google Scholar
[17]Soshnikov, A., The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), 13531370.CrossRefGoogle Scholar
[18]Weil, A., Sur les Courbes Algébriques et les Variétés qui s’en Déduisent (Hermann, Paris, 1948).Google Scholar
[19]Wieand, K., Eigenvalue distributions of random unitary matrices, Probab. Theory Related Fields 123 (2002), 202224.CrossRefGoogle Scholar