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One Level Density for Cubic Galois Number Fields

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  04 January 2019

Patrick Meisner
Patrick Meisner*
Affiliation:
Tel Aviv University, 6997801 Tel Aviv, Israel Email: meisner@mail.tau.ac.il
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Abstract

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Katz and Sarnak predicted that the one level density of the zeros of a family of L-functions would fall into one of five categories. In this paper, we show that the one level density for L-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.

Keywords

L-functionone level density

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 149 - 167
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 320755.

References

Bucur, A., Costa, E., David, C., Guerreiro, J., and Lowry-Duda, D., Traces, high powers and one level density for families of curves over finite fields . Math. Proc. Cambridge Philos. Soc. (2017). https://doi.org/10.1017/S030500411700041X.Google Scholar
Cassels, J. W. S. and Fröhlich, A., Algebraic number theory. London Mathematical Society, London, 1967.Google Scholar
Häberle, L., On cubic Galois field extensions . J. Number Theory 130(2010), no. 2, 307–317. https://doi.org/10.1016/j.jnt.2009.09.001.Google Scholar
Katz, N. and Sarnak, P., Zeroes of zeta functions and symmetry . Bull. Amer. Math. Soc. 36(1999), no. 1, 1–26. https://doi.org/10.1090/S0273-0979-99-00766-1.Google Scholar
Katz, N. and Sarnak, P., Random matrices, Frobenius eigenvalues, and monodromy. American Mathematical Society Colloquium Plublications, 45, American Mathematical Society, Providence, RI, 1999.Google Scholar
Koval’chik, F. B., Density theorems for sectors and progressions . Lithuanian Math. J. 15(1975), no. 4, 618–631.Google Scholar
Rudnick, Z., Traces of high powers of the Frobenius class in the hyperelliptic ensemble . Acta Arith. 143(2010), no. 1, 81–99. https://doi.org/10.4064/aa143-1-5.Google Scholar
Rudnick, Z. and Sarnak, P., Zeros of principal l-functions and random matrix theory . Duke Math. J. 81(1996), no. 2, 269–322. https://doi.org/10.1215/S0012-7094-96-08115-6.Google Scholar
Weil, A., Sur les courbes algébriques et les variétés qui s’ en déduisent. Actualités Sci. Ind., 1041, Hermann, Paris,, pp. 19–48.Google Scholar
Wright, D. J., Distribution of discriminants of abelian extensions . Proc. London Math. Soc. 3(1989), no. 1, 17–50. https://doi.org/10.1112/plms/s3-58.1.17.Google Scholar
Yang, A., Distribution problems associated to zeta functions and invariant theory. Ph.D. Thesis, Princeton University, 2009.Google Scholar