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Traces, high powers and one level density for families of curves over finite fields

Published online by Cambridge University Press:  31 July 2017

Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093, U.S.A. e-mail:
Department of Mathematics, Dartmouth College, 27 N Main Street, 6188 Kemeny Hall, Hanover, NH 03755-3551, U.S.A. e-mail:
Concordia University, 1455 de Maisonneuve West, Montréal, Quebec, CanadaH3G 1M8. e-mail:
Max–Planck–Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany. e-mail:
Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, RI 02912, U.S.A. e-mail:


The zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix ΘC. We develop and present a new technique to compute the expected value of tr(ΘCn) for various moduli spaces of curves of genus g over a fixed finite field in the limit as g is large, generalising and extending the work of Rudnick [Rud10] and Chinis [Chi16]. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given in [BDF+16] and [Zha]. We extend [BDF+16] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet L-functions L(1/2 + it, χ). As applications, we compute the one-level density for hyperelliptic curves, cyclic ℓ-covers, and cubic non-Galois covers.

Research Article
Copyright © Cambridge Philosophical Society 2017 

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[AT14] Altuğ, S. A. and Tsimerman, J. Metaplectic Ramanujan conjecture over function fields with applications to quadratic forms. Int. Math. Res. Not. IMRN 13 (2014), 34653558.Google Scholar
[BDF+16] Bucur, A., David, C., Feigon, B., Kaplan, N., Lalín, M., Ozman, E. and Wood, M. The distribution of 𝔽q-points on cyclic ℓ-covers of genus g. Int. Math. Res. Not. IMRN 14: (2016), 42974340.Google Scholar
[BF16] Bui, H. M. and Florea, A. Zeros of quadratic Dirichlet L-functions in the hyperelliptic ensemble. Preprint (2016). arXiv:1605.07092.Google Scholar
[BG01] Brock, B. W. and Granville, A. More points than expected on curves over finite field extensions. Finite Fields Appl. 7 (1): (2001), 7091. Dedicated to Professor Chao Ko on the occasion of his 90th birthday.Google Scholar
[CC11] Carneiro, E. and Chandee, V. Bounding ζ(s) in the critical strip. J. Number Theory 131 (3) (2011), 363384.Google Scholar
[Chi16] Chinis, I. J. Traces of high powers of the Frobenius class in the moduli space of hyperelliptic curves. Res. Number Theory 2:Art. 13, 18 (2016).Google Scholar
[CS11] Chandee, V. and Soundararajan, K. Bounding $\vert \zeta(\frac12+it)\vert $ on the Riemann hypothesis. Bull. Lond. Math. Soc. 43 (2) (2011), 243250.Google Scholar
[CV10] Carneiro, E. and Vaaler, J. D. Some extremal functions in Fourier analysis. II. Trans. Amer. Math. Soc. 362 (11) (2010), 58035843.Google Scholar
[DS94] Diaconis, P. and Shahshahani, M. On the eigenvalues of random matrices. J. Appl. Probab. 31A: (1994), 4962. Studies in applied probability.Google Scholar
[DW88] Datskovsky, B. and Wright, D. J. Density of discriminants of cubic extensions. J. Reine Angew. Math. 386 (1988), 116138.Google Scholar
[Flo16] Florea, A. The fourth moments of quadratic Dirichlet L-functions over function fields. Preprint (2016), arXiv:1609.01262.Google Scholar
[FPS16] Fiorilli, D., Parks, J. and Södergren, A. Low-lying zeros of quadratic Dirichlet L-functions: Lower order terms for extended support. Preprint (2016), arXiv:1601.06833.Google Scholar
[FR10] Faifman, D. and Rudnick, Z. Statistics of the zeros of zeta functions in families of hyperelliptic curves over a finite field. Compos. Math. 146 (1) (2010), 81101.Google Scholar
[Kat01] Katz, N. M. Frobenius-Schur indicator and the ubiquity of Brock-Granville quadratic excess. Finite Fields Appl. 7 (1) (2001), 4569. Dedicated to Professor Chao Ko on the occasion of his 90th birthday.Google Scholar
[KR09] Kurlberg, P. and Rudnick, Z. The fluctuations in the number of points on a hyperelliptic curve over a finite field. J. Number Theory 129 (3) (2009), 580587.Google Scholar
[KS99] 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
[Ros02] Rosen, M. Number theory in function fields, Graduate Texts in Mathematics. vol. 210 (Springer-Verlag, New York, 2002).Google Scholar
[Rud10] Rudnick, Z. Traces of high powers of the Frobenius class in the hyperelliptic ensemble. Acta Arith. 143 (1) (2010), 8199.Google Scholar
[TX14] Thorne, F. and Xiong, M. Distribution of zeta zeroes for cyclic trigonal curves over a finite field. Preprint (2014).Google Scholar
[Yan09] Yang, A. Distribution problems associated to zeta functions and invariant theory. PhD. Thesis Princeton University (2009).Google Scholar
[Zha] Zhao, Y. On sieve methods for varieties over finite fields. Preprint.Google Scholar