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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

ALINA BUCUR
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0112, La Jolla, CA 92093, U.S.A. e-mail: alina@math.ucsd.edu
EDGAR COSTA
Affiliation:
Department of Mathematics, Dartmouth College, 27 N Main Street, 6188 Kemeny Hall, Hanover, NH 03755-3551, U.S.A. e-mail: edgarcosta@math.dartmouth.edu
CHANTAL DAVID
Affiliation:
Concordia University, 1455 de Maisonneuve West, Montréal, Quebec, CanadaH3G 1M8. e-mail: cdavid@mathstat.concordia.ca
JOÃO GUERREIRO
Affiliation:
Max–Planck–Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany. e-mail: guerreiro@mpim-bonn.mpg.de
DAVID LOWRY–DUDA
Affiliation:
Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, RI 02912, U.S.A. e-mail: djlowry@math.brown.edu

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

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