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Traces, high powers and one level density for families of curves over finite fields. (English) Zbl 1451.11128

Summary: The zeta function of a curve \(C\) over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix \(\Theta_C\). We develop and present a new technique to compute the expected value of \(\mathrm{tr}(\Theta_C^n)\) 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 Z. Rudnick [Acta Arith. 143, No. 1, 81–99 (2010; Zbl 1260.11057)] and I. J. Chinis [Res. Number Theory 2, Paper No. 13, 18 p. (2016; Zbl 1411.11058)]. 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 [A. Bucur et al., Int. Math. Res. Not. 2016, No. 14, 4297–4340 (2016; Zbl 1404.11088)] and [Y. Zhao, “On sieve methods for varieties over finite fields”, Preprint]. We extend [Bucur et al., loc. cit.] by describing explicit dependence on the place and give an explicit proof of the Lindelöf bound for function field Dirichlet \(L\)-functions \(L(1/2+it,\chi)\). As applications, we compute the one-level density for hyperelliptic curves, cyclic \(\ell\)-covers, and cubic non-Galois covers.

MSC:

11R59 Zeta functions and \(L\)-functions of function fields
11G20 Curves over finite and local fields
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