1. Traces, high powers and one level density for families of curves over finite fields.
- Author
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BUCUR, ALINA, COSTA, EDGAR, DAVID, CHANTAL, GUERREIRO, JOÃO, and LOWRY–DUDA, DAVID
- Subjects
FINITE fields ,ZETA functions ,POLYNOMIALS ,EXPECTED returns ,MODULI theory - 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 Θ
C . We develop and present a new technique to compute the expected value of tr(ΘC 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 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. [ABSTRACT FROM AUTHOR]- Published
- 2018
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