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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(Θ 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 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.
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Desingularization and p-Curvature of Recurrence Operators
Linear recurrence operators in characteristic p are classified by their p-curvature. For a recurrence operator L, denote by χ(L) the characteristic polynomial of its p-curvature. We can obtain information about the factorization of L by factoring χ(L). The main theorem of this paper gives an unexpected relation between χ(L) and the true singularities of L. An application is to speed up a fast algorithm for computing χ(L) by desingularizing L first. Another contribution of this paper is faster desingularization.
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- Award ID(s):
- 2007959
- PAR ID:
- 10385968
- Date Published:
- Journal Name:
- ISSAC'2022
- Page Range / eLocation ID:
- 111 to 118
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3476446.3535478
