Vol. 4, No. 1, 2020

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Lifting low-gonal curves for use in Tuitman's algorithm

Wouter Castryck and Floris Vermeulen

Vol. 4 (2020), No. 1, 109–125
Abstract

Consider a smooth projective curve C̄ over a finite field 𝔽q, equipped with a simply branched morphism C̄ 1 of degree d 5. Assume char 𝔽q > 2 if d 4, and char 𝔽q > 3 if d = 5. In this paper we describe how to efficiently compute a lift of C̄ to characteristic zero, such that it can be fed as input to Tuitman’s algorithm for computing the Hasse–Weil zeta function of C̄𝔽q. Our method relies on the parametrizations of low rank rings due to Delone and Faddeev, and Bhargava.

Keywords
point counting, Tuitman's algorithm, Delone–Faddeev correspondence, Bhargava correspondence
Mathematical Subject Classification 2010
Primary: 11G20, 11Y99, 14Q05
Milestones
Received: 23 February 2020
Revised: 19 July 2020
Accepted: 29 August 2020
Published: 29 December 2020
Authors
Wouter Castryck
Department ESAT, imec-COSIC
KU Leuven
Leuven
Belgium
Department of Mathematics: Algebra and Geometry
Ghent University
Belgium
Floris Vermeulen
Department of Mathematics
KU Leuven
Leuven
Belgium