Vol. 2, 2019

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Computing zeta functions of cyclic covers in large characteristic

Vishal Arul, Alex J. Best, Edgar Costa, Richard Magner and Nicholas Triantafillou

Vol. 2 (2019), No. 1, 37–53
Abstract

We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic p that runs in time p12+o(1). We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gonçalves’s generalization of Kedlaya’s algorithm for cyclic covers, and Harvey’s work on Kedlaya’s algorithm for large characteristic.

Keywords
computational number theory, superelliptic, hyperelliptic, arithmetic geometry, Monsky–Washnitzer cohomology, zeta functions, p-adic
Mathematical Subject Classification 2010
Primary: 11G20
Secondary: 11M38, 11Y16, 14G10
Milestones
Received: 2 March 2018
Revised: 17 September 2018
Accepted: 18 September 2018
Published: 13 February 2019
Authors
Vishal Arul
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Alex J. Best
Department of Mathematics
Boston University
Boston, MA
United States
Edgar Costa
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States
Richard Magner
Department of Mathematics
Boston University
Boston, MA
United States
Nicholas Triantafillou
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States