#### 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 ${p}^{1∕2+o\left(1\right)}$. 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