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2021 | OriginalPaper | Buchkapitel

Square Root Time Coleman Integration on Superelliptic Curves

verfasst von : Alex J. Best

Erschienen in: Arithmetic Geometry, Number Theory, and Computation

Verlag: Springer International Publishing

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Abstract

Since Kedlaya first introduced a p-adic algorithm for computing zeta functions of hyperelliptic curves, many related algorithms for computing both zeta functions and Coleman integrals on various classes of algebraic curves have been studied. These algorithms compute in the Monsky-Washnitzer cohomology or the rigid cohomology of the curve to determine the action of Frobenius on this cohomology.
We give a new algorithm for explicitly computing Coleman integrals on superelliptic curves over unramified extensions of p-adic fields. The runtime is softly linear with respect to the square root of the size of the residue field, bringing the runtime in line with that of the corresponding zeta function algorithms. We also describe the implementation of this algorithm in Nemo, a new package for the Julia programming language, which adds functionality for computational number theory. We compare Nemo with other systems in use in this area.
Vorheriges Kapitel Computing Classical Modular Forms for Arbitrary Congruence Subgroups
Nächstes Kapitel Computing Classical Modular Forms
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Metadaten
Titel
Square Root Time Coleman Integration on Superelliptic Curves
verfasst von
Alex J. Best
Copyright-Jahr
2021
Buch
Arithmetic Geometry, Number Theory, and Computation

Print ISBN: 978-3-030-80913-3

Electronic ISBN: 978-3-030-80914-0

Copyright-Jahr: 2021

DOI
https://doi.org/10.1007/978-3-030-80914-0_3

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