Explicit Coleman integration in larger characteristic

Best, Alex J.

doi:10.2140/obs.2019.2.85

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…From knowledge of M ij , f i , and the action of φ on points R and S, one is able to solve for R S ω i . We note that Best [Bes19] has improved the complexity of the integration algorithms introduced in [BBK10].…”

Section: Hyperelliptic Curvesmentioning

confidence: 99%

p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

Kaya

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For curves over the field of p-adic numbers, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally, and Vologodsky integrals with desirable number-theoretic properties. These integrals have the advantage of being insensitive to the reduction type at p, but are known to coincide with Coleman integrals in the case of good reduction. Moreover, there are practical algorithms available to compute Coleman integrals.Berkovich-Coleman and Vologodsky integrals, however, differ in general. In this thesis, we give a formula for passing between them. To do so, we use combinatorial ideas informed by tropical geometry. We also introduce algorithms for computing Berkovich-Coleman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by basic wide open spaces, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the Berkovich-Coleman integrals into Vologodsky integrals using our formula.As an application, we provide an algorithm for computing Coleman-Gross p-adic heights on Jacobians of bad reduction hyperelliptic curves, whose definition relies on Vologodsky integration. This algorithm, for instance, can be used in the quadratic Chabauty method to find rational points on hyperelliptic curves of genus at least two.Finally, I owe an enormous debt of gratitude towards my family and friends. The times I have shared with them make my life worthwhile. Teşekkürler! Some of the work in this thesis was carried out during my two visits to the Ohio State University. I thank the mathematics cluster DIAMANT for partially supporting the visit.During my PhD, I have been partially supported by the Dutch Research Council (NWO) grant 613.009.124.

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Section: Hyperelliptic Curvesmentioning

confidence: 99%

p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

Kaya

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p-adic Integration on Bad Reduction Hyperelliptic Curves

Katz

Kaya

2020

International Mathematics Research Notices

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In this paper, we introduce an algorithm for computing $p$-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of $p$-adic integration: Berkovich–Coleman integrals, which can be performed locally, and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve with basic wide-open sets, we reduce the computation of Berkovich–Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya and to Balakrishnan and Besser for regular and meromorphic $1$-forms, respectively. We then employ tropical geometric techniques due to the 1st-named author with Rabinoff and Zureick-Brown to convert the Berkovich–Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish.

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Explicit Coleman integration in larger characteristic

Cited by 2 publications

References 18 publications

p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications

p-adic Integration on Bad Reduction Hyperelliptic Curves

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