Work

p-adic integration on bad reduction hyperelliptic curves

Year: 2020

Type: article

Abstract: 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 by annuli and 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 on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-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.

Source: University of Groningen research database (University of Groningen / Centre for Information Technology)

DOI: 10.48550/arxiv.2003.03400

Authors: Katz, Eric, Kaya, Enis

Institutions: The Ohio State University, University of Groningen

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Ohio State University, Rijksuniversiteit Groningen, Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Awards: CAREER: Tropical and Diophantine Geometry, p-adic heights and p-adic BSD for modular abelian varieties with split purely toric reduction