2020/03/06 by Eric Katz, Katz, Eric, Enis Kaya +1
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.2003.03400
Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions
openalex publication_date 2020/03/06 · arxiv created 2020/08/03 · arxiv updated 2020/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.