Explicit Vologodsky integration for hyperelliptic curves

Kaya, Enis

doi:10.1090/mcom/3720

Explicit Vologodsky integration for hyperelliptic curves
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by Enis Kaya;

Math. Comp. 91 (2022), 2367-2396

DOI: https://doi.org/10.1090/mcom/3720

Published electronically: June 30, 2022

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Abstract:

Vologodsky’s theory of $p$-adic integration plays a central role in computing several interesting invariants in arithmetic geometry [Mosc. Math. J. 3 (2003), pp. 205–247, 260]. In contrast with the theory developed by Coleman [Invent. Math. 69 (1982), pp. 171–208; Duke Math. J. 52 (1985), pp. 765–770; Ann. of Math. (2) 121 (1985), pp. 111–168; Invent. Math. 93 (1988), pp. 239-266], it has the advantage of being insensitive to the reduction type at $p$. Building on recent work of Besser and Zerbes [Vologodsky integration on curves with semi-stable reduction, to appear in Israel J. Math], we describe an algorithm for computing Vologodsky integrals on bad reduction hyperelliptic curves. This extends previous joint work with Katz [Int. Math. Res. Not. IMRN 8 (2022), pp. 6038–6106] to all meromorphic differential forms. We illustrate our algorithm with numerical examples computed in Sage.

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