Conductors and minimal discriminants of hyperelliptic curves: A\n comparison in the tame case

2019/10/17 by Padmavathi Srinivasan, Srinivasan, Padmavathi
Computer Science · Mathematics · #11G20 #14H25 #14J17 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1910.08228

openalex publication_date 2019/10/17 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Let C be a hyperelliptic curve of genus g over the fraction field K of\na discrete valuation ring R. Assume that the residue field k of R is\nperfect and that \char k > 2g+1. Let S = \Spec R. Let X\nbe the minimal proper regular model of C over S. Let \Art (C/K)\ndenote the Artin conductor of the S-scheme X and let \ν (\Δ_C)\ndenote the minimal discriminant of C. We prove that -\Art (C/K)\n\≤ \ν (\Δ_C). The key ingredients are a combinatorial refinement of the\ndiscriminant introduced in this paper (called the metric tree) and a recent\nrefinement of Abhyankar's inversion formula for studying plane curve\nsingularities. We also prove combinatorial restrictions for -\Art \n(C/K) = \ν (\Δ_C).\n

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