2019/05/31 by Lercier, Reynald, Elisa Lorenzo García, Christophe Ritzenthaler +2
Computer Science · Mathematics · #11G20 #14D05 #14H42 #14H45 #14H50 #14K25 #14Q05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.1906.00795
openalex publication_date 2019/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the existence of a special plane quartic model and in terms of the valuations of the Dixmier-Ohno invariants of C. The last one gives in particular an easy computable criterion for the reduction type. However, it does not produce a stable model, even in the case of good reduction. In this paper we give an algorithm to obtain (an approximation of) the stable model when the reduction of the latter is hyperelliptic and the characteristic of the residue field is not 2. This is based on a new criterion giving the reduction type in terms of the valuations of the theta constants of C. Some examples of the computation of these models are given.