Picard curves with small conductor

2017/01/08 by Michel Börner, Irene I. Bouw, Börner, Michel +3
Arts and Humanities · Mathematics · Social Sciences · #11G30 #14H25 #14H45 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Number Theory (math.NT) #Vietnamese History and Culture Studies

paper · pdf · doi:10.48550/arxiv.1701.01986

openalex publication_date 2017/01/08 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We study the conductor of Picard curves over ℚ, which is a product of local factors. Our results are based on previous results on stable reduction of superelliptic curves that allow to compute the conductor exponent f_p at the primes p of bad reduction. A careful analysis of the possibilities of the stable reduction at p yields restrictions on the conductor exponent f_p. We prove that Picard curves over ℚ always have bad reduction at p=3, with f₃≥ 4. As an application we discuss the question of finding Picard curves with small conductor.

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