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arXiv · 1701.01986

Picard curves with small conductor

Michel Börner·Irene I. Bouw·Stefan Wewers
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Abstract

We study the conductor of Picard curves over $\mathbb{Q}$, 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 $\mathbb{Q}$ always have bad reduction at $p=3$, with $f_3\geq 4$. As an application we discuss the question of finding Picard curves with small conductor.

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Michel Börner, Irene I. Bouw, Stefan Wewers. 2017-01-08. Picard curves with small conductor. https://arxiv.org/abs/1701.01986

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