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 one 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 ≥ 4. As an application we discuss the question of finding Picard curves with small conductor.
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Notes
- 1.
By k-linear action we mean that the action is compatible with the structure of \(\bar {Y}\) as a k-scheme.
- 2.
When working in a local context, \(f_{\mathfrak {p}}\) is often simply called the conductor of Y .
- 3.
The precise meaning of an effective proof is that it provides an explicitly computable bound on the height of the curve or abelian variety in question.
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Börner, M., Bouw, I.I., Wewers, S. (2017). Picard Curves with Small Conductor. In: Böckle, G., Decker, W., Malle, G. (eds) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-70566-8_4
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