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Picard Curves with Small Conductor

In: Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

Author

Listed:
  • Michel Börner

    (Universität Ulm, Institut für Reine Mathematik)

  • Irene I. Bouw

    (Universität Ulm, Institut für Reine Mathematik)

  • Stefan Wewers

    (Universität Ulm, Institut für Reine Mathematik)

Abstract

We study the conductor of Picard curves over ℚ $$\mathbb {Q}$$ , 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 ℚ $$\mathbb {Q}$$ 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.

Suggested Citation

  • Michel Börner & Irene I. Bouw & Stefan Wewers, 2017. "Picard Curves with Small Conductor," Springer Books, in: Gebhard Böckle & Wolfram Decker & Gunter Malle (ed.), Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, pages 97-122, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-70566-8_4
    DOI: 10.1007/978-3-319-70566-8_4
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