Arithmetic of superelliptic curves

• Applicant: Professor Dr. Ulf Kühn
• Subject Area: Mathematics
• Term: from 2011 to 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 203526262

• Final Report Year: 2017

Final Report Abstract

An overall aim of this project was to use superelliptic curves as a testing field for conjectures in arithmetic geometry. At first a mayor portion of this project has been devoted the study of arithmetic self intersection number ω^2-Ar of the relative dualizing sheaf of arithmetic surfaces. Jan-Steffen Müller and I were able to prove a general lower bound that depends only on data related to the bad reduction reduction. Moreover for particular superelliptic curves lower and upper bounds were calculated within the PhD thesis of Malte Moos, who was partially supported by this project. In the context of Beilinson conjectures it was proved by Vinzenz Busch in his PhD thesis that only for a small and computational inaccessible class of superelliptic curves enough elements in K2 can be obtained by the methods of Dejeu, Dokchitzer and Zagier. In the joint work with Müller some new ideas for finding integral elements of K2 of curves were introduced. Müller and I were able to derive consequences of the abc-conjecture to elliptic divisibility sequences. We weren’t able to find similar results for superelliptic curves.

Publications

• A height inequality for rational points on elliptic curves implied by the abc-conjecture. Functiones et Approximatio Commentarii Mathematici , 52, 127–132 (2015)
  Kühn Ulf; Müller Jan Steffen
  (See online at https://doi.org/10.7169/facm/2015.52.1.10)
• Abschätzungen von Arakelov Schnittzahlen für eine Familie von superelliptischen Kurven, Dissertation Universität Hamburg (2016)
  Moos Malte
  
• Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves. J. Reine Angew. Math. 720, 51-79 (2016)
  Jennifer S. Balakrishnan, Amnon Besser, J. Steffen Müller
  (See online at https://doi.org/10.1515/crelle-2014-0048)
• Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces. Trans. Amer. Math. Soc. 369, 1869– 1894 (2017)
  Kühn Ulf; Müller Jan Steffen
  (See online at https://doi.org/10.1090/tran/6787)