Abstract
Szpiro and Tucker recently proved that, under mild conditions, the valuation of the minimal discriminant of an elliptic curve with semistable reduction over a discrete valuation ring can be expressed in terms of intersections between n-torsion and 2-torsion, where n tends to infinity. The argument of Szpiro and Tucker is geometric in nature. We give a proof based on the arithmetic of division polynomials, and generalize the result to the case of hyperelliptic curves.
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Acknowledgments
The research done for this paper was supported by VENI grant 639.033.402 from the Netherlands Organisation for Scientific Research (NWO). Part of the research was done at the Max Planck Institute in Bonn, whose hospitality is greatly acknowledged.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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de Jong, R. One half log discriminant and division polynomials. Arch. Math. 97, 251–257 (2011). https://doi.org/10.1007/s00013-011-0295-5
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DOI: https://doi.org/10.1007/s00013-011-0295-5