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The Quarterly Journal of Mathematics Advance Access published online on February 6, 2008
The Quarterly Journal of Mathematics, doi:10.1093/qmath/ham048
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© The author 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ON DAVENPORT–STOTHERS INEQUALITIES AND ELLIPTIC SURFACES IN POSITIVE CHARACTERISTIC

Matthias Schütt{dagger}

Mathematics Department, Harvard University, Science Center, 1 Oxford Street, Cambridge, MA 02138, USA

Andreas Schweizer{ddagger}

Mathematical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF

{dagger} E-mail: mschuett{at}math.harvard.edu

{ddagger} E-mail: A.Schweizer{at}exeter.ac.uk

Received 20 April 2007; revised 2 October 2007
  Abstract

We show that the Davenport–Stothers inequality from characteristic 0 fails in any characteristic p>3. The proof uses elliptic surfaces over P1 and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti and Szpiro. For characteristics 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P1. Our ideas are also related to supersingular surfaces (in Shioda's sense).


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