On Davenport-Stothers inequalities and elliptic surfaces in positive characteristic

Details On Davenport-Stothers inequalities and elliptic surfaces in positive characteristic On Davenport-Stothers inequalities and elliptic surfaces in positive characteristic

2006-08-16

arxiv.org/abs/math/0608427v3

Quarterly J. Math. 59 (2008), 499-522

Mathematics

Number Theory

24 pages, 1 table; v3: refereed version with minor changes; some typos corrected and section 2 extended

Scientific paper

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

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