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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$


Author: Andreas Schweizer
Journal: Trans. Amer. Math. Soc. 357 (2005), 1047-1059
MSC (2000): Primary 11G05, 14J27
Published electronically: May 10, 2004
MathSciNet review: 2110432
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Abstract: We study the extension generated by the $x$-coordinates of the $p^e$-torsion points of an elliptic curve over a function field of characteristic $p$. If $S\to C$ is a non-isotrivial elliptic surface in characteristic $p$ with a $p^e$-torsion section, then for $p^e>11$ our results imply restrictions on the genus, the gonality, and the $p$-rank of the base curve $C$, whereas for $p^e\le 11$ such a surface can be constructed over any base curve $C$. We also describe explicitly all occurring $p^e$ in the cases where the surface $S$ is rational or $K3$ or the base curve $C$ is rational, elliptic or hyperelliptic.


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Additional Information

Andreas Schweizer
Affiliation: Korea Institute for Advanced Study (KIAS), 207-43 Cheongnyangni 2-dong, Dong- daemun-gu, Seoul 130-722, Korea
Email: schweiz@kias.re.kr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03520-2
PII: S 0002-9947(04)03520-2
Keywords: Elliptic curve, non-isotrivial elliptic surface, $p$-primary torsion, uniform bound, Hasse invariant, Igusa curve, gonality, $K3$ surface
Received by editor(s): August 5, 2002
Received by editor(s) in revised form: August 25, 2003
Published electronically: May 10, 2004
Article copyright: © Copyright 2004 American Mathematical Society