On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$
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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.References
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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
- Received by editor(s): August 5, 2002
- Received by editor(s) in revised form: August 25, 2003
- Published electronically: May 10, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1047-1059
- MSC (2000): Primary 11G05, 14J27
- DOI: https://doi.org/10.1090/S0002-9947-04-03520-2
- MathSciNet review: 2110432