On the torsion of elliptic curves and elliptic surfaces in characteristic
Author:
Andreas Schweizer
Journal:
Trans. Amer. Math. Soc. 357 (2005), 10471059
MSC (2000):
Primary 11G05, 14J27
Published electronically:
May 10, 2004
MathSciNet review:
2110432
Fulltext PDF Free Access
Abstract 
References 
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Abstract: We study the extension generated by the coordinates of the torsion points of an elliptic curve over a function field of characteristic . If is a nonisotrivial elliptic surface in characteristic with a torsion section, then for our results imply restrictions on the genus, the gonality, and the rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.
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D. Cox and W. Parry, Torsion in elliptic curves over , Compositio Math. 41 (1980), 337354. MR 81k:14035
 [DoKe]
I. Dolgachev and J. Keum, Wild cyclic actions on surfaces, J. Algebraic Geom. 10 (2001), 101131. MR 2001i:14049
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D. Goldfeld and L. Szpiro, Bounds for the order of the TateShafarevich group, Compositio Math. 97 (1995), 7187. MR 97a:11102
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M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419450. MR 89k:11044
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J. Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96106. MR 39:1457
 [Ito1]
H. Ito, On unirationality of extremal elliptic surfaces, Math. Annalen 310 (1998), 717733. MR 99f:14045
 [Ito2]
H. Ito, On extremal elliptic surfaces in characteristic and , Hiroshima Math. J. 32 (2002), 179188. MR 2003g:14050
 [Ke]
J. Keum, Wild cyclic actions on smooth surfaces with , J. Algebra 244 (2001), 4558. MR 2002g:14060
 [La1]
W. Lang, Extremal rational elliptic surfaces in characteristic . I: Beauville surfaces, Math. Z. 207 (1991), 429438. MR 92f:14032
 [La2]
W. Lang, Extremal rational elliptic surfaces in characteristic . II: Surfaces with three or fewer singular fibres, Ark. Mat. 32 (1994), 423448. MR 96d:14034
 [Le]
M. Levin, On the group of rational points of elliptic curves over function fields, Amer. J. Math. 90 (1968), 456462. MR 37:6283
 [NgSa]
K. V. Nguyen and M.H. Saito, gonality of modular curves and bounding torsions, preprint arXiv:alggeom/9603024, 29Mar96.
 [OgSh]
K. Oguiso and T. Shioda, The MordellWeil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991), 8399. MR 92g:14036
 [Ro]
M. Rosen, Some remarks on the rank of an algebraic curve, Arch. Math. (Basel) 41 (1983), 143146. MR 85b:14030
 [Si1]
J. H. Silverman, The Arithmetic of Elliptic Curves, Springer GTM 106, BerlinHeidelbergNew York, 1986. MR 87g:11070
 [Si2]
J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer GTM 151, BerlinHeidelbergNew York, 1994. MR 96b:11074
 [Ta]
J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Functions of One Variable IV, Springer LNM 476, BerlinHeidelbergNew York, 1975, pp. 3352. MR 52:13850
 [Vo]
J.F. Voloch, Explicit descent for elliptic curves in characteristic , Compositio Math. 74 (1990), 247258. MR 91f:11042
 [CoPa]
 D. Cox and W. Parry, Torsion in elliptic curves over , Compositio Math. 41 (1980), 337354. MR 81k:14035
 [DoKe]
 I. Dolgachev and J. Keum, Wild cyclic actions on surfaces, J. Algebraic Geom. 10 (2001), 101131. MR 2001i:14049
 [GoSz]
 D. Goldfeld and L. Szpiro, Bounds for the order of the TateShafarevich group, Compositio Math. 97 (1995), 7187. MR 97a:11102
 [HiSi]
 M. Hindry and J. H. Silverman, The canonical height and integral points on elliptic curves, Invent. Math. 93 (1988), 419450. MR 89k:11044
 [Ig]
 J. Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96106. MR 39:1457
 [Ito1]
 H. Ito, On unirationality of extremal elliptic surfaces, Math. Annalen 310 (1998), 717733. MR 99f:14045
 [Ito2]
 H. Ito, On extremal elliptic surfaces in characteristic and , Hiroshima Math. J. 32 (2002), 179188. MR 2003g:14050
 [Ke]
 J. Keum, Wild cyclic actions on smooth surfaces with , J. Algebra 244 (2001), 4558. MR 2002g:14060
 [La1]
 W. Lang, Extremal rational elliptic surfaces in characteristic . I: Beauville surfaces, Math. Z. 207 (1991), 429438. MR 92f:14032
 [La2]
 W. Lang, Extremal rational elliptic surfaces in characteristic . II: Surfaces with three or fewer singular fibres, Ark. Mat. 32 (1994), 423448. MR 96d:14034
 [Le]
 M. Levin, On the group of rational points of elliptic curves over function fields, Amer. J. Math. 90 (1968), 456462. MR 37:6283
 [NgSa]
 K. V. Nguyen and M.H. Saito, gonality of modular curves and bounding torsions, preprint arXiv:alggeom/9603024, 29Mar96.
 [OgSh]
 K. Oguiso and T. Shioda, The MordellWeil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991), 8399. MR 92g:14036
 [Ro]
 M. Rosen, Some remarks on the rank of an algebraic curve, Arch. Math. (Basel) 41 (1983), 143146. MR 85b:14030
 [Si1]
 J. H. Silverman, The Arithmetic of Elliptic Curves, Springer GTM 106, BerlinHeidelbergNew York, 1986. MR 87g:11070
 [Si2]
 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer GTM 151, BerlinHeidelbergNew York, 1994. MR 96b:11074
 [Ta]
 J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular Functions of One Variable IV, Springer LNM 476, BerlinHeidelbergNew York, 1975, pp. 3352. MR 52:13850
 [Vo]
 J.F. Voloch, Explicit descent for elliptic curves in characteristic , Compositio Math. 74 (1990), 247258. MR 91f:11042
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Additional Information
Andreas Schweizer
Affiliation:
Korea Institute for Advanced Study (KIAS), 20743 Cheongnyangni 2dong, Dong daemungu, Seoul 130722, Korea
Email:
schweiz@kias.re.kr
DOI:
http://dx.doi.org/10.1090/S0002994704035202
PII:
S 00029947(04)035202
Keywords:
Elliptic curve,
nonisotrivial 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
