On the -torsion of elliptic curves and elliptic surfaces in characteristic

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 non-isotrivial 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.

**[CoPa]**David A. Cox and Walter R. Parry,*Torsion in elliptic curves over 𝑘(𝑡)*, Compositio Math.**41**(1980), no. 3, 337–354. MR**589086****[DoKe]**I. Dolgachev and J. Keum,*Wild 𝑝-cyclic actions on 𝐾3-surfaces*, J. Algebraic Geom.**10**(2001), no. 1, 101–131. MR**1795552****[GoSz]**Dorian Goldfeld and Lucien Szpiro,*Bounds for the order of the Tate-Shafarevich group*, Compositio Math.**97**(1995), no. 1-2, 71–87. Special issue in honour of Frans Oort. MR**1355118****[HiSi]**M. Hindry and J. H. Silverman,*The canonical height and integral points on elliptic curves*, Invent. Math.**93**(1988), no. 2, 419–450. MR**948108**, 10.1007/BF01394340**[Ig]**Jun-ichi Igusa,*On the algebraic theory of elliptic modular functions*, J. Math. Soc. Japan**20**(1968), 96–106. MR**0240103****[Ito1]**Hiroyuki Ito,*On unirationality of extremal elliptic surfaces*, Math. Ann.**310**(1998), no. 4, 717–733. MR**1619752**, 10.1007/s002080050168**[Ito2]**Hiroyuki Ito,*On extremal elliptic surfaces in characteristic 2 and 3*, Hiroshima Math. J.**32**(2002), no. 2, 179–188. MR**1925896****[Ke]**JongHae Keum,*Wild 𝑝-cyclic actions on smooth projective surfaces with 𝑝_{𝑔}=𝑞=0*, J. Algebra**244**(2001), no. 1, 45–58. MR**1856530**, 10.1006/jabr.2001.8789**[La1]**William E. Lang,*Extremal rational elliptic surfaces in characteristic 𝑝. I. Beauville surfaces*, Math. Z.**207**(1991), no. 3, 429–437. MR**1115175**, 10.1007/BF02571400**[La2]**William E. Lang,*Extremal rational elliptic surfaces in characteristic 𝑝. II. Surfaces with three or fewer singular fibres*, Ark. Mat.**32**(1994), no. 2, 423–448. MR**1318540**, 10.1007/BF02559579**[Le]**Martin Levin,*On the group of rational points on elliptic curves over function fields*, Amer. J. Math.**90**(1968), 456–462. MR**0230723****[NgSa]**K. V. Nguyen and M.-H. Saito,*-gonality of modular curves and bounding torsions*, preprint arXiv:alg-geom/9603024, 29Mar96.**[OgSh]**Keiji Oguiso and Tetsuji Shioda,*The Mordell-Weil lattice of a rational elliptic surface*, Comment. Math. Univ. St. Paul.**40**(1991), no. 1, 83–99. MR**1104782****[Ro]**Michael Rosen,*Some remarks on the 𝑝-rank of an algebraic curve*, Arch. Math. (Basel)**41**(1983), no. 2, 143–146. MR**719417**, 10.1007/BF01196870**[Si1]**Joseph H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210****[Si2]**Joseph H. Silverman,*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368****[Ta]**J. Tate,*Algorithm for determining the type of a singular fiber in an elliptic pencil*, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Springer, Berlin, 1975, pp. 33–52. Lecture Notes in Math., Vol. 476. MR**0393039****[Vo]**J. F. Voloch,*Explicit 𝑝-descent for elliptic curves in characteristic 𝑝*, Compositio Math.**74**(1990), no. 3, 247–258. MR**1055695**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
11G05,
14J27

Retrieve articles in all journals with MSC (2000): 11G05, 14J27

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

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