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On the size of the Shafarevich—Tate group of elliptic curves over function fields

Published online by Cambridge University Press:  04 December 2007

C. S. RAJAN
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai – 400 005, India. e-mail: rajan@math.tifr.res.in
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Abstract

Let ${E}$ be a nonconstant elliptic curve, over a global field ${K}$ of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of ${E}$, we show that the order of the Shafarevich-Tate group of ${E}$, is given by${O}({N}^{1/2+6\,\log(2)/\log({q})})$, where ${N}$ is the conductor of ${E}, {q}$ is the cardinality of the finite field of constants of ${K}$, and where the constant in the bound depends only on ${K}$. The method of proof is to work with the geometric analog of the Birch-Swinnerton Dyer conjecture for the corresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.

Type
Research Article
Copyright
© 1997 Kluwer Academic Publishers

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On the size of the Shafarevich—Tate group of elliptic curves over function fields
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