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Zbl 0731.14010
Kuwata, Masato
The field of definition of the Mordell-Weil group of an elliptic curve over a function field.
(English)
[J] Compos. Math. 76, No.3, 399-406 (1990). ISSN 0010-437X; ISSN 1570-5846/e

Let us consider a curve C defined over a perfect field K and an elliptic curve E defined over the function field K(C). Assume the j-invariant of E non constant. Let $\bar K$ be an algebraic closure of K. By the Mordell- Weil theorem, the group of $\bar K(C)$-rational points of E is a finitely generated abelian group. Under the hypothesis that $E(\bar K(C))\sb{tor}\simeq {\bbfZ}/m\sb 1{\bbfZ}\oplus {\bbfZ}/m\sb 2{\bbfZ}$, $m\sb 2\vert m\sb 1,\ m\sb 2\ne 1$ or the characteristic of $K,$ and that all these torsion elements are defined over K, which can be achieved extending the field K if necessary, the author proves that there is an explicitly computable integer $m>0$ and an explicitly computable finite extension L/K such that $mE(\bar K(C))=m(E(L(C))$. Then, if $E(L(C))$ can be computed, one can find $E(\bar K(C))$ itself. \par Let $\Delta =\sum n\sb iP\sb i$ be the discriminant of E, which is a divisor of C. Define $K(\Delta)$ as the smallest finite extension of K such that all these $P\sb i's$ are defined over $K(\Delta)$. Then $L=K((1/\ell)\Delta)$ is the smallest finite extension of $K(\Delta)$ such that all the $\ell$-roots of the $P\sb i's$ in the jacobian $J(C)$ are defined over L. The idea of the proof is to consider the Galois action of Gal$(\bar K/L)$ on $E(F)/\ell E(F)$, where $F=\bar K(C)$ and $\ell$ is a prime divisor of $m\sb 2$ different from the characteristic of K, and to review the proof of the weak Mordell-Weil theorem. \par A similar result is obtained when $m\sb 2=1$ and $m\sb 1>1$ is divisible for two different primes neither of them is equal to the characteristic of K. If E does not have torsion points at all, the author chooses a finite cover $C'\to C$ and a finite extension L/K such that $E(L(C'))$ has the necessary torsion points.
[P.Bayer (Barcelona)]
MSC 2000:
*14G05 Rationality questions, rational points
14H52 Elliptic curves
14G40 Arithmetic varieties and schemes
14A05 Relevant commutative algebra

Keywords: field of definition of the Mordell-Weil group; elliptic curve; Mordell- Weil theorem; rational points; jacobian; Galois action

Cited in: Zbl 0786.14010

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