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Torsion points on elliptic curves over all quadratic fields. (English) Zbl 0599.14029
Let p be one of the primes 17, 19, 23, 29 or 31. Let X be the completion of the curve over \({\mathbb{Q}}\) that classifies isomorphism classes of elliptic curves together with a rational point of order p. It is proved that X has a nonhyperelliptic quotient whose jacobian has finite Mordell- Weil group over \({\mathbb{Q}}\). From this the author deduces that for any quadratic field K, there is no elliptic curve defined over K containing a K-rational point of order p.
[See also part II of this paper (Zbl 0599.14030 below).]
Reviewer: K.Lai

MSC:
14H45 Special algebraic curves and curves of low genus
14G05 Rational points
11R11 Quadratic extensions
14H25 Arithmetic ground fields for curves
14H52 Elliptic curves
14H40 Jacobians, Prym varieties
Citations:
Zbl 0599.14030
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