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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.
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
Zbl 0599.14030
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