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Stein, William A.

There are genus one curves over \({\mathbb Q}\) of every odd index. (English) Zbl 1002.11050

J. Reine Angew. Math. 547, 139-147 (2002).
Let \(K\) be a number field and let \(r\) be an integer not divisible by 8. In this paper the author proves that there are infinitely many curves \(X\) of genus one defined over \(K\) such that \(r\) is the smallest of all degrees of extensions of \(K\) over which \(X\) has a point. The proof of this result relies on the study of Kolyvagin’s Euler system of Heegner points combined with explicit computations on the modular curve \(X_0(17)\).
Reviewer: D.Poulakis (Thessaloniki)

Cited in 4 Documents

MSC:

11G05 Elliptic curves over global fields
14G05 Rational points
14H25 Arithmetic ground fields for curves

Keywords:

elliptic curve; Galois representation; Kolyvagin’s Euler system of Heegner points; modular curve

Software:

ecdata
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References:

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