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Homomorphisms from the group of rational points on elliptic curves to class groups of quadratic number fields. (English) Zbl 0811.14035

Let \(E: Y^ 2= X^ 3+ a_ 2 X^ 2+ a_ 4 X+ a_ 6\) be an elliptic curve defined over a field \(k\) (of characteristic \(\neq 2\)). The author exhibits a homomorphism from a subgroup of the \(k\)-rational points of \(E\), called the group of primitive points on \(E\), to the ideal class group of the order \(\mathbb{Z}+ \mathbb{Z} \sqrt{a_ 6}\). A key remark is that an integral point \((x,y)\) on \(E\) gives birth to the quadratic form \((x+n)S^ 2+ 2y ST+ (n^ 2+ (a_ 2+ x)n+ x^ 2+ a_ 2 x+a_ 4) T^ 2\) of discriminant \(D= 4(-n^ 3+ a_ 2n^ 2- a_ 4 n+ a_ 6)\). This looks like a promising direction.

MSC:

14H52 Elliptic curves
14G05 Rational points
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11G05 Elliptic curves over global fields
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