On the class numbers of certain number fields obtained from points on elliptic curves. II
In [Osaka J. Math. 38, No. 4, 811--825 (2001; Zbl 1023.11058)], the author studied the family of quadratic number fields \(\mathbb Q(\sqrt{4\xi^3 - 27}\,)\) and showed that, for \(\xi \in \mathbb Q\), their class number is divisible by \(3\) with probability \(> \frac 34\). This theorem is extended in the present paper to other families of quadratic fields. To this end, the author considers an elliptic curve \(E\) defined over \(\mathbb Q\) with a rational point of prime order \(\ell\) (for \(\ell = 3, 5, 7\)). The existence of such a torsion point implies the existence of a rational \(p\)-isogeny \(\lambda: E \to E^*\), and using formulas obtained by \textit{J. Vélu} [C. R. Acad. Sci., Paris, Sér. A 273, 238--241 (1971; Zbl 0225.14014)] the author finds an explicit formula \[ y^2 + A_1 xy + A_3 y = x^3 + A_2 x^2 + A_4 x + A_6 \] for \(E^*\). Now fix a number field \(k\), define \[ F(X) = 4X^3 + (A_1^2 + 4A_2)X^2 + 2(A_1A_2 + 2A_4)X + A_3^2 + 4A_6, \] and set \(K_\xi = k(\sqrt{F(\xi)}\,)\) for \(\xi \in k\). For an affine point \(Q\) on \(E^*\) with \(x\)-coordinate \(\xi\) it turns out that \(K_\xi = k(Q)\), the field obtained by adjoining the \(x\)- and \(y\)-coordinates of \(Q\). The main result states that for all \(\xi \in k\) satisfying a couple of technical conditions, the field \(k(\lambda^{-1}(Q))\) is a cyclic unramified extension of \(k(Q)\) with degree \(\ell\). This is then used to find families of quadratic number fields whose class numbers are divisible by \(3\), \(5\) or \(7\) with high probability. Remark: The idea of using fields \(\mathbb Q(\lambda^{-1}(Q))\) in the case of \(\ell = 3\) goes back to an article by \textit{R. Bölling} [Math. Nachr. 73, 155--170 (1976; Zbl 0357.12005)].
scientific article; zbMATH DE number 5309632
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