On the class numbers of certain number fields obtained from points on elliptic curves. II
type: Item
instance of: scholarly article
title: On the class numbers of certain number fields obtained from points on elliptic curves. II
zbMATH Open document ID: 1197.11148
author: Atsushi Sato
published in: Osaka Journal of Mathematics
publication date: 2008-08-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1216151104
review text: 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)].
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5309632
zbMATH Keywords: class groups
zbMATH Keywords: quadratic number fields
zbMATH Keywords: elliptic curves
zbMATH Keywords: isogenies
zbMATH Keywords: class field theory
MaRDI profile type: Publication
cites work: Q4486426
cites work: Q3847902
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cites work: On the class numbers of certain number fields obtained from points of elliptic curves
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cites work: The Hardy-Littlewood property of flag varieties
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description: scientific article; zbMATH DE number 5309632