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On the class numbers of certain number fields obtained from points on elliptic curves III

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  • Mathematics

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Sato, A. Osaka J. Math. 48 (2011), 809–826 ON THE CLASS NUMBERS OF CERTAIN NUMBER FIELDS OBTAINED FROM POINTS ON ELLIPTIC CURVES III ATSUSHI SATO (Received October 6, 2009, revised March 23, 2010) Abstract We study the ramifications in the extensions of number fields arising from an isogeny of elliptic curves. In particular, we start with an elliptic curve with a ra- tional torsion point, and show that the extension is unramified if and “only if” the point which generates the extension is reduced into a nonsingular point (we need to assume certain conditions in order to prove the “only if” part). We also study a characterization of quadratic number fields with class numbers divisible by 5. 1. Introduction The ideal class groups of number fields have been studied for a long time. One studies the ideal class groups by using certain Diophantine equations, especially the arithmetic theory of elliptic curves. For example, T. Honda [2] (see also [3]) used el- liptic curves to construct infinitely many (real and imaginary) quadratic number fields with class numbers divisible by 3. He also studied a characterization of such number fields (cf. [5]). In [10] and [11] (see also [12]), the author gave a geometric interpreta- tion for Honda’s work, and introduced a way to construct, from an elliptic curve with a rational torsion point of order l 2 {3, 5, 7}, infinitely many quadratic number fields with class numbers divisible by l. Let k be a number field of finite degree, and let E be an elliptic curve defined over k which has a k-rational point T0 of prime order l. Then there exist an elliptic curve E� and an isogeny �W E ! E�, which are defined over k, such that Ker�D hT0i. Here hT0i denotes the subgroup of E(k) generated by T0. Such a pair (E�,�) is unique up to k-isomorphism, and E� is often denoted by E=hT0i. Taking certain equation for E and using Vélu’s formulas, the author studied, in [10] and [11], the ramification in the extension k(��1(Q))=k(Q) for a point Q on E� with X (Q) 2 k,

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