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Arithmetic of Pell surfaces. (English) Zbl 1211.14026

Let \(\Delta\in\mathbb{Z}\) be a fundamental discriminant, and let \(Q(X,Y)=x^2-mY^2\) or \(X^2+XY-mY^2\) be the corresponding principal form, so that \(\Delta=4m\) or \(4m+1\). The “Pell surface” of degree \(n\) is given by \(Q(X,Y)=Z^n\). Let \(S(n,\Delta)\) be the set of integer points with \((X,Y)=1\). It is then shown that there is a group structure on \(S(n,\Delta)\), and a surjective homomorphism to the \(n\)-torsion part of the narrow class group of \(\mathbb{Q}(\sqrt{\Delta})\). Y. Yamamoto [Osaka J. Math. 7, 57–76 (1970; Zbl 0222.12003)] has used points of \(S(n,\Delta)\) to construct real quadratic fields with class number divisible by \(n\), and the present paper interprets his procedure in terms of the group structure.

MSC:

14G05 Rational points
11R11 Quadratic extensions
11E16 General binary quadratic forms
11R29 Class numbers, class groups, discriminants
11D41 Higher degree equations; Fermat’s equation
14J25 Special surfaces

Citations:

Zbl 0222.12003
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