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.


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


Zbl 0222.12003
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