Integral points on norm-form varieties
Let \(k_ j\), \(j=1,...,r\) be finite, totally complex Galois extensions of \({\mathbb{Q}}\) which are arithmetically independent, i.e. for every rational prime its ramification indices in \(k_ j\) and in \(k_ i\) are coprime whenever \(1\leq i<j\leq r\). Let \(f_ j(x_ j)\), \(x_ j=(x_{j1},...,x_{jn_ j})\) be full norm forms corresponding to ideal classes of the maximal orders in \(k_ j\), \(j=1,...,r\). Let \(V_ j(h)=\{a: f_ j(a)=h\), \(a\in {\mathbb{R}}^{n_ j}\}\), \(j=1,...,r\), \(h\in {\mathbb{R}}\), \(h\geq 0\) and \(V_ 0(h)=V_ 1(h)\times...\times V_ r(h)\). For \(a\in V_ j(h)\), \(j=1,...,r\) take \(\pi_ j(a)=0\), when \(h=0\), \(\pi_ j(a)=ah^{-1/n_ j}\), when \(h>0\), and \(\pi =\pi_ 1\times...\times \pi_ r\). Denote by N(U,X) the cardinality of the set \(\{\) \(a: a\in {\mathbb{Z}}^{n_ 1+...+n_ r}\), \(f_ 1(a_ 1)=...=f_ r(a_ r)<X\), \(\pi\) (a)\(\in U\}\), where \(a=(a_ 1,...,a_ r)\), \(a_ r\in {\mathbb{Z}}^{n_ j}\) for each j and \(U\subseteq V_ 0(1).\) If U is a measurable smooth subset with respect to a suitable defined positive Borel measure \(\mu\) on \(V_ 0(1)\) then it is proved that there exist two positive numbers b and \(\gamma\) depending only on the fields \(k_ j\), \(j=1,...,r\) and a constant C(U) depending on U and \(\mu\) such that \[ N(U,X)=b\mu (U)X+O(C(U)X^{1-\gamma}). \]
arithmetically independent Galois extensions
scientific article; zbMATH DE number 3961770