Integral points for multi-norm tori (Q2888912)

From the authors' introduction: The integral points on homogeneous spaces of semisimple and simply connected linear algebraic groups of non-compact type were studied by \textit{M. Borovoi} and \textit{Z. Rudnick} [Invent. Math. 111, 37--66 (1995; Zbl 0917.11025)] and by Colliot-Thélène and the second named author [Compos. Math. 145, 309--363 (2009; Zbl 1190.11036)] by using the strong approximation theorem and the Brauer-Manin obstruction. Recently, \textit{D. Harari} [Algebra Number Theory 2, No. 5, 595--611 (2008; Zbl 1194.14067)] showed that the Brauer-Manin obstruction accounts for the nonexistence of integral points. Colliot-Thélène noticed that a finite subgroup of the Brauer group is enough to account for the nonexistence of integral points by the compactness argument. However, this result is nonconstructive: one does not know which finite subgroup to use and cannot use it to determine the existence of integral points on the specific equations. In this paper, we give some explicit construction for such finite subgroups for multi-norm tori. NEWLINENEWLINEThe paper is also inspired by Colliot-Thélène's suggestion of studying Gauss'NEWLINEidea for determining integers represented by positive definite binary quadratic forms,NEWLINEwhich is beautifully explained by \textit{D. A. Cox} [Primes of the form \(x^2 +ny^2\). New York etc.: John Wiley (1989; Zbl 0701.11001)], from the point of view of Brauer-Manin obstruction. The advantage of using Brauer-Manin obstruction is to provide more perspective. For example, one can determine the solvability of the negative Pell equations by using the class field theory instead of the continued fractional method (the quadratic Diophantine approximation).NEWLINENEWLINEThe paper is organized as follows. In Section 1, we construct the idele groups whichNEWLINEare the so-called \(X\)-admissible groups for determining the integral points for some integral model \(X\). In Section 2, we interpret the \(X\)-admissible subgroup in terms of finite Brauer-Manin obstruction and also explain that there is no finite Brauer-Manin obstruction to detect all separated integral models of finite type. In Section 3, we apply our construction to study the classical binary quadratic Diophantine equations. Such classical quadratic Diophantine equations have been studied for a long time by various methods. This construction is the natural extension of Gauss' method for determining integers represented by a positive definite integral binary quadratic form and one can further determine all integers represented by a given binary inhomogeneous quadratic Diophantine equation. In Section 4, we provide several examples of 1-dimensional non-split tori where the splitting fields are imaginary quadratic fields. In Section 5, some examples of 1-dimensional non-split tori where the splitting fields are real quadratic fields are studied. In Section 6, we explain how to apply our construction to study the high dimensional multi-norm tori by providing some more explicit examples.