A local-global principle for norms from cyclic extensions of \(\mathbb{Q}(t)\) (a direct, constructive and quantitative approach) (Q1594956)

If \(L/K\) is a cyclic extension of number fields, then the well-known Hasse local-global principle establishes that an element \(a\in K^\ast\) is a norm from \(L^\ast\) iff for every prime divisor \({\mathcal P}\) in \(L\) and \({\wp}={\mathcal P}\cap K\), \(a\in K^\ast_{\wp}\) is a norm from \(L^\ast_{\mathcal P}\) where \(K_{\wp}\) and \(L^\ast_{\mathcal P}\) denote the completions of \(K\) at \(\wp\) and of \(L\) at \(\mathcal P\) respectively. Since Hasse's result holds for any cyclic extension of global fields, it is natural to study what happens in other cases. In the paper under review, the author investigates the case \(K=k(t)\) where \(k\) is a number field and \(L/K\) is a cyclic extension. Let \(k={\mathbb{Q}}\) be the field of rational numbers. Let \({\mathbb{P}}\) be a set of positive lower Dirichlet density, that is, \[ \liminf _{s\to 1^+}\Biggl(\sum _{p \in {\mathbb{P}}} p^{-s} /\log \biggl(\frac{1}{s-1}\biggr)\Biggr) > 0. \] Define \(A=\bigcup _{p\in{\mathbb{P}}} \{x\in{\mathbb{Q}}\mid v_p(x-r_p)=1\}\) where \(v_p\) is the \(p\)-adic valuation and \(r_p \in {\mathbb{Z}}\). We say that a subset of \({\mathbb{Q}}\) is rare if it is disjoint from a set \(A\). For a function \(f\in {\mathbb{Q}}(t)\), let \(N _f\) be the set of rational numbers \(s\) such that \(f(s)\) is of the form \(N(s,x_1,\ldots,x_d)\) where \[ N(t,X_1,\ldots,X_d) = N_{L/K}(X_1 w_1+ \ldots+ X_d w_d)\in {\mathbb{Q}}[t][X_1,\ldots,X_d] \] with \(w_1,\ldots, w_d\) being an integral basis for \(L\) over \({\mathbb{Q}}[t]\). Finally a subset \(T\subseteq {\mathbb{Q}}\) is called \textit{thin} if \(T\) is contained in a finite union of sets of type \(\varphi(X({\mathbb{Q}}))\) where \(X/{\mathbb{Q}}\) is a curve and \(\varphi\colon X\to {\mathbb{P}}^1\) is a rational map of degree \(\geq 2\). The main result of the paper is that if \(f\in{\mathbb{Q}}(t)^\ast\) is not a norm from \(L^\ast\) then \(N_f\) is contained in a union of a thin and a rare set. A corollary of this result is that if \(f\) is not a norm from \(L\), then the complement of \(N_f\) in \({\mathbb{Q}}\) contains an arithmetical progression. It is also obtained that if \(\Omega\) is a finite set of places of \(\mathbb{Q}\) such that, for \(v\not\in\Omega\), \(f\in {\mathbb{Q}}(t)\) is a norm from \({\mathbb{Q}}_v L\) to \({\mathbb{Q}}_v(t)\) and for all \(v\in \Omega\) there exist \(a_v, b_{i,v}\in{\mathbb{Q}}\) with \(f(a_v) = N(a_v,b_{1,v},\ldots,b_{d,v}) \in {\mathbb{Q}}_v^\ast\), then \(f\) necessarily is a norm from \(L\). The paper runs as follows. First some facts from cohomology are recalled. Next the theorem and its corollaries are proved. Then the author gives a simple counterexample when \(\text{ Gal}(L/K)\) is a noncyclic group of order \(4\). In the last section the author discusses how to find effectively a possible representation of \(f\) by \(N\). The paper presents a generalization of a result of \textit{H. Davenport, D. Lewis} and \textit{A. Schinzel} [Acta Arith. 11, 353-358 (1966; Zbl 0139.27101)]. The langauge used avoids any reference to Brauer groups and to Faddeev exact sequence in Galois cohomology. However, as the author remarks, the main ideas are in fact implicitly near to these concepts.