A new local-global principle for quadratic function fields (Q600737)

Let \(K\) be a number field, \([K:{\mathbb Q}]<\infty\) and let \(V^ K\) be the set of all valuations of \(K\). Let \(f(x)\in K[x]\) be a monic square free polynomial of even degree. Let \(D_ f:=K[x](\sqrt{f})\). A unit \(u\in D_ f\) is called trivial if \(u\in K^ \ast\). If \(D_ f\) has nontrivial units, then \(D_ f^ \ast=K^ \ast\times \langle u \rangle\) with \(\langle u\rangle\) an infinite cyclic group, and \(u\) is a fundamental unit. For \(v\in V^ K\) let \(k_ v\) be the residue field. We have that \(f(x)\) is integral at \(v\) for almost all \(v\in V^ K\). Let \(f_ v:=f\bmod v\in k_ v[x]\) and \(D_ {f_ v}=\{\alpha_ v+\beta_ v \sqrt{f_ v}\mid \alpha_ v, \beta_ v\in k_ v[x]\}\). We have that \(D_ {f_ v}\) contains nontrivial units and \(D^ \ast _{f_ v}=k_ v^ \ast\times \langle u_ v\rangle\) where \(u_ v\) is a fundamental unit. In [\textit{V. P. Platonov}, Dokl. Math. 81, No. 1, 55--57 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 430, No. 3, 318--320 (2010; Zbl 1217.11094)] the following local--global principle was proved by the first author using Jacobian varieties: \textbf{Theorem 1}. The ring \(D_ f\) has a nontrivial unit if and only if there exists a constant \(C\) such that \(\deg u_ v:=\deg \alpha _ v<C\) for almost all \(v\in V^ K\). The goal of this paper is to give a new proof of Theorem 1 without using Jacobian varieties. Theorem 1 is a consequence of the following result: Let \(|\cdot |_ \infty\) be an infinite valuation on \(K[x]\) and let \(\overline{K(x)}\) be the completion of \(K(x)\) with respect to \(|\cdot |_ \infty\). Given the Laurent expansion of \(\sqrt{f}\) in \(\overline{K(x)}\): \(\sqrt{f}=x^s+d_{s-1}x^ {s-1} +d_ {s-2}x^{s-2}+\cdots\), we consider the matrix \[ H_r=\left(\begin{matrix} d_ {-1} & d_ {-2}&\cdots&d_ {-r}\cr d_ {-2}&d_{-3}&\cdots&d_{-r-1}\cr \vdots&\vdots&\ddots&\vdots\cr d_{-r-s+2}&d_{-r-s+1}&\cdots&d_{-2r-s+3} \end{matrix} \right). \] Then: \textbf{Theorem 2}. The ring \(D_ f\) has a fundamental unit \(u=\alpha+\beta \sqrt{f}\), \(\alpha,\beta\in K[x]\) and \(\deg \beta=r\), if and only if the rank of \(H_ {r+1}\) is less than \(r+1\) and the rank of \(H_ m\) is \(m\) for \(m<r+1\). In this case \(\deg u:=\deg \alpha =r+s\). Finally, the authors give a complete answer to the question of for which polynomials \(f(x)=x^ 4+bx+c\), the ring \(D_ f\) has nontrivial units. Indeed they prove that \(D_ f\) has a fundamental unit of degree \(n\) for: (i) \(n=2, b=0, c\in {\mathbb Q}^ \ast, u=x^ 2+\sqrt{f}\); (ii) \(n=3, c=0, b\in {\mathbb Q}^ \ast, u=x^3+{{b}\over{2}}+ x\sqrt{f}\); (iii) \(n=4, b=t^ 3, c={{t^ 4}\over {2}}\) where \(t\in {\mathbb Q}^\ast, u=x^ 4-tx^ 3+{{t^ 2}\over {2}}x^ 2+ {{t^ 3}\over {2}}x-{{t^ 4}\over {4}}+\big(x^ 2-tx+{{t^ 2} \over {2}}\big) \sqrt{f}\).