Hilbert specialization results with local conditions (Q1683395)

Let \(k(T)\) be a rational function field with \(k\) a field of characteristic \(0\). A \textit{specialization} at \(t_0\in {\mathbb P}^1(k)\) of a finite extension \(E/k(T)\) of degree \(n\) is a \(k\)-étale algebra of degree \(n\), that is, a finite product \(\prod_i F_i/k\) of finite extensions of \(k\) such that \(\sum_i [F_i:k]=n\). This paper deals with the construction of specialization points such that the specialization of \(E/k(T)\) at \(t_0\) consists of a single degree \(n\) field extension \(E_{t_0}/k\) and the extension \(E_{t_0}/k\) has specific local behavior. The unramified case was approached in [\textit{P. Dèbes} and the author, Adv. Stud. Pure Math. 63, 141--162 (2012; Zbl 1321.11114)] and the ramified one in [the author, Isr. J. Math. 214, No. 2, 621--650 (2016; Zbl 1380.12004)] in the special case where the extension \(E/k(T)\) is \(k\)-regular, that is, when \(E/k(T)\) is a geometric extension. The main objective of the paper is two fold. First, it deals with the ramified case for arbitrary finite extensions and, second, it provides unifying results. In this way, it is proved that if \(k\) is Hilbertian and if \(\{t_1,\ldots, t_r\}\subseteq {\mathbb P}^1(\bar{k})\) is its branch point set, then for infinitely many points \(t_0\in {\mathbb P}^1(k)\setminus \{t_1,\ldots, t_r\}\), the specialization of \(E/k(T)\) at \(t_0\) consists of a single degree \(n\) field extension \(E_{t_0}/k\) and the set of ramification indices of \(E_{t_0}/k\) at each \({\mathcal P}\in{\mathcal S}\) is the set of all lengths of disjoint cycles involved in the decomposition of \(v(g_{i_{\mathcal P}}^{a_{\mathcal P}})\). Here \({\mathcal S}\) is certain finite set and \(g_{i_{\mathcal P}}^{a_{\mathcal P}}\) (\({\mathcal P}\in{\mathcal S}\)) generates the inertia group at \({\mathcal P}\). Sometimes the Hilbertianity assumption can be relaxed (Theorem 4.3). The non-Galois case is treated in Theorem 4.2 Section 5 provides a unifying result in the number field case (Theorem 5.1). The result for the special case \(k={\mathbb Q}\) is the following. Let \(E/{\mathbb Q}(T)\) be a finite extension of degree \(n\) with at least one \({\mathbb Q}\)-rational branch point and such that the Galois closure \(\hat{E}/{\mathbb Q}(T)\) is \({\mathbb Q}\)-regular. Then there exist three positive constants \(m_0,\alpha\) and \(\beta\) satisfying the following. Given two disjoint finite sets \({\mathcal S}_{ \mathrm{ur}}\) and \({\mathcal S}_{\mathrm{ra}}\) of prime numbers \(p\geq m_0\), there exist rational numbers \(t_0\) such that: (1) the specialization of \(E/{\mathbb Q}(T)\) at \(t_0\) consists of a single degree \(n\) extension \(E_{t_0}/{\mathbb Q}\) and \(\mathrm{ Gal}(\hat{E}_{t_0}/{\mathbb Q})=\mathrm{Gal}(\hat{E}/k(T))\); (2) no prime \(p\in {\mathcal S}_{\mathrm{ur}}\) ramifies in \(E_{t_0} /{\mathbb Q}\); (3) each prime number \(p\in {\mathcal S}_{\mathrm{ra}}\) ramifies in \(E_{t_0}/{\mathbb Q}\); (4) if \(d_{E_{t_0}}\) is the discriminant of \(E_{t_0}/{\mathbb Q}\), then \(\prod_{p\in {\mathcal S}_{\mathrm{ra}}} p \leq |d_{E_{t_0}}|\leq \alpha \prod_{p\in {\mathcal S}_{\mathrm{ur}} \cup {\mathcal S}_{\mathrm{ra}}} p^{\beta}\).