On Stoll's criterion for the maximality of quadratic arboreal Galois representations
Let \(f(x)=x^2+a\in\mathbb{Z}[x]\) and let \(f^n\) be the \(n\)-th iterate of \(f\). Assume that \(a\neq -2\) and \(-a\) is not a square in \(\mathbb Z\), then all iterates of \(f\) are irreducible over \(\mathbb Q\). Let \(T_n\) be the graph whose vertex set is \(\bigcup_{0\leq i\leq n}f^{-i}(0)\) and which has an edge between \(\alpha\) and \(\beta\) if \(f(\alpha)=\beta\). The graph \(T_n\) is a complete binary rooted tree on which the Galois group of \(f^n\) acts faithfully; the representation is said to be maximal if the Galois group is isomorphic to \(\mathrm{Aut}(T_n)\). Extending ideas of \textit{R. W. K. Odoni} [Mathematika 35, No. 1, 101--113 (1988; Zbl 0662.12010)], \textit{M. Stoll} [Arch. Math. 59, No. 3, 239--244 (1992; Zbl 0758.11045)] proved that an arboreal representation is maximal if there are no squares in a sequence of integers \(\beta_n=|b_n|\) and he could find some conditions for this to happen. \textit{M. Stoll} proved that the representation is maximal if \(a>0\) and \(a\equiv 1\pmod{4}\) or \(a\equiv 2\pmod{4}\) and if \(a<0\) and \(a\equiv 0\pmod{4}\) (and \(-a\) is not a square in \(\mathbb Q\)). The main results in the present paper are proved combining the techniques of \textit{R. W. K. Odoni} and \textit{M. Stoll} with a new criterion for \(\beta_n\) not to be a square. More concretely the author proves that the representation is maximal if \(f(x)=x^2+a\) and \(a=-(8k+2)(8k+3)\) or \(a=-(4k+1)(4k+2)-1\) with \(k\in\mathbb N\) (including \(k=0\)).
scientific article; zbMATH DE number 7374889
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