Existence of nongeometric pro-\(p\) Galois sections of hyperbolic curves
Let \(X\) be a complete hyperbolic curve over a number field \(k\) and let \(1\to \pi_1(U_{\bar k}) \to \pi_1(U) \to G_k \to 1\) be the fundamental exact sequence associated to the surjection of the etale fundamental group of \(U\) to the absolute Galois group \(G_k\) of \(k\) with kernel the geometric part of the fundamental group. Fix a prime \(p\) and consider the surjection to \(G_k\) from \(\pi_1^{(p)}(U)\), the pushforward quotient of \(\pi_1(U)\) obtained by taking the maximal pro-\(p\) quotient of \(\pi_1(U_{\bar k})\). In this paper, the author shows that in certain cases of \(X\), say, the Fermat curve over \(\mathbb{Q}(\mu_p)\) of degree \(p\) with \(p\) a regular prime, the natural mapping \(X(k)\to Sect(\pi_1^{(p)}(U)/G_k)\) from the set of \(k\)-rational points on \(X\) to the conjugacy classes of the homomorphic sections of the above surjection is not surjective, i.e., the pro-\(p\) analog of Grothendieck's section conjecture (bijectivity of the issued mapping) fails to be true. (The injectivity has been known to follow from \textit{S. Mochizuki} [Invent. Math. 138, No. 2, 319--423 (1999; Zbl 0935.14019)]). The main idea is, in the case when \(X\) is realized as the smooth compactification of \(p\)-Galois covers of \(\mathbb{P}^1-\{0,1,\infty\}\), to construct a cartesian diagram descending the surjection \(\pi_1^{(p)}(U)\to G_k\) to a surjection \(\pi_1'\to \mathrm{Gal}(L/k)\) so that \(\mathrm{Gal}(L/k)\) comes to be free pro-\(p\) as the Galois group of the maximal pro-\(p\) extension \(L\) over \(k\) unramified outside \(p\), using the key fact that the outer Galois representation in the pro-\(p\) fundamental group of a curve \(X\) has kernel containing that arises from \(\mathbb{P}^1-\{0,1,\infty\}\) which has been studied in depth by \textit{Y. Ihara} [Proc. Symp. Pure Math. 70, 247--273 (2002; Zbl 1065.14025)]. The freeness of \(\mathrm{Gal}(L/k)\) gives rise to infinitely many sections of \(\pi_1'\to \mathrm{Gal}(L/k)\) and those induced in \(Sect(\pi_1^{(p)}(U)/G_k)\), while \(X(k)\) is a finite set when the genus is large enough by the well known Faltings theorem.
scientific article; zbMATH DE number 5834707