Trading degree for dimension in the section conjecture: the non-abelian Shapiro lemma. (English) Zbl 1190.14028

Stix, Jakob

Math. J. Okayama Univ. 52, 29-43 (2010).

Let \(X\) be a geometrically connected variety over a field \(K\). We have the exact sequence \[ 1 \to \pi_{1}(X\otimes K^{\mathrm{sep}}) \to \pi_{1}(X) \to \mathrm{Gal}(K^{\mathrm{sep}}/K) \to 1 \] and every \(K\)-rational point gives rise to a section of the last map. Actually, if \(K\) is a number field and \(X/K\) a geometrically connected smooth and projective curve of genus at least, Grothendieck conjectured that \(K\)-rational points are sent bijectively to \(\pi_{1}(X\otimes K^{\mathrm{sep}})\)-conjugacy classes of sections. We will denote this last set by \(S_{\pi_{1}(X/K)}\).
The aim of the paper is to provide evidence for the section conjecture by studying its behavior with regards to Weil restriction. Let \(L/K\) be a finite separable field extension and \(X/L\) be a projective variety (or quasi-projective in characteristic 0). The author proves that the map \(X(L) \to S_{\pi_{1}(X/L)}\) (defined as above) is a bijection if and only if the map \(R_{L/K} X(K) \to S_{\pi_{1}(R_{L/K}X/K)}\) is a bijection. From this result, the author deduces that Grothendieck’s section conjecture holds for smooth projective curves over number fields if it holds for certain smooth projective algebraic \(K(\pi,1)\) spaces over \(\mathbb{Q}\), hence the title of the paper.
The proof of the result goes through group theory and a study of non-abelian induction with a special emphasis on induction of extensions of groups (Shapiro’s lemma). Those results are used to identify the fundamental group \(\pi_{1}(R_{L/K} X/K)\) of the Weil restriction: under the same hypotheses as above, it is isomorphic to the non-abelian induction \(\mathrm{Ind}^{\mathrm{Gal}_{K}}_{\mathrm{Gal}_{L}} \pi_{1}(X/L)\).

Reviewer: Jérôme Poineau (Strasbourg)

MSC:

14H30	Coverings of curves, fundamental group
14G05	Rational points
20E22	Extensions, wreath products, and other compositions of groups

Keywords:

section conjecture; rational points; anabelian geometry; Shapiro’s lemma