On the birational section conjecture over finitely generated fields (Q2027528)

Let \(k\) be a field of characteristic zero and \(X\) a smooth geometrically connected, projective curve over \(k\). We denote by \(k(X)\) the function field of \(X\) and by \(\overline{k(X)}\) an algebraic closure of \(k(X)\). The absolute Galois group of \(X\) is defined to be the group \(G_X = \mathrm{Gal}(\overline{k(X)}/k(X))\). Let \(\bar{k}\) be the algebraic closure of \(k\) in \(\overline{k(X)}\), \(G_k = \mathrm{Gal}(\bar{k}/k)\) and \(G_{X_{\bar{k}}}= \mathrm{Gal}(\overline{k(X)}/k(X)\cdot \bar{k})\). Then we have the short exact sequence \(1 \rightarrow G_{X_{\bar{k}}} \rightarrow G_X \rightarrow G_k \rightarrow 1\). A splitting of this sequence is called a section of \(G_X\). Further, if \(x \in X(k)\), then we denote by \(\tilde{x}\) the valuation of \(\overline{k(X)}\) extending the valuation \(v_x\) of \(k(X)\) corresponding to \(x\) and by \(D_{\tilde{x}}\) the decomposition group of \(\tilde{x}\). We call that a section \(s\) of \(G_X\) geometric if its image \(s(G_k)\) is contained in a decomposition group \(D_{\tilde{x}}\) for some point \(x \in X(k)\) and some extension \(\tilde{x}\) of \(x\) to \(\overline{k(X)}\). In this case, we say that the section \(s\) arises from the point \(x\). The birational section conjecture (BSC) asserts that if \(X\) is a smooth, projective, geometrically connected curve over a field \(k\) which is finitely generated field over \(\mathbb{Q}\), then every section of \(G_X\) is geometric and arises from a unique \(k\)-rational point \(x\in X(k)\). This statement can be considered also for more general fields \(k\). The paper under review studies the birational section conjecture for curves over function fields of characteristic zero. It is proved that for a certain class of fields \(k\) of characteristic zero, and under the condition of finiteness of certain Shafarevich-Tate groups, the proof of BSC over function fields over \(k\) can be reduced to the proof of BSC over finite extensions of \(k\). Further, it is shown that for finitely generated extensions of \(\mathbb{Q}\), this statement is independent of finiteness of the Shafarevich-Tate groups. It follows that BSC holds over finitely generated fields over \(\mathbb{Q}\) if it holds over number fields.