Rational points and arithmetic of fundamental groups. Evidence for the section conjecture
For an algebraic variety \(X\) over a field \(k\) the étale fundamental group \(\pi_1(X)\) is an extension of the Galois-group \(\text{Gal}(k)\) by the geometric fundamental group \(\pi_1(\overline X)\). Rational points in \(X(k)\) induce \(\pi_1(\overline X)\)-conjugacy classes of sections of this extension. The section conjecture (due to Grothendieck) states that for suitable \(X\) and \(k\) this induces a bijection. ``Suitable'' of course implies that \(\pi_1(\overline X)\) and \(\text{Gal}(k)\) are sufficiently big, for example \(X\) could be a hyperbolic curve and \(k\) a number field. Except in cases where there are no sections the truth of conjecture is unknown. In the present book the author gives an overview of mathematics around the conjecture. For example it is shown that many aspects of rational points generalise to sections.
scientific article; zbMATH DE number 6046477