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The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group…mehr

Produktbeschreibung
The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves.
This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by passing to a characteristic quotient of the fundamental group extension or its birational analogue.
Rezensionen
From the book reviews:
"The book under review, resulting from the author's dissertation ... is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendieck's section conjecture from the foundations to the current state of the art. ... It will be useful not only to specialists, as it is accessible to anyone familiar with the basics of modern algebraic geometry and the theory of algebraic fundamental groups." (Marco A. Garuti, Mathematical Reviews, May, 2014)