EAN: 9783642306730

RATIONAL POINTS AND ARITHMETIC OF FUNDAMENTAL G...

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 using a characteristic quotient or its
birational analogue in lieu of the fundamental group extension.

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