03399nam a22004215i 4500001001800000003000900018005001700027007001500044008004100059020001800100024003500118050001400153072001700167072002300184082001500207100002600222245013700248264008200385300003300467336002600500337002600526338003600552347002400588490005300612505101100665520098901676650001702665650002502682650001902707650001702726650002402743650001902767710003402786773002002820776003602840830005302876856004802929978-3-642-30674-7DE-He21320150803155057.0cr nn 008mamaa121026s2013 gw | s |||| 0|eng d a97836423067477 a10.1007/978-3-642-30674-72doi 4aQA564-609 7aPBMW2bicssc 7aMAT0120102bisacsh04a516.352231 aStix, Jakob.eauthor.10aRational Points and Arithmetic of Fundamental Groupsh[electronic resource] :bEvidence for the Section Conjecture /cby Jakob Stix. 1aBerlin, Heidelberg :bSpringer Berlin Heidelberg :bImprint: Springer,c2013. aXX, 249 p.bonline resource. atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aLecture Notes in Mathematics,x0075-8434 ;v20540 aPart I Foundations of Sections -- 1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid -- 3 Basic Geometric Operations in Terms of Sections -- 4 The Space of Sections as a Topological Space -- 5 Evaluation of Units -- 6 Cycle Classes in Anabelian Geometry -- 7 Injectivity in the Section Conjecture -- Part II Basic Arithmetic of Sections -- 7 Injectivity in the Section Conjecture -- 8 Reduction of Sections -- 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers -- Part III On the Passage from Local to Global -- 10 Local Obstructions at a p-adic Place -- 11 Brauer-Manin and Descent Obstructions -- 12 Fragments of Non-abelian Tateâ€“Poitou Duality -- Part IV Analogues of the Section Conjecture -- 13 On the Section Conjecture for Torsors -- 14 Nilpotent Sections -- 15 Sections over Finite Fields -- 16 On the Section Conjecture over Local Fields -- 17 Fields of Cohomological Dimension 1 -- 18 Cuspidal Sections and Birational Analogues. aThe 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. 0aMathematics. 0aGeometry, algebraic. 0aNumber theory.14aMathematics.24aAlgebraic Geometry.24aNumber Theory.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9783642306730 0aLecture Notes in Mathematics,x0075-8434 ;v205440uhttp://dx.doi.org/10.1007/978-3-642-30674-7