Homotopy sections and rational points on algebraic varieties

Details Homotopy sections and rational points on algebraic varieties Homotopy sections and rational points on algebraic varieties

2010-02-08

arxiv.org/abs/1002.1731v2

Mathematics

Number Theory

revised, but still preliminary version, new theorems added, 27 pages

Scientific paper

We study a generalisation of the anabelian section conjecture of Grothendieck by substituting the arithmetic fundamental group with the 'etale homotopy type. We show that the map associating homotopy sections to rational points factors though $R$-equivalence for projective varieties defied over fields of characteristic zero. We prove a homotopy version of the section conjecture for algebraic varieties defined over the real and complex number fields. We show that a natural homotopy version of the section and Shafarevich-Tate conjectures over $p$-adic and number fields are equivalent to their well-established analogues in the special case of curves and abelian varieties. We also prove this conjecture for Ch\^atalet surfaces and varieties over number fields for which the descent obstruction on torsors under linear algebraic groups explains the failure of the Hasse principle.

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