Injectivity in the Section Conjecture
We recall the well known injectivity of the profinite Kummer map in the arithmetically relevant cases. There are at least two approaches towards injectivity. The abelian approach relies on the determination of the Kummer map for abelian varieties and thei
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Injectivity in the Section Conjecture
We recall the well known injectivity of the profinite Kummer map in the arithmetically relevant cases. There are at least two approaches towards injectivity. The abelian approach relies on the determination of the Kummer map for abelian varieties and their arithmetic, see Corollary 71, and also on the computation of the maximal abelian quotient extension 1ab .X=k/, see Proposition 69, which for later use in Sect. 13.5 we carefully revise also for smooth projective varieties of arbitrary dimension. The second approach is intrinsically anabelian and due to Mochizuki, see Theorem 76. In general, for a geometrically connected variety, the injectivity of the Kummer map (after arbitrary finite scalar extension) implies that the fundamental group must be large in the sense of Koll´ar, see Proposition 77, which imposes strong geometric constraints on possible higher dimensional anabelian varieties.
7.1 Injectivity via Arithmetic of Abelian Varieties This approach exploits essentially the cohomology of the Kummer sequence of an abelian variety. The crucial arithmetic input in the case of a number field as base field comes form the Mordell–Weil Theorem. Injectivity of the profinite Kummer map for smooth projective curves of genus 2 based on the Mordell–Weil Theorem was known to Grothendieck, see [Gr83] p. 4. The abelian weight 1 quotient. Let U=k be a smooth geometrically connected variety with a smooth proper completion X=k. Such a completion always exists in characteristic 0 by Nagata’s embedding theorem and Hironaka’s resolution of singularities, or at least if dim U 2. The birational invariance of the fundamental group [SGA1] X Corollary 3.4 shows that the quotient 1 .U / 1 .X / is independent of the choice of X . J. Stix, Rational Points and Arithmetic of Fundamental Groups, Lecture Notes in Mathematics 2054, DOI 10.1007/978-3-642-30674-7 7, © Springer-Verlag Berlin Heidelberg 2013
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7 Injectivity in the Section Conjecture
Definition 68. We define the weight 1 quotient of 1ab .U / as the quotient W1 1ab .U / D 1ab .X / and the weight 1 quotient extension of 1 .X=k/ as W1 1ab .U=k/ D 1ab .X=k/: The abelianized fundamental group. Let AlbU be the Albanese variety of U . The Albanese torsor map, see Wittenberg [Wi08], ˛ D ˛U W U ! Alb1U
(7.1)
factors over the inclusion U X into a proper smooth completion, and the resulting map ˛X W X ! Alb1U agrees with the Albanese torsor map of X , see [Mi86] Theorem 3.1. We recall the connection of the Albanese torsor map with W1 1ab .U /. We define the N´eron–Severi group scheme NSX over Spec.k/ of X=k by the exact sequence (7.2) 0 ! Pic0X red ! PicX ! NSX ! 0: N is a finitely generated abelian group, hence the torsion It is known, that NSX .k/ subgroup scheme NSX;tors , namely the preimage of the finite e´ tale torsion subgroupscheme of 0 .NSX / under the projection NSX ! 0 .NSX /; is a finite flat commutative group scheme over Spec.k/. We denote by G D the Cartier dual of a finite flat group scheme G, and
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