The motivic anabelian geometry of local heights on abelian varieties (Q6640524)

``The author studies the problem of describing local components of Néron-Tate height functions on abelian varieties over characteristic \(0\) local fields as functions on spaces of torsors under various realisations of the \(2\)-step unipotent motivic fundamental group of the \(\mathbf G_m\)-torsor corresponding to the defining line bundle. He presents three main theorems (Theorems 1.6, 1.12, and 1.8) giving such a description in terms of the \(\mathbf Q\) - and \(\mathbf Q_p\)-pro-unipotent étale realisations when the base field is \(p\)-adic, and in terms of the \(\mathbf R\)-pro-unipotent Betti-de Rham realization when the base field is archimedean.''\N\N``The proofs of Theorems 1.6 and 1.8 proceed by showing that the Néron log-metrics are uniquely characterized by a certain list of properties, and then verifying these properties for the maps described in the statements of the respective theorems.''\N\N``In the course of proving the \(p\)-adic instance of these theorems, the author develops a new technique for studying local nonabelian Bloch-Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable him to construct a pro-unipotent generalization of the Bloch-Kato exponential sequence under minimal assumptions.''\N\N\N\NThe book consists of eight Chapters.\N\NIn Chapter 1, the author gives a general introduction, including discussion on Grothendieck's section conjecture, on a conditional anabelian proof of the Siegel-Faltings theorem on the finiteness of the set of \(S\)-integral points of hyperbolic curves over a number field. Also, the main theorems are formulated, and methods of proofs are sketched.\N\NAfter recalling some basic facts about pro-unipotent groups and filtrations in Chapter 2, he proves in Chapter 3 the \(l\)-adic version of the main theorem, Theorem 1.6. The proof is relatively simple, but the details will serve to motivate many of the lemmas necessary in the proof of its much harder cousin, Theorem 1.12. The following chapters serve to build up to the proof of this latter theorem, developing first the basic language of local Galois representations on finitely generated pro-unipotent groups in Chapter 4, then studying the local Bloch-Kato Selmer sets and quotients in Chapter 6 using the homotopical-algebraic language reviewed in Chapter 5.\N\NThis theoretical basis being established, the proof in Chapter 7 of the \(p\)-adic main theorem then follows exactly the blueprint of the adic proof, and indeed is the shortest section of the paper. Then, in Chapter 8 he proves the archimedean version of the main theorem, developing a mixed-Hodge-theoretic analogue of the representation-theoretic language of the rest of the paper, and relating the Archimedean nonabelian Kummer map to the higher Albanese maps of \textit{R. M. Hain} and \textit{S. Zucker} [Invent. Math. 88, 83--124 (1987; Zbl 0622.14007)]. The proof of the Archimedean main theorem then follows exactly the blueprint of the previous two proofs.\N\N\NThis book will certainly be useful to graduate students and experts working in this beautiful and difficult area of arithmetic algebraic geometry. This is a welcome addition to the literature in a field.