Bost J.-B. Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves

Basel: Birkhäuser, 2020. — 395 p.This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves. It contains a complete description of the theta invariants which give rise to a closer parallel with the geometric case. The author then unfolds his theory of infinite Hermitian vector bundles over arithmetic curves and their theta invariants, which provides a conceptual framework to deal with the sequences of lattices occurring in many diophantine constructions.
The book contains many interesting original insights and ties to other theories. It is written with extreme care, with a clear and pleasant style, and never sacrifices accessibility to sophistication.Front Matterx
Hermitian Vector Bundles over Arithmetic Curves
θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves
Geometry of Numbers and θ-Invariants
Countably Generated Projective Modules and Linearly Compact Tate Spaces over Dedekind Rings
Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves
θ-Invariants of Infinite-Dimensional Hermitian Vector Bundles: Definitions and First Properties
Summable Projective Systems of Hermitian Vector Bundles and Finiteness of θ-Invariants
Exact Sequences of Infinite-Dim. Hermitian Vector Bundles and Subadditivity of Their θ-Invariants
Infinite-Dimensional Vector Bundles over Smooth Projective Curves
Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization