Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves - Info and Reading Options

The Open Library:

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.

Open Data:

Intro -- Contents -- Preface -- Introduction -- 0.1 Pro-Euclidean Lattices and θ-Invariants -- 0.1.1 -- 0.1.2 -- 0.1.3 -- 0.1.4 -- 0.2 Contents -- 0.2.1 -- 0.2.2 -- 0.3 Acknowledgments -- 0.4 Notation and Conventions -- Chapter 1 Hermitian Vector Bundles over Arithmetic Curves -- 1.1 De nitions and Basic Operations -- 1.1.1 Hermitian vector bundles over arithmetic curves. -- 1.1.2 Pullback. Tensor operations. -- 1.1.3 The Hermitian line bundles O(δ). -- 1.2 Direct Images. The Canonical Hermitian Line Bundle ϖOK/Z over Spec OK -- 1.2.1 -- 1.2.2 -- 1.3 Arakelov Degree and Slopes -- 1.3.1 De nition and basic properties. -- 1.3.2 Arakelov degree and direct image. -- 1.3.3 Slopes. -- 1.4 Morphisms and Extensions of Hermitian Vector Bundles -- 1.4.1 The filration Hom≤OK (E -- F) on Hom OK(E, F). -- 1.4.2 Injective and surjective admissible morphisms. -- 1.4.3 Heights of morphisms. -- 1.4.4 Admissible short exact sequences of Hermitian vector bundles. -- 1.4.5 Arithmetic extensions. -- 1.4.6 Admissible short exact sequences and arithmetic extensions. -- 1.4.7 Direct images of arithmetic and admissible extensions. -- Chapter 2 θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves -- 2.1 The Poisson Formula -- 2.1.1 The Poisson formula for Schwartz functions. -- 2.1.2 The Poisson formula for Gaussian functions. -- 2.2 The The θ-Invariants h0θ and h1θ and the Poisson-{Riemann-Roch Formula -- 2.2.1 The θ-invariants of a Euclidean lattice and the Poisson formula. -- 2.2.2 The θ-invariants of Hermitian vector bundles over a general arithmetic curve and the Poisson-Riemann-Roch formula. -- 2.3 Positivity and Monotonicity -- 2.3.1 Some elementary estimates. -- 2.3.2 Comparing the θ-invariants of generically isomorphic Hermitian vector bundles. -- 2.4 The Functions τ and η

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