Introduction.- 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: denitions and first properties.- Summable projective systems of Hermitian vector bundles and niteness of θ-invariants.

- Exact sequences of infinite dimensional Hermitian vector bundles and subadditivity of their θ-invariants.- Infinite dimensional vector bundles over smooth projective curves.- Epilogue: formal-analytic arithmetic surfaces and algebraization.- Appendix A. Large deviations and Cramér's theorem.- Appendix B. Non-complete discrete valuation rings and continuity of linear forms on prodiscrete modules.- Appendix C.

Measures on countable sets and their projective limits.- Appendix D. Exact categories.- Appendix E. Upper bounds on the dimension of spaces of holomorphic sections of line bundles over compact complex manifolds.- Appendix F. John ellipsoids and finite dimensional normed spaces.