1 Valued Fields and Normed Spaces -- 1.1 Valued fields -- 1.2 Banach spaces and Banach algebras -- 2 The Projective Line -- 2.1 Some definitions -- 2.2 Holomorphic functions on an affinoid subset -- 2.3 The residue theorem -- 2.4 The Grothendieck topology on P -- 2.5 Some sheaves on P -- 2.6 Analytic subspaces of P -- 2.7 Cohomology on an analytic subspace of P -- 3 Affinoid Algebras -- 3.1 Definition of an affinoid algebra -- 3.2 Consequences of the Weierstrass theorem -- 3.3 Affinoid spaces, Examples -- 3.4 Properties of the spectral (semi-)norm -- 3.5 Integral extensions of affinoid algebras -- 3.6 The differential module ?A/kf -- 3.7 Products of affinoid spaces, Picard groups -- 4 Rigid Spaces -- 4.1 Rational subsets -- 4.2 The weak G-topology and Tateโs theorem -- 4.3 General rigid spaces -- 4.4 Sheaves on a rigid space -- 4.5 Coherent analytic sheaves -- 4.6 The sheaf of meromorphic functions -- 4.7 Rigid vector bundles -- 4.8 Analytic reductions and formal schemes -- 4.9 Analytic reductions of a subspace of Pk1, an -- 4.10 Separated and proper rigid spaces -- 5 Curves and Their Reductions -- 5.1 The Tate curve -- 5.2 Nรฉron models for abelian varieties -- 5.3 The Nรฉron model of an elliptic curve -- 5.4 Mumford curves and Schottky groups -- 5.5 Stable reduction of curves -- 5.6 A rigid proof of stable reduction for curves -- 5.7 The universal analytic covering of a curve -- 6 Abelian Varieties -- 6.1 The complex case -- 6.2 The non-archimedean case -- 6.3 The analytification of an algebraic torus -- 6.4 Lattices and analytic tori -- 6.5 Meromorphic functions on an analytic torus -- 6.6 Analytic tori and abelian varieties -- 6.7 Nรฉron models and uniformization -- 7 Points of Rigid Spaces, Rigid Cohomology -- 7.1 Points and sheaves on an affinoid space -- 7.2 Explicit examples in dimension 1 -- 7.3 $$ \mathcal{P} $$(X) and the reductions of X -- 7.4 Base change for overconvergent sheaves -- 7.5 Overconvergent affinoid spaces -- 7.6 Monsky-Washnitzer cohomology -- 7.7 Rigid cohomology -- 8 Etale Cohomology of Rigid Spaces -- 8.1 Etale morphisms -- 8.2 The รฉtale site -- 8.3 Etale points, overconvergent รฉtale sheaves -- 8.4 Etale cohomology in dimension 1 -- 8.5 Higher dimensional rigid spaces -- 9 Covers of Algebraic Curves -- 9.1 Introducing the problem -- 9.2 I. Serreโs result -- 9.3 II. Rigid construction of coverings -- 9.4 III. Reductions of curves modulo p -- References -- List of Notation

1. Mathematics
2. Algebraic geometry
3. Functions of complex variables
4. Geometry
5. Number theory
6. Mathematics
7. Geometry
8. Algebraic Geometry
9. Several Complex Variables and Analytic Spaces
10. Number Theory