Several approaches to non-archimedean geometry

Let k be a non-archimedean field: a field that is complete with respect to a specified nontrivial non-archimedean absolute value | · |. There is a classical theory of k-analytic manifolds (often used in the theory of algebraic groups with k a local field), and it rests upon versions of the inverse and implicit function theorems that can be proved for convergent power series over k by adapting the traditional proofs over R and C. Serre’s Harvard lectures [S] on Lie groups and Lie algebras develop this point of view, for example. However, these kinds of spaces have limited geometric interest because they are totally disconnected. For global geometric applications (such as uniformization questions, as first arose in Tate’s study of elliptic curves with split multiplicative reduction over a non-archimedean field), it is desirable to have a much richer theory, one in which there is a meaningful way to say that the closed unit ball is “connected”. More generally, we want a satisfactory theory of coherent sheaves (and hence a theory of “analytic continuation”). Such a theory was first introduced by Tate in the early 1960’s, and then systematically developed (building on Tate’s remarkable results) by a number of mathematicians. Though it was initially a subject of specialized interest, in recent years the importance and power of Tate’s theory of rigid-analytic spaces (and its variants, due especially to the work of Raynaud, Berkovich, and Huber) has become ever more apparent. To name but a few striking applications, the proof of the local Langlands conjecture for GLn by Harris–Taylor uses étale cohomology on non-archimedean analytic spaces (in the sense of Berkovich) to construct the required Galois representations over local fields, the solution by Raynaud and Harbater of Abyhankar’s conjecture concerning fundamental groups of curves in positive characteristic uses the rigidanalytic GAGA theorems (whose proofs are very similar to Serre’s proofs in the complex-analytic case), and recent work of Kisin on modularity of Galois representations makes creative use of rigid-analytic spaces associated to Galois deformation rings. The aim of these lectures is to explain some basic ideas, results, and examples in Tate’s theory and its refinements. In view of time and space constraints, we have omitted most proofs in favor of examples to illustrate the main ideas. To become a serious user of the theory it is best to closely study a more systematic development. In particular, we recommend [BGR] for the “classical” theory due