Classical algebraic geometry dealt with complex varieties defined by a system of polynomial equations in several variables. One might, for example, study the surface in three-dimensional space defined by the equation x5 + y5 + z5 – 5xyz = 1. In the classical approach, one thinks of this as a complex manifold (possibly with some singularities) and studies the resulting geometry. Complex analysis turns out to be an important tool, even if we are mostly interested in algebraic questions. To connect analytic and algebraic questions, there are GAGA theorems, named after a famous paper of Serre called “Géométrie Algébrique et Géometrie Analytique” (Annales de l'Institut Fourier 6: 1–42; there is a Wikipedia page discussing the main results).
Given the importance and usefulness of p-adic methods in number theory, it is inevitable that we would like to consider the analogous situation in the p-adic setting. The p-adic numbers Qp are analogous to the reals: they are the completion of Q with respect to a natural (but different) notion of distance. There is a field Cp that is both complete with respect to the p-adic distance and algebraically closed, so it can play the role of the complex numbers. It seems we’re ready to go.
Unfortunately, if we stick to the natural way of building them, manifolds over Cp are nothing like complex manifolds. The p-adic topology has so many open sets that it is totally disconnected, i.e., the only connected subsets are those with only one point. As a result, there are far too many functions that are given by a power series locally at every point. For example, there are locally analytic functions that are constant in an open set but not constant everywhere, so analytic continuation doesn’t work. We end up with a theory which looks nothing like algebraic geometry. It has its uses and beauties, but one can’t hope for GAGA theorems.
So we need something better, known as “rigid geometry.” It was started by John Tate in 1961 in a famous paper entitled “Rigid Analytic Spaces” (it was only published in 1971). Tate’s idea was to restrict the allowed open sets to what he called “affinoids” and then define rigid spaces by gluing them together.
Since then, the theory has been used to prove many important results, but for a while we had a pattern where each mathematician who needed it went back to Tate’s paper and then built on it in his or her own way. There was no standard reference. Even worse, several different ways to build the theory were found, and it was not clear whether they were actually equivalent. We are now in the much better situation of having some good accounts of the theory, the best known being Rigid Analytic Geometry and its Applications, by Fresnel and van der Put, and Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry, by Bosch, Güntzer, and Remmert. The two books under review add to that list, each in a different way.
Éléments de Géométrie Rigide signals its approach by its name: this is a book whose ambition is to live next to the famous EGA: Éléments de Géométrie Algébrique, by Grothendieck and Dieudonné. In fact, the author’s avant-propos says that it is impossible to read this book without a good knowledge of EGA (and SGA 1 to 4). A high-level exposition, then. Abbes prefers to develop the theory from Raynaud’s point of view based on formal schemes. Like EGA, the approach uses category theory: the category of rigid spaces is obtained by starting from formal schemes and then identifying objects that differ by “admissible” coverings. This is the first of two volumes; the promised second volume will deal with étale cohomology of rigid spaces.
p-adic Geometry is a very different book. It contains written versions of short courses taught at the Arizona Winter School, which is aimed at equipping graduate students for research. There is an article by Brian Conrad comparing the various foundations for p-adic geometry and an introduction to Berkovich’s version of the theory by Matthew Baker. Other chapters deal with the p-adic analogue of the complex upper halfplane and p-adic cohomology. The articles are well written and are more concerned to get quickly to what is useful than to give a full account of the foundations.
Both books are likely to be important references for those who want to learn p-adic geometry. Most of us will want to read p-adic Geometry first, to get an idea of the subject and what can be done with it.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.