Hello, everyone.
I want to ask some questions about rigid geometry.
1.what is the motivation of rigid geometry?
2.what is the applications of rigid geometry for solving arithmetic problems, especially for studying the fundamental groups of algebraic curves? what the beautiful theorems which were first proved by rigid geometry method?
thank you very much
I am really not an expert in the field, so I apologize in advance for omissions or mistakes - I would indeed be glad to get corrections. But let me try, anyhow...
You are asking for a motivation for rigid geometry and here, I guess, Kevin is right when saying that the first historical motivation was may be Tate's theory of uniformization of elliptic curves (with additive reduction): it says that every elliptic curve $E$ over $\mathbb{C}_p$ whose $j$ invariant $j_E$ verifies $|j_E|>1$ is isomorphic to $\mathbb{C}_p^\times/q(j_E)^\mathbb{Z}$, where $q(j_E)$ is the unique solution of $j(q(j_E))=j_E$ for the classical (i. e. complex-theoretic) modular function $j(q)$. The problem is in writing ''isomorphic'': Tate's starting point was to develop a sheaf theory on (roughly speaking) subquotients of $\mathbb{C}_p^n$ endowed with a certain Grothendieck topology that could be compared to the usual algebraic theory, pretty much the same way one can do with proper varieties over $\mathbb{C}$, and define the category or rigid spaces by means of this sheaf-theoretic description. Then $\mathbb{C}_p^\times/q(j_E)^\mathbb{Z}$ falls in this category and one can play with this more explicit object to grasp information on $E$ better: for instance, $N$-torsion points in this quotient are easy to understand. One other advantage is that, unlikely $\mathbb{C}$ where there is no change-the-base field-game to be played, rigid geometry is well-suited to define analytic objects over every complete ultrametric field, like $\mathbb{Q}_p$. Then you can look at points of rigid analytic spaces over extensions of the base field, in a much more agebro-geometric flavour. I guess the best references for these are (as Kevin says) the book by Bosch, Güntzer and Remmert and that by Fresnel-van der Put. Nowadays Tate's approach seems a little bit too ad hoc and Berkovich' one is perhaps preferable – the definition of spaces itself remembers that of schemes much more than Tate's definition did. You can look in his book Theorem 3.4.1 for an a beautiful and extremely concise statement of GAGA in this setting.
Another key point for developing rigid geometry was Dwork's proof of the rationality of zeta function for varieties over finite fields. The very rough idea is that if you have an affine scheme over a finite field $k$, you might try to lift it (i.e. its equations) to the Witt vectors $W(k)$ and now look at the $W(k)$-scheme defined by those. With some luck, it will not depend on the lifting, but one is rarely lucky. On the other hand, let $K=\mathrm{Frac}(W(k))$: one can look at the zero-locus of the family of polynomial as a subspace of $K^n$ - it comes endowed with its natural $p$-adic topology ($p=\mathrm{char}(k)$). Monsky and Washnitzer were the first to realize that imposing a certain overconvergency condition on functions over this space would provide for a cohomology theory which is
1. Close enough to de Rham cohomology to be meaningful/interesting
2. Independent of your liftings, so you can read information about your starting affine $k$-scheme in it.
Along these lines is Dwork's proof of the rationality of the zeta function over a finite field. But then one can look for more general situation, may be relative ones, in which one is given an $S$-scheme $X$ to start with, where $S$ is in turn a $k$-scheme. Crsytalline cohomology provides the answer in the proper and smooth case: but then rigid cohomology (as developed by Berthelot) is the key if one wants to study more general schemes. It is called rigid because it is defined in terms of rigid geometry – let me be vague (mainly due to my ignorance): try to mimic Monsky-Washnitzer idea, but without having ''equations''! The best you can do is to see your $k$-scheme $S$ as a $\mathrm{Spf}(W(k))$-scheme: since rigid geometry allows you to attach in a pretty functorial way a $K$-rigid analytic space to a $W(k)$-formal scheme, you have a reasonable definition of cohomology at your disposal by simply taking de Rham cohomology of this rigid analytic space (needless to say, many things need to be checked - but it works). I find it really nice - you should read Berthelot's introduction both to his book with Ogus ''Notes on crystalline cohomology'' and to his preprint ''Cohomologie rigide et cohomologie rigide à support propre'' as well as Le Stum's ''Rigid Cohomology''.
As for applications, Kevin is certainly right in saying that many are of arithmetic nature: Katz used rigid geometry to define $p$-adic modular forms in his Antwerpen paper, where he plays with the rigid analytic space attached to the usual modular curve and ''removes disks around points representing supersingular curves''. You see immediately that this would make no sense in a purely algebraic setting (you have no radii!) but it turns out to be an extremely fruitful idea - as Kevin Buzzard suggested in his answer to Geometric interpretation of Hida isomorphism, Gouvea's thesis is a nice place to read about this stuff. Abhyankar's conjecture has already been quoted by Niels, and Langlands conjecture over function field is another striking application. But I am sure I am forgetting/ignoring many...
It is known that both the Jacquet-Langlands correspondence and the local Langlands correspondence for GLn can be realized in the étale cohomology of a Lubin-Tate tower (or, more precisely, in the étale cohomology of the Berkovich space attached to the rigid analytic space which is the generic fibre of the Lubin-Tate tower).
