Rigid Analytic Geometry And Its Applications
Sanny Olafeso
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?
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 tale cohomology of a Lubin-Tate tower (or, more precisely, in the tale cohomology of the Berkovich space attached to the rigid analytic space which is the generic fibre of the Lubin-Tate tower).
"The book under review gives a very complete and careful introduction into the technical foundations of the theory and also treats in detail the rigid analytic part of some of the important applications which the theory has found in recent years in number theory and geometry. The exposition is self contained, the authors only assume some familarity with basic algebraic geometry. . . Many of the subjects treated in this book are not easily available from the literature. The book which contains an extensive bibliography is a very valuable source for everyone wishing to learn about rigid geometry or its applications."
This is the first in a series of 5 posts whose goal is to briefly introduce rigid geometry with a focus on providing a big picture between the interactions between different perspectives of rigid geometry.
One of the most fruitful interactions in the semi-classical theory of algebraic geometry is that between the theory of schemes and the theory of complex analytic varieties. Namely, there is an analytification functor
It turns out that this functor preserves many of the properties of -schemes algebraic geometers are interested in (cf. [SGA1, Expos XII)). This statement is no better exemplified than in the GAGA results of Serre et al. (e.g. as in [Ser]) as well as the relationship betwen the tale topology of and the analytic topology of exhibited by Grothendieck, Artin, et al. (e.g. see [SGA1, Expos XII Thorme 5.1] and [SGA4, Expos XI Thorme 4.4]).
Now, while the the reduction to is useful for proving purely geometric results, the passage from something like to via the Lefschetz principle completely forgets arithmetic. Namely, if is a variety over then the variety has an action of the absolute Galois group which is of great interest to number theorists. Such information is completely lost in the passage from to .
All of these theories are relatively different in their presentation even though they generally capture, at their heart, the same theory (except for the theory of diamonds as already mentioned). In fact, one of the many strengths of rigid analytic geometry is that one can move between these various theories to use their different strengths. For example, there might be some powerful argument utilizing the well-behaved topology of Berkovich spaces that is completely invisible (or certainly much less obvious) from the perspective Huber.
These posts are mainly meant to be high-level introductions, focusing on giving some major definitions, major results, quirky examples and observations, but no real proofs. In particular, we will (of course) not go into any real detail here. For this, I highly suggest the following sets of notes/books:
(which, by the magic of non-archimedean theory, are open in the topology given by ) and, of course, combinations of these two types of opens (which one might call something like semi-algebraic open subsets).
Before we discuss the fix to this topological problem we actually first discuss the analogue of Step 2 in the theory of rigid spaces. Namely, if we are to think of as then the analogue of Step 2 should be the following.
But, there is an instrinic way to describe these sets as well. Namely, for any we know that (again by [Bos, 3.1 Proposition 4]) is a finite extension of and therefore carries a unique absolute value extending that of . One can then define to be the real number given by evaluating this absolute on on the element . One can then check that the set from agrees with the set
One then shows that this is a sheaf on a base (e.g. see [Vak, Theorem 4.1.2]). A pivotal step in this procedure is essentially to use the quasi-compactness of to reduce to showing that a finite distinguished open cover satisfies the sheaf property.
Now that we have singled out the correct analogue of for we want to say what the value of our presheaf on these distinguished opens is. The guess is quite clear: the ring should be the ring of analytic functions on that are convergent on . Namely, we could embed inside of and say that is
In other words, the presheaf is a sheaf with respect to rational domains and finite coverings by rational domains. The idea of proof is to do explicit computations much as in the case of [Vak, Theorem 4.1.2] (for a full proof see the references later on).
In fact, two stronger versions of this theorem can be shown to hold which are often stated instead of Theorem 4. Namely, one can replace rational domains and finite rational coverings by affinoid domains (resp. admissible opens) and finite affinoid coverings (resp. admissible coverings).
Well, let us call an open subset admissible if there exists an affinoid (in ) cover (possibly infinite) of such that for any map of affinoid -algebras such that the induced map has one has that the affinoid cover of has a refinement (cf. [Bos, Pg. 83]) by a finite affinoid open cover. Similarly, an admissible covering of the admissible open is a collection of admissible open subsets of which cover and have the property that for any -algebra map of -affinoids such that one has that the cover has a refinement by a finite affinoid open cover.
One needs to assume that the above colletions have reasonable enough properties to talk about sheaf theory (for a rigorous definition see [Bos, 5.1] and [Bos, 5.2]). Namely, one calls a contravariant functor
where the permissible opens are the admissible opens (resp. affinoid domains), and the permissible covers are the admissible ones (resp. finite affinoid covers). We call this the strong (resp. weak) -topology on . Unless specified otherwise, the -topology on will always taken to be the strong one. We call a locally ringed -space over of the form (with the strong -topology) an affinoid space over . While is really a pair, we shall, in practice conflate and .
Now, given any locally ringed -space over and any open permissible subset one naturally gets a -topology on (e.g. is the set of elements of contained in ) and one can restrict to get a sheaf on so that the pair is also a locally ringed -space over .
So, with all of this setup we can finally defined rigid analytic varieties a la Tate. Namely, a rigid analytic variety is a pair which is a locally ringed -space over such that there is a permissible open cover of such that is isomorphic to an affinoid space. A morphism of rigid spaces is a morphism of locally ringed -spaces over .
Let us finally note that the set of affinoid open subspaces of a rigid space form a distinguished base for (in the sense of [Vak, 2.5]) and thus if one wants to define a sheaf on it suffices to define a sheaf on the distinguished base of affinoid opens (e.g. by [Vak, Theorem 2.5.1]).
We start with the basic object which we have already seen many times above. Namely, the closed unit disk (or closed unit ball) over is the rigid analytic variety . Let us start to prove some of the basic properties of .
Proof: We tailor build admissible covers so to force the quasi-compactness. Namely, if is an admissible cover of then letting be the identity map (in the definition of admissible) one sees by definition that must admit a finite affinoid refinement.
To see the second claim we proceed as usual. If with admissible then and so has non-trivial idempotents as soon as and are not the zero ring. To see this note that if is a non-empty rigid space then is non-zero. Indeed, let and let be an affinoid neighborhood of . Then, we have a ring map and, in particular, if is the zero ring this forces to be the zero ring which forces to be empty, which is a contradiction.
Conversely, suppose that has non-trivial idempotents so that with not zero rings. Let and . Then I leave it to the reader to check that the non-vanishing locus of and , called and , are open, non-intersecting, and that is an admissible cover of .
are admissible opens (we leave this to the reader, or see [Con1, Example 2.2.8]) which intersect disjointly. If one wants to soup this up to an example consisting only of affinoid opens one can take two non-equal points and let . Then,
One then sees as the global sections of this sheaf. Intuitively one imagines that is the set of elements of such that (e.g. see [Tia, Proposition 1.4.13] for a rigorous statement) from which Proposition 13 then makes intuitive sense.
In particular, let us note that since is a ring (as we leave to the reader to check) one has that is a ring object in the category of rigid spaces over . This is perhaps not too surprising since is meant to be the rigid geometry avatar of which is itself a ring.
Let us also try and say something slightly more interesting about . Namely, let us define a line bundle on a rigid space as a sheaf of -modules which locally (for the -topology) is isomorphic to (cf. [FdvP, Definition 4.5.1]). As per usual the group of (isomorphism classes of) line bundles on forms a group under tensor product which is called the Picard group of and written . What is the Picard group of ?
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