As Quomodocumque once said concerning the most recent set of courses at Arizona Winter School, “Jared Weinstein [gives] a great lecture.” On that note, I am delighted to welcome our first guest post, by the man himself. Note that it has been converted from LaTeX into “wordpress” flavour of LaTeX, so any errors were probably introduced by me in the conversion.
In any treatment of -adic Hodge theory, one inevitably encounters a procedure for passing from fields of characteristic 0 to fields of characteristic . In modern parlance, if is a perfectoid field of characteristic 0, one has the tilt , a perfectoid field of characteristic . (For instance, the construction of Fontaine’s period ring has as an intermediate step.) What happens when you try to un-tilt? That is, given a perfectoid field , can we describe the set of all perfectoid fields with ? This question leads us right into the subject of our post, the remarkable “fundamental curve of -adic Hodge theory”, due to Fargues and Fontaine. (See here, and also an English summary here.)
First we review the tilting procedure. Start with a field of characteristic 0 which is complete with respect to a nonarchimedean absolute value, whose residue field is perfect of characteristic . (Pedantic note: The absolute value itself doesn’t come packaged with , only the topology does.) Let be its ring of integers. Then form
Then is a ring in characteristic . It’s not hard to see that is a domain. Let be its fraction field.
Typically we only care about the case when is a perfectoid field, which means that it satisfies the following two properties:
- The value group is non-discrete.
- The Frobenius map is surjective on .
The field satisfies the second property but not the first, and . If is prime to , then the completion of satisfies the first property but not the second, and also. In these examples the passage from to is intolerably lossy. But if is a perfectoid field, then it turns out that is another perfectoid field.
So suppose is perfectoid. To see where the topology on comes from, one has to observe the isomorphism of multiplicative monoids (not rings)
(This is a good exercise if you haven’t seen this before. Also, for this it is important that be complete.)
Given an element which corresponds to in the above bijection, we put . If is an absolute value which defines the topology on , we can define a corresponding absolute value on by setting .
For instance, let be the completion of the field obtained by adjoining all th power roots of unity to . Then contains an element which is topologically nilpotent, as well as a system of roots . This means that must contain the ring , this being the -adic completion of . In fact , the -adic completion of . Similarly, if , we may set , and then once again .
Probably the most striking relationship between and is this:
Theorem A: The absolute Galois groups of and are
The precise statement of this theorem is that there is an equivalence of categories between finite étale -algebras and finite étale -algebras. Applied to , Theorem A lies at the heart of the construction of -modules attached to -adic Galois representations. Scholze’s work on perfectoid spaces provides a version of Theorem A that works in families; using this he was able to prove Deligne’s weight-monodromy conjecture for a hypersurface over by wrestling it into characteristic , where the conjecture was known previously.
Anyway, we promised to talk about how to un-tilt. Thus suppose we are given , a perfectoid field in characteristic . Does there exist a valued field with ? If so, is unique? Evidently not: We have just seen that the fields and both have tilt . So there are at least two ways to un-tilt the latter field. Let’s make precise what we mean by un-tilt:
Definition: An un-tilt of is an isomorphism class of pairs , where is a perfectoid field of characteristic 0 and is an embedding of topological fields, such that is a finite extension. (Two such pairs and are isomorphic if there is an isomorphism making the obvious diagram commute.) The degree of is the degree of .
Let be the set of un-tilts of .
The idea behind these definitions is that there ought to be some kind of geometric object (something like a rigid space) whose set of closed points parametrize un-tilts. I should explain why I’m including un-tilts of degree , rather than using a stricter definition requiring that be an isomorphism. The reason is that if is a Galois extension with group , then we want -orbits of to be in bijection with , and the definition is exactly what is necessary to make this happen. Note that a -orbit of size constisting of points of of degree corresponds to a single point of of degree .
Let be the th power Frobenius automorphism. Then there is an action of on , given by sending to . We wanted to parametrize the un-tilts of , but it seems like two un-tilts which differ by aren’t all that different. Let us call two un-tilts of equivalent if they differ by some power of Frobenius, so that the set of equivalence classes of un-tilts of is .
This is one of the main theorems of Fargues-Fontaine:
Theorem B: There exists a complete curve* whose closed points are naturally in bijection with equivalence classes of un-tilts of . If corresponds to the class of the un-tilt , then is the residue field of .
I now have to explain the asterisk and the dagger, and in doing so I will try to get across just how strange the object is. First, the asterisk. A “curve” is a separated integral noetherian scheme which is regular of dimension 1. In other words, a curve is built by gluing together spectra of Dedekind rings. Thus is a curve, and so are the affine line and the projective line over any field . In the latter two examples, the residue fields of closed points are finite extensions of the base field . But un-tilts of don’t seem to lie over any common base field–recall that the fields and are both un-tilts of , and these fields are certainly not finite over any common subfield.
So perhaps is more like , in the sense that it doesn’t admit a finite type morphism to any , for a field. Fine, except that is also complete. What does complete mean here, if not that it admits a proper morphism to some ? Fargues and Fontaine define it this way: A complete curve is a curve admitting a map , such that the degree of any principal divisor is 0. Theorem B then says that is complete with respect to the degree map which has already been defined on un-tilts.
Is there an analogue of Theorem B for ? Not quite. The situation is analogous to the situation of the Tate curve in rigid-analytic geometry. Let be the multiplicative group , considered as a rigid-analytic space over , and let have positive valuation. Then acts discontinously on without fixed points, one can form the quotient , which ends up being the analytification of an elliptic curve with -invariant . So is a complete curve and is not.
Let’s sketch the construction of . Since is supposed to parametrize un-tilts of , which are in characteristic 0, perhaps it is not surprising that Witt vectors get involved. Say corresponds to . Then the sharp map induces an honest ring homomorphism , characterized by . This extends to a surjective homomorphism , which we also write as . The kernel of is a maximal ideal, so we get a map . Unfortunately I highly doubt this map is a bijection– probably has complicated maximal ideals whose residue fields aren’t un-tilts of .
It seems that is not the full ring of functions on . We’ll construct a ring which contains which has the property that closed maximal ideals of are in bijection with . The construction is analytic in nature. Let be an absolute value on which induces its topology. For , define a norm on by
Now suppose is an un-tilt of . Let be the absolute value on for which . This absolute value induces on , which is a finite extension of via . The two absolute values on must be equivalent, in the sense that there exists for which for all . Thus .
Given , we can compare and . If , then
It follows from this inequality that if is a Cauchy sequence in with respect to , then converges in .
Definition: Let be the Fréchet completion of with respect to the norms for . That is, is the ring of sequences in which are Cauchy with respect to every , modulo those sequences which converge to 0 with respect to every .
In light of the foregoing discussion, if , then makes sense for any . Thus for every un-tilt of , we get a continuous surjection extending , whose kernel is a closed maximal ideal of .
Theorem C: Closed maximal ideals of are in bijection with .
This tempts us to define as a rigid space by setting , except that isn’t anything like a Tate algebra. It turns out that can be given a meaningful definition as an adic space, but this is the topic for another post.
At this point we can link to classical -adic Hodge theory. If is a perfectoid field in characteristic 0, and , then we get a point , and a maximal ideal . Then the completion of with respect to is , the de Rham period ring associated to . This is a complete DVR with residue field .
Now we turn to , which ought to be the quotient . This is supposed to be something like a projective curve, and I would like to motivate the construction of with projective curves in mind. To that end, suppose is a curve which is proper over a field, and you would like to give some kind of explicit presentation for . (For intance, could be a smooth curve of genus 1, and you would like to show that is isomorphic to a plane cubic.) The usual thing to do is to find a very ample line bundle on , in which case
In the case of the Fargues-Fontaine curve , what should be? Whatever it is, it must pull back to a line bundle on which is -equivariant. Since is the ring of analytic functions on , this should be the same as giving a free -module of rank 1 together with a -semilinear map . Let , where . This corresponds to a line bundle on the (not yet defined) . Then we ought to have, for any ,
This prompts the following definition.
Definition: , where is the graded -algebra with
To convince you this was the right thing to do, let me list the following facts, which hold when is algebraically closed:
- is a graded factorial ring, whose irreducible homogeneous elements are exactly the nonzero elements of degree 1.
- If is nonzero, then its divisor in is , for a point of degree 1.
- Conversely, if then there exists whose divisor is .
- More generally, the divisor of a nonzero element of has degree .
(I didn’t say exactly what the divisor of an element is, but it’s what you think: a formal sum of points in , weighted with multiplicities. Since each is a DVR, the multiplicities make sense.) Thus resembles nothing so much as the projective line over a field! In fact, Fargues and Fontaine show that (again under the assumption that is algebraically closed) is simply connected.
Let me close with an amusing observation. Consider the field , which is of course a perfectoid field. Let . (I would call it , but was taken!) Of course one of the un-tilts of is , but what are the others? Are they all isomorphic to ?
I don’t know. But suppose instead we took to be the field of Malcev-Neumann series , where is algebraically closed and is a divisible ordered abelian group. Elements of are “power series” in with coefficients in and exponents in , where the only restriction is that the support of each power series be a well-ordered subset of . You get a valuation on by looking at the least exponent of that appears in such a series.
is a maximally complete field, meaning that any valued extension field either has a larger residue field or else a larger value group. There is also a characteristic 0 construction, which I’d like to call , and then . A result of Bjorn Poonen is that any maximally complete field with residue field and value group has to be isomorphic or , depending on its characteristic. (His proof uses the axiom of choice in an essential way.)
(Sometimes I feel that the true -adic analogue of the complex numbers isn’t but rather . The field is spherically complete, meaning that any nested sequence of balls has nonempty intersection. doesn’t have this property, but does. Furthermore, elements of have decimal expansions. Elements of also do, in the sense that every element is a power series in . How do you write down a generic element of ? You basically can’t.)
Let be any un-tilt of . Then is maximally complete. (Exercise. Hint: Tilting preserves both residue field and value group.) By Poonen’s result, there is an isomorphism . This induces an isomorphism . Composing with gives an automorphism of (as a topological field).
This argument shows that acts transitively on the set of un-tilts . The un-tilt of gives a point , and it is almost tautological to see that the stabilizer of is the image of under the natural map . Thus there is a bijection
Similarly, you get a description of as . I find this rather amazing, since and are unimaginably huge groups with no obvious geometric structure, while is a proper curve. I’m not sure if this observation is useful to the study of , but given the rising role of maximally complete fields in -adic Hodge theory, it’s worth a look.