NON-ARCHIMEDEAN GEOMETRY

JOHANNES NICAISE

Abstract. These are the lecture notes for an LSGNT/TCC course in the Winter term of 2017. Please report any typos or other errors you find!

1. What is non-archimedean geometry?

1.1. Non-archimedean fields.

(1.1.1) Let k be a field. An on k is a multiplicative norm, that is, a function | · | : k → R≥0 : x 7→ |x| that satisfies the following axioms: (1) |0| = 0 and |1| = 1; (2) |xy| = |x| · |y| for all x, y in k; (3) triangle inequality: |x + y| ≤ |x| + |y| for all x, y in k. We say that | · | is non-archimedean if a stronger version of (3) holds: (3’): ultrametric triangle inequality: |x+y| ≤ max{|x|, |y|} for all x, y in k. We say that | · | is archimedean if it is not non-archimedean. Exercise 1.1.2. Show that the inequality in (30) is always an equality if |x|= 6 |y|. Exercise 1.1.3. Show that an absolute value | · | on k is archimedean if and only if it satisfies the archimedean property: for every r there exists an integer n such that |n| ≥ r. Thus | · | is non-archimedean if and only if | · | is uniformly bounded on the image of Z in k. Hint: consider the binomial expansion of (x + y)m and send m to ∞. Definition 1.1.4. A non-archimedean field is a field k endowed with a non- archimedean absolute value | · | such that k is complete with respect to the metric induced by | · |.

(1.1.5) Let (k, | · |) be a non-archimedean field. We will use the following notation (due to Berkovich): ko = {x ∈ k | |x| ≤ 1} koo = {x ∈ k | |x| < 1} ek = ko/koo 1 2 JOHANNES NICAISE

Exercise 1.1.6. Show that ko is a valuation in k, that is, it is a subring such that for every non-zero element x of k, either x or x−1 belongs to ko. Show that koo is the unique maximal ideal in ko; thus ek is a field.

We call ko the of k and ek the residue field of k. The set ∗ |k | of absolute values of non-zero elements in k is a subgroup of (R>0, ·), called the value of k.

Example 1.1.7. (1) On any field k we can consider the trivial absolute value |·|0, defined ∗ by |0|0 and |x|0 = 1 for all x in k . The pair (k, | · |0) is a non- archimedean field, with k = ko = ek and koo = (0). Although this construction seems empty, analytic geometry over the trivally valued field actually has some deep and surprising applications! (2) Let F be any field and set k = F ((t)), the field of Laurent series over F . We endow it with a t-adic absolute value, given by |0| = 0 ordt(x) and |x| = ε where ordt(x) is the t-adic valuation of x (the smallest exponent of t with non-zero coefficient in the Laurent series expansion). Here ε is a fixed value in (0, 1). As ε varies all these absolute values are equivalent in the sense that they give rise to the same metric topology on k, but they are different as absolute values. We have ko = F [[t]], koo = (t) and ek = F . (3) Let p be a prime number and let k = Qp be the field of p-adic numbers. We endow it with a p-adic absolute value, given by ordp(x) |0| = 0 and |x| = ε where ordp(x) is the p-adic valuation of x (the smallest exponent of p with non-zero coefficient in the p-adic expansion). Here ε is a fixed value in (0, 1). In number theory, it is customary to take ε = 1/p (for instance to get a so-called product formula). As ε varies all these absolute values are equivalent in the sense that they give rise to the same metric topology on k, but they o oo are different as absolute values. We have k = Zp, k = (p) and ek = Fp, the field with p elements. (4) If (k, | · |) is a non-archimedean field, then | · | extends uniquely to an absolute value on any algebraic extension k0 of k (here we use the assumption that k is complete). The field k0 is complete if it is a finite extension of k but not in general; its completion is denoted by kb0 and is again a non-archimedean field. If k0 is an algebraic closure of k, then kb0 is still algebraically closed. These considerations allow us to produce new examples of non-archimedean fields. For instance, if k = C((t)) then

a [ 1/n k = C((t )) n>0 is an algebraic closure of k, called the field of complex Puiseux series. Exercise: describe the extensions of the t-adic absolute values to ka a a and determine the completion of k . If k = Qp and k is an algebraic a closure, the completion kc is denoted by Cp and called the field of complex p-adic numbers. It is difficult to describe this field explicitly NON-ARCHIMEDEAN GEOMETRY 3

because the extensions of Qp of degree divisible by p are hard to classify.

(1.1.8) The absolute values on Q have been completely classified by Ostrowski (1916). r • Archimedean: | · |∞ where | · |∞ denotes the usual absolute value and r is an element of (0, 1] (note that the triangle inequality is violated for r > 1). These are all equivalent and the completion of Q with respect to any of these is R. • Non-archimedean: the trivial absolute value | · |0 and the p-adic absolute values, for all primes p and bases ε ∈ (0, 1). For fixed p, the p-adic absolute values are all equivalent and give rise to the completion Qp. 1.2. Why do we care about non-archimedean fields? Non- archimedean fields pop up in the “local” study of families of objects in number theory and algebraic geometry. Let us discuss some representative examples.

(1.2.1) Number theory: diophantine problems (finding solutions of equations) over Q are hard! It is often useful to consider the problem over R and all the p-adic fields Qp (where we can exploit the analytic structure given by the absolute value) and then try to glue these “local” data together to an answer over Q. This is called the “local-to- global principle”. The archetypal example is the Hasse-Minkowski theorem for quadratic forms: two quadratic forms over Q are equivalent (i.e., they coincide up to a linear coordinate change) if and only if they are equivalent over R and all the p-adic fields Qp. These “local” equivalences can be checked explicitly by computing some basic invariants. One can deduce that a quadratic form over Q has a non-trivial zero if and only if this holds over R and all the Qp.

(1.2.2) Algebraic geometry: suppose that we want to understand the geometry of a smooth projective variety X over C. A standard technique consists in placing X in a family that degenerates into a union of irreducible components with a simpler structure; for instance, we can try to let a curve of positive genus degenerate into a union of projective lines. We try to find a variety X together with a morphism 1 f : X → AC = Spec C[t] −1 −1 such that f (1) is isomorphic to the curve X and X0 = f (0) is a union of projective lines. In order to draw conclusions about the shape of X we need to understand what happens to the fibers of f as t approaches 0. The correct way to do this in algebraic geometry is to consider the base change

X ×C[t] C((t)). This is a projective variety over the non-archimedean field k = C((t)) that can be thought of as an infinitesimal punctured tubular neighbourhood around X0. We can exploit the analytic structure given by the absolute value on k to analyze the geometry of our degeneration (this is closely related to tropical geometry). 4 JOHANNES NICAISE

1.3. What is wrong with k-analytic manifolds? (1.3.1) Non-archimedean geometry is the theory of analytic spaces (manifolds) over non-archimedean fields. Let us first decide what to expect from such a theory by looking at the classical setting over the complex numbers. Let X be a smooth complex algebraic variety. Then, through a process of analytification, we can turn X into a complex manifold Xan (we will recall the construction below). The important points about this process are the following: (1) We can apply new tools to Xan compared to X, namely, those coming from complex analysis, differential geometry and algebraic topology (for instance, Hodge theory). (2) If X is proper, we do not lose information by passing to Xan. This is guaranteed by Serre’s GAGA principle (GAGA=g´eom´etrie alg´ebriqueet g´eom´etrieanalytique). It says, among other things, that two smooth and proper complex varieties X and Y are isomorphic if and only if their analytifications are isomorphic as complex manifolds. Thus we can reconstruct X from Xan, at least in principle. Beware that this fails if X is not proper. Point (2) is essential and we will see below that it fails miserably if we adopt the na¨ıve definition of analytic manifold over a non-archimedean field.

(1.3.2) Let us briefly recall the construction of Xan. Its underlying set is the set X(C) of points of X with complex coordinates (that is, the closed points of the scheme X). To define its analytic structure, we can follow two approaches, one more explicit and one more intrinsic. For the explicit approach, cover X by affine opens U and embed each of these opens as n a closed subvariety into some affine space A . Then U is described by n C polynomial equations inside A and the same equations define a complex an n C an manifold U inside C . Gluing these together we get X .

(1.3.3) More intrinsically, we can endow X(C) with the weakest topology such that the following properties hold: (1) it is stronger than the Zariski topology, in the sense that the inclusion ι : X(C) → X is continuous with respect to the Zariski topology on X; (2) for every Zariski-open U in X and every regular function f ∈ OX (U), we can consider the induced map an −1 f : ι (U) = U(C) → C : u 7→ f(u) and its absolute value |f an|. We require that |f an| is continuous with respect to the usual topology on R. The resulting topology on X(C) is called the complex topology. If V is an open in X(C), we say that a function V → C is analytic if it can be locally written as a uniform limit of regular functions. This defines a sheaf of analytic functions OXan on X(C). One can show that the locally ringed an space X = (X(C), OXan ) is a complex manifold and that this construction is equivalent with the first one. NON-ARCHIMEDEAN GEOMETRY 5

(1.3.4) We want to set up a similar construction over a non-archimedean field k. We endow k with its metric topology. For every open V in kn, we call a function V → k analytic if it can locally be written as a converging power series in the coordinates on kn with coefficients in k. Then we can define k-analytic manifolds in the usual way, either via charts and atlases or as locally ringed spaces in k-algebras that are locally isomorphic to a n pair of the form (V, OV ) with V an open in k and OV the sheaf of analytic functions on V . This theory is developed in detail in [J.-P. Serre, Lie algebras and Lie groups, 1965]. For every smooth algebraic variety X over k, we can put a structure of k-analytic manifold on the set X(k) by simply copying either of the two constructions over C. This construction is perfectly well- defined, but the problem is that we lose almost all the geometric structure in the analytification process because the geometry of k-analytic manifolds is quite poor. In particular, the GAGA principle breaks down in a spectacular way1. This is due to the pathological nature of the metric topology on k, as we will now discuss.

(1.3.5) An “open”, resp. “closed” ball in k is a subset of the form B−(a, r) = {x ∈ k | |x − a| < r} B+(a, r) = {x ∈ k | |x − a| ≤ r} with a in k and r in R>0. We put “open” and “closed” between quotation marks because we will see that these balls are actually simultaneously open and closed in the metric topology. We call a a center of the ball and r a radius. The radius is not always uniquely determined, except when the ∗ value group |k | is dense in R>0 (for instance, if k is non-trivially valued and algebraically closed). As for the center, we have the following surprising property. Proposition 1.3.6. Any point of a ball (“open” or “closed”) is a center of the ball. Proof. Exercise. 

(1.3.7) This property is of course very different from what we are used to over R and C. It depends in a crucial way on the ultrametric property of the absolute value, and it has profound consequences for the metric topology on k. Corollary 1.3.8. If two balls in k intersect, then one is included in the other. Corollary 1.3.9. Every ball in k is both open and closed in the metric topology. Corollary 1.3.10. The metric topology on k is totally disconnected, that is, the only connected subsets are the singletons and the empty set. Exercise 1.3.11. Prove these corollaries.

1A superficial issue is that we only considered k-rational points on X, and this set may be too small and even empty, but this is easy to fix; however, even if k is algebraically closed, much more serious problems remain. 6 JOHANNES NICAISE

(1.3.12) This is very bad news for the geometry of k-analytic manifolds: they can be broken up into arbitrarily small disjoint open pieces which behave independently from each other, and there is no global geometric structure at all! Let us look at a few instances of this problem. Consider the function f : k → k that maps x to 1 if |x| ≤ 1 and to 0 otherwise. This function is locally constant, thus certainly analytic according to our na¨ıve definition. But we could have given any other constant value to the points x with |x| > 1. Thus the restriction of f to the unit ball B+(0, 1) knows nothing at all about the shape of f on the complement of this ball. This is very different from what happens over the complex numbers, where we have the principle of analytic continuation. There are simply too many analytic functions according to our definition; in particular, we would much prefer a definition such that the analytic functions k → k are those that are globally given by converging power series. As we have just seen, this property can not be checked locally in the metric topology.

(1.3.13) The plethora of analytic functions also leads to too many isomorphisms between k-analytic manifolds, and thus too few isomorphism classes. This is the death of the GAGA principle: many non-isomorphic proper and smooth k-varieties have isomorphic analytifications, so that we cannot hope to recover the geometry of the variety from its analytification. An extreme illustration is given by the following theorem by Serre. A non- archimedean field is called a local field if the absolute value is non-trivial and the metric topology is locally compact; one can show that the local fields are precisely the finite extensions of p-adic fields and the Laurent series fields over finite fields. In particular, the residue field of a local field is always finite. Theorem 1.3.14 (Serre). Let k be a local field and denote the cardinality of its residue field by q. Then every non-empty compact k-analytic manifold of dimension n is isomorphic to a disjoint union of s copies of B+(0, 1)n, for some unique element s in {1, . . . , q − 1}. Proof. We only give a brief sketch. It is not difficult to see that every non- empty compact manifold is isomorphic to a disjoint union of s copies of B+(0, 1)n, for some positive integer s. We can always arrange that s lies in {1, . . . , q − 1} by observing that B+(0, 1) itself is isomorphic to a disjoint union of q copies of B+(0, 1) (look at all the cosets of koo in ko; by translation and homothety, each of these is isomorphic to B+(0, 1)). The subtle point is the uniqueness of s. Serre gave a direct but cumbersome argument in [Lie algebras and Lie groups] and a much more elegant proof in [Topology 1965], involving p-adic integration.  Thus we need to be (much) more clever in defining analytic spaces over non-archimedean fields, to overcome the problem of total disconnectedness. We will discuss some approaches in the next lecture, and then focus on one of them, developed by Vladimir Berkovich at the end of the 1980s. NON-ARCHIMEDEAN GEOMETRY 7

2. Berkovich spaces

2.1. The development of non-archimedean geometry.

(2.1.1) The first systematic theory of non-archimedean geometry was developed by John Tate in the 1960s, following ideas by Grothendieck. His results were first informally distributed in the form of lecture notes and then published in Inventiones Mathematicae as the foundational paper Rigid analytic spaces (1971). Remarkably, the publication was an initiative of the editorial board, as Tate had never submitted the paper. Tate called the analytic manifolds that we considered in the first lecture wobbly, and his new spaces rigid, as the whole point of the theory was to give them a stronger global structure. The key idea is to modify the topology on analytic manifolds by only allowing certain types of open covers. This gives rise to a Grothendieck topology, a generalization of the classical notion of a topology2. One of Tate’s main motivations was to explain certain calculations on elliptic curves in terms of a non-archimedean uniformization theory. The class of elliptic curves where these methods apply are now called tate curves.

(2.1.2) An alternative viewpoint on Tate’s theory was proposed by Michel Raynaud in the early 1970s. Raynaud’s idea was to describe Tate’s rigid spaces as generic fibers of so-called formal schemes. The theory of formal schemes, developed by Grothendieck, is not too different from ordinary scheme theory and some significant technology is available to study them; the advantage of Raynaud’s viewpoint is that we can now also apply this technology to rigid spaces. These ideas were systematically developed by Siegfried Bosch and Werner L¨utkebohmert in the 1990s.

(2.1.3) A different approach was taken by Vladimir Berkovich around 1990. He put forward the idea that the topology itself is not the problem, but rather that there are many points missing on non-archimedean manifolds according to the na¨ıve definition. Adding these points gives a more solid structure to the manifold and gives rise to a satisfactory theory of non- archimedean geometry. Berkovich’s theory is in several respects the closest to the theory of real and complex manifolds; this is the approach that we will follow in this course.

(2.1.4) We should also mention the theory of adic spaces, introduced by Roland Huber in the 1990s. This theory looks much less like real or complex geometry but it has important and profound applications in arithmetic geometry, particularly in the context of Peter Scholze’s work on perfectoid spaces and the Langlands program. Huber’s theory can be applied to a wider range of algebraic objects than Berkovich’s formalism; in particular, it allows one to deal with objects that are not locally of finite type over a base field.

2In this case, the generalization is relatively mild, because the admissible opens are still subsets of the manifold. 8 JOHANNES NICAISE

(2.1.5) Today, rigid geometry has largely been overtaken by the theories of Berkovich and Huber, which exist in parallel; depending on the applications you are interested in, one of them may be better suited than the other. Nevertheless, the use of formal schemes is still an important aspect of both theories.

2.2. Berkovich’s analytification functor.

(2.2.1) In a first approach, we will only discuss analytifications of algebraic schemes over a non-archimedean field k. These only form a small portion of the objects in Berkovich’s category of analytic spaces, but will already yield several interesting applications. We will discuss a more general set-up later in the course (if time permits). Let X be a k-scheme of finite type. Then the an Berkovich analytification X is defined as the set of couples x = (ξx, | · |x) where ξx is a (scheme-theoretic) point of X and | · |x is an absolute value on the residue field κ(x) that extends the given absolute value on k. It is essential that we allow ξx to be a non-closed point: if ξ is a closed point of X then κ(ξ) is a finite extension of k and there is a unique extension of the absolute value on k to κ(ξ). Thus there is only one point x in Xan with ξx = ξ and we get nothing new. If ξ is not closed, κ(ξ) is a transcendental extension of k and there will be infinitely many extensions of the absolute value on k to κ(ξ). These absolute values supported at non-closed points constitute the main part of the Berkovich space Xan.

(2.2.2) We can endow Xan with a topology by copying the characterisation of the complex topology from the first lecture. It is the weakest topology that satisfies the following properties. (1) It is stronger than the Zariski topology, in the sense that the forgetful map an ι : X → X : x 7→ ξx is continuous with respect to the Zariski topology on X; (2) For every Zariski-open U in X and every regular function f ∈ OX (U), we can consider the induced map −1 |f| : ι (U) → R : x 7→ |f(x)| := |f(ξx)|x.

This definition makes sense because ξx lies in U so that f is defined at ξx and its evaluation is an element of the residue field κ(ξx). We require that |f| is continuous with respect to the usual topology on R. Exercise 2.2.3. Assume that X is affine. (1) Show that we can also describe the set Xan as the set of all multiplicative seminorms O(X) → R≥0 that extend the absolute value on k. (2) Show that we can also characterize the topology on Xan as the an weakest topology such that the function |f| : X → R is continuous for every f in O(X). NON-ARCHIMEDEAN GEOMETRY 9

Exercise 2.2.4. Assume that k is algebraically closed. Then we can embed the set X(k) into Xan by sending each ξ ∈ X(k) to the unique point in ι−1(ξ). Show that the induced topology on X(k) is the metric topology.

(2.2.5) For every point x of Xan, we define the residue field H (x) of Xan at x as the completion of κ(ξx) with respect to the absolute value |·|x. This is again a non-archimedean field, and the inclusion k → H (x) is an isometric embedding; we say that H (x) is a valued extension of k. Just like any other non-archimedean field, H (x) has a valuation ring H (x)o with maximal ideal H (x)oo and residue field Hf(x) (notice the clash of terminology here: we consider the residue field of the residue field at x).

(2.2.6) Finally, we can define a sheaf of analytic functions on Xan. An analytic function f on an open V of Xan is a rule that assigns to every point x in V an element f(x) of the residue field H (x) such that f is locally a unform limit of regular functions on X. More precisely, every point x of V there should have an open neighbourhood W in V such that the restriction of f to W is a uniform limit of a sequence of rational functions on X without poles in ι(W ). The analytic functions form a sheaf on Xan that is called the an structure sheaf of X and denoted by OXan .

(2.2.7) If g : Y → X is a morphism of k-schemes of finite type then it induces a natural continuous map gan : Y an → Xan and a local morphism of sheaves in k-algebras an OXan → g∗ OY an . (Exercise: work out the details.) In this way, we obtain an analytification functor (·)an from the category of k-schemes of finite type to the category of locally ringed spaces in k-algebras. Exercise 2.2.8. Show that gan is an open embedding if g is an open embedding.

(2.2.9) It is interesting to remark that, if we replace k by the field C with its usual absolute value, the above definition of Xan is equivalent to the usual complex analytification of X. The reason is that every valued extension of C is trivial, by the Gelfand-Mazur theorem. Thus if ξ is a point in X then −1 an ι (ξ) is empty except when ξ is closed. Hence, X = X(C).

(2.2.10) We should point out right away that we have not said the final word about the definition of the analytic space Xan. A fundamental complication (which also arises in complex analytic geometry) is that the algebras of analytic functions on open subsets of Xan have bad algebraic properties; for instance, they are not noetherian. In order to apply techniques from commutative algebra it is much more convenient to work with noetherian rings. For this reason, one extends the sheaf OXan by also considering analytic functions on certain closed subsets of Xan (for instance, 1,an the closed unit disc in Ak ). Doing this in a systematic way requires the introduction of a Grothendieck topology, called the G-topology, where such 10 JOHANNES NICAISE subsets are treated as opens. We may come back to this toward the end of the course. 2.3. Basic topological properties. Proposition 2.3.1. Let X be a k-scheme of finite type. Then Xan is locally compact and locally path-connected. Moreover, we have the following equivalences: (1) X is connected if and only if Xan is connected; (2) X is separated if and only if Xan is Hausdorff; (3) X is proper if and only if Xan is compact. This indicates that the Berkovich topology has reasonable properties and reflects the basic geometric properties of X quite well. The proofs of these properties are not straightforward and will not be discussed here, but they are feasible once one has developed the basic aspects of the theory of analytic spaces. The following result lies much deeper; the only known proof depends heavily on techniques from model theory (mathematical logic). Theorem 2.3.2 (Hrushovski–Loeser, 2016). The topological space Xan is locally contractible. If X is smooth over k, this had previously been proven by Berkovich using de Jong’s theory of alterations.

1,an (2.3.3) As a first example, we will describe the topological space Pk , the Berkovich analytification of the projective line. One can show that, for any k-scheme X of finite type and every algebraic closure ka of k, the a a an absolute Galois group Gal(k /k) acts on (X ×k kc) and the space X is canonically homeomorphic to the quotient space of this action. This reduces the problem to the case where k = ka; for the remainder of this section, we will therefore assume that k is algebraically closed. We have already seen that the fiber of the analytification map 1,an 1 ι : Pk → Pk 1 over any closed point of Pk is a singleton, consisting of the closed point 1 together with the given absolute value on its residue field k. Thus Pk(k) = 1,an k ∪ {∞} injects naturally into Pk . It remains to classify the absolute values on the function field k(T ) that extend the absolute value on k, and 1,an 1,an to describe the topology on the space Pk . Note that we can identify Ak 1,an with Pk \ {∞}.

2.4. The projective line – trivially valued case. (2.4.1) We first consider the case where the absolute value on k is trivial. 1,an 1 Then Pk has a canonical point (invariant under all automorphisms of Pk), namely, the trivial absolute value | · |0 on k(T ). This point is called the Gauss point and denoted by ηG; its residue field H (ηG) is simply k(T ) with the trivial absolute value. Next, for every element a of k ∪ {∞}, we can consider the valuation ∗ orda : k(T ) → Z NON-ARCHIMEDEAN GEOMETRY 11 that measures the order of the zero or pole of a at a. More precisely, if a ∈ k then we can write every non-zero element R(T ) of k(T ) as (T − a)vQ(T ) where Q(T ) is a rational function without zero or pole at a; then ordaR(T ) = v. If a = ∞ then we set ordaR(T ) = −degR(T ). This valuations defines a family of absolute values | · |a,ε on k(T ), with ε ∈ (0, 1), by setting ordaR(T ) |R(T )|a,ε = ε for every non-zero element R(T ) of k(T ). Proposition 2.4.2. Every absolute value on k(T ) that is trivial on k is of the form | · |0 or | · |a,ε. Proof. An absolute value | · | on k(T ) that is trivial on k is completely determined by its values on linear T − a with a ∈ k, because k is algebraically closed and absolute values are multiplicative. If |T − a| = 1 for all a ∈ k then | · | is the trivial absolute value. If |T − a| > 1 for all a ∈ k then by applying the ultrametric triangle inequality to b − a = (T − a) + (b − T ) we see that |T − a| = |T − b| for all a, b ∈ k; we call the inverse of this value ε (recall that the ultrametric triangle inequality is always an equality if the terms in the right hand side have different absolute values). Then it is clear that | · | = | · |∞,ε. Thus we may assume that |T − a| < 1 for some a ∈ k. We set ε = |T − a|. This time, applying the ultrametric inequality to b − a = (T − a) + (b − T ) shows that |T − b| = 1 for all b ∈ k different from a. It follows that | · | = | · |a,ε. 

1,an (2.4.3) We denote by ηa,ε the point of Pk corresponding to the absolute value | · |a,ε on k(T ). Then the residue field H (ηa,ε) is the Laurent series field k((T − a)) if a ∈ k, and it is k((T −1)) if a = ∞. Now let us find out how all of these points ηa,ε, together with the Gauss point ηG and the points 1,an of k ∪ {∞}, sit in the topological space Pk . We introduce the notations ηa,1 = ηG and ηa,0 = a for all a ∈ k ∪ {∞}. This notation expresses that |R(T )|a,ε converges to |R(T )|0 as ε → 1 and converges to |R(a)|0 as ε → 0, for every element R(T ) of k(T ). Proposition 2.4.4. For every a ∈ k ∪ {∞}, the map an γa : [0, 1) → X : ε 7→ ηa,ε is a homeomorphism onto its image, and this image is an open subset of an 1,an X . A subset S of Pk is open if and only if it satisfies the following conditions:

(1) the intersection of S with the image of γa is open, for every a ∈ k ∪ {∞}; (2) if S contains ηG, then it contains all but finitely many of the segments Im(γa). 12 JOHANNES NICAISE

Proof. For notational convenience, we assume that a 6= ∞; this can always 1 be arranged by means of an automorphism of Pk. The map γa is obviously 1 injective. It is very easy to check that ι ◦ γa : [0, 1) → Pk is continuous, and that −1 |f| ◦ γa : γa (U) → R 1 is continuous for every Zariski-open U in Pk and every regular function f on U. Thus γa is continuous. It is also open, because for all real numbers 1,an δ1 < δ2 ≤ 1, γa maps (δ1, δ2) ∩ [0, 1) to the open subset of Ak defined by δ1 < |T − a| < δ2. This proves the first part of the statement. To finish the argument, it suffices to describe the open neighbourhoods of the Gauss point ηG. By the definition of the Berkovich topology, a pre-basis of open neighbourhoods is given by the subsets of the form 1,an V = {x ∈ Ak | δ1 < |P (x)| < δ2} where P (T ) is a non-zero polynomial in k[T ] and δ1 < 1 < δ2 (otherwise ηG does not lie in this set). The set V contains the image of γa as soon as P (T ) does not have a zero at a. Thus the image of γa lies in V for all but finitely many a. 

(2.4.5) Note that Proposition 2.4.4 completely characterizes the topology 1,an on Pk . The induced topology on k ∪ {∞} coincides with the metric 1 topology on Pk(k), which is the discrete topology because the absolute value on k is trivial. In particular, it is totally disconnected. Nevertheless, by adding all the absolute values on k(T ) to interpolate between these points we have obtained a path-connected compact Hausdorff space (path-connected and Hausdorff are obvious; you can prove that the space is compact as an 1,an 1,an exercise). Observe that Ak = Pk \ {∞} is not compact; this reflects the 1 lack of properness of Ak. Exercise 2.4.6. Describe the k-algebras of analytic functions on the opens 1,an of Pk . Remark 2.4.7. You may have noticed a similarity between Proposition 2.4.2 and Ostrowski’s classification of absolute values on Q. In fact, one can also define the Berkovich analytification of the arithmetic scheme Spec (Z), 1,an and the picture is quite similar to the one for Pk . The role of the points in k is played by the primes in Z, and the role of ∞ is played by the usual absolute value | · |∞. This analogy between number fields and function fields is one of the cornerstones of arithmetic geometry and provides a very powerful framework to think about number fields and their rings of integers in a geometric way.

2.5. The projective line – non-trivially valued case. (2.5.1) Now we consider the case where k is not trivially valued (but we still assume that it is algebraically closed). Then the picture becomes more 1,an complicated. The points of Pk have been classified by Berkovich into four types. The points of k ∪ {∞} are called type 1 points. The remaining points correspond to the absolute values on k(T ) that extend the given absolute NON-ARCHIMEDEAN GEOMETRY 13 value on k. Fix an element a in k and a positive real number r. For every polynomial f(T ) in k[T ], we write

X i f(T ) = ci(T − a) i=0 where the coefficients ci belong to k, and we set

i |f(T )|a,r = max |ci|r . i

It is an easy exercise to check that this defines an absolute value | · |a,r on k(T ). We can similarly define absolute values | · |∞,r that are characterized by the formula d X i i | ci(1/T ) |∞,r = max |ci|r . i i=0 The following proposition provides a geometric description of these absolute values.

Proposition 2.5.2. Let a be an element of k and let r be a positive real number. Consider the “closed” ball

B+(a, r) = {z ∈ k | |z − a| ≤ r} in k. Let f(T ) be a rational function in k(T ) that does not have any poles in B+(a, r). Then

|f(T )|a,r = sup |f(z)| z∈B+(a,r) and if r lies in |k∗|, the supremum in the right hand side is always attained (the latter statement is called the maximum modulus principle).

Proof. By means of an affine change of coordinate, we can arrange that a = 0. We first observe that, when g(T ) is a polynomial in k[T ] without zeroes in B+(0, r), we have |b| = |z−b| > r for every root b of g(T ) and every z in B+(0, r). It follows that |g(T )| is constant on B+(0, r), and that the equality in the statement is satisfied (both sides equal |g(0)|). Applying this observation to the denominator of f(T ), we can reduce to the case where f(T ) is a polynomial. ∗ The fact that k is algebraically closed implies that |k | is dense in R>0 (it is a divisible group). Thus we can approximate r from above and below by elements in |k∗|, so that it is enough to prove the result when r lies in |k∗|. By means of a linear change of coordinate, we can then arrange that r = 1. Rescaling f, we can also arrange that |f(T )|0,1 = 1. Writing f(T ) d as c0 + ... + cdT , this means that maxi |ci| = 1. We must show that there exists an element z of ko = B+(0, 1) such that |f(z)| = 1. Since one of the coefficients of f(T ) has absolute value 1, the reduction fe of f module koo is non-zero. Let z be any element of ko such that its reduction modulo koo is oo not a root of fe. Then f(z) does not lie in k , and thus |f(z)| = 1.  14 JOHANNES NICAISE

(2.5.3) Here are a few properties that follow immediately from the definition and will be used repeatedly in the proofs below. If z is an element + of k, then |T − z|a,r = r if and only if z lies in B (a, r). Otherwise, + |T − z|a,r = |z − a| > r. Moreover, if z does not lie in B (a, r), and B+(a0, r0) is contained in B+(a, r) for some a0 ∈ k and r > 0, then 0 |z − a | = |T − z|a0,r0 = |T − z|a,r = |z − a|.

(2.5.4) Note that the ball B+(a, r) and the radius r are uniquely determined by the absolute value | · |a,r, since r is the minimal value of + |T − z|a,r as z ranges through k, and B (a, r) is the set of points z in k where this minimal value is attained. It follows that | · |a,r = | · |b,r0 if and 0 only if r = r and |a − b| ≤ r, for all a, b ∈ k. We also have | · |∞,r = | · |0,1/r for every r > 0. For every a ∈ k ∪ ∞ and every positive real number r we 1,an denote by ηa,r the point of Pk corresponding to |·|a,r. We also set ηa,0 = a. ∗ We say that ηa,r is of type 2 if r belongs to the value group |k |, and of type 3 if r is non-zero and does not lie in |k∗|.

(2.5.5) The final class of points, those of type 4, are related to a curious property of non-achimedean fields called spherical (in)completeness.A non-archimedean field F is called spherically complete if every decreasing sequence of “closed” balls has non-empty intersection. This is guaranteed by completeness of the field if the radii of the balls tend to 0, but not otherwise; in fact, most of the non-archimedean fields we encounter in practice are not spherically complete (an important exception are the local fields, that is, those with finite residue field). One can show that a non-archimedean field is spherically complete if and only if it has no non-trivial immediate extensions. An immediate extension is an extension of valued fields that induces isomorphisms between the value groups and the residue fields. Exercise 2.5.6. (1) Show that non-archimedean local fields are spherically complete. Hint: recall that these are precisely the locally compact non- archimedean fields. (2) Show that the completion of the field of complex Puiseux series is not spherically complete, for instance by constructing an immediate extension. Hint: If you do not manage to come up with a construction yourself, google “Hahn series”.

(2.5.7) If k is not spherically complete, every decreasing sequence B = + (Bn)n∈N of “closed” balls Bn = B (an, rn) with empty intersection ∩nBn gives rise to a new kind of absolute value, characterized by

|f(T )| = inf |f(T )|a ,r B n n n for every f(T ) ∈ k[T ]. Note that |f(T )|B is non-zero when f(T ) 6= 0: for every a ∈ k we have |T − a|am,rm ≥ rn for all m as soon as a does not lie 1,an in Bn. We denote the corresponding point of Pk by ηB, and we call ηB a point of type 4. We are now ready to state Berkovich’s classification theorem 1,an for points on Pk . NON-ARCHIMEDEAN GEOMETRY 15

Theorem 2.5.8 (Berkovich). Let B = (Bn)n∈N be a decreasing sequence of + “closed” balls Bn = B (an, rn) in k such that

r := inf{rn | n ∈ N} > 0. We set |f(T )| = inf |f(T )|a ,r B n n n for every polynomial f(T ) in k[T ]. This defines an absolute value on k(T ), and every absolute value on k(T ) that extends the given absolute value on k 0 is of this form. If B is another such sequence, then | · |B = | · |B0 if and only if one of the following holds: (1) both sequences have non-empty intersections, and these intersections are the same; (2) both sequences have empty intersections, and for every n ≥ 0, we 0 0 can find m ≥ 0 such that Bm ⊂ Bn and Bm ⊂ Bn. We will express this condition by saying that the sequences are interlaced. If B has non-empty intersection, then this intersection is equal to B+(a, r) for any a ∈ ∩nBn, and | · |B = | · |a,r.

Proof. For every a ∈ k, the point a lies in ∩B if and only if |T − a|B = r, because |T − a|am,rm > rn is independent of m, for all m ≥ n, if a does not lie in Bn. If a ∈ ∩B then we have ∩B = B+(a, r) and | · |B = | · |a,r. This implies, in particular, that two sequences B and B0 that define the same absolute value have the same intersection, and that the converse holds if this intersection is non-empty. Now assume that B and B0 are two sequences with empty intersection. We must show that they define the same absolute value if and only if they are interlaced. The “if” part of the assertion is obvious. So assume that |·|B = |·|B0 and let n be an 0 element of N. By symmetry, it suffices to find m ∈ N such that Bm ⊂ Bn. Choose an element a in k such that |T − a|B < rn. Then when m is large 0 0 enough, we must also have |T − a|am,rm < rn and |T − a|am,rm < rn, where 0 + 0 0 0 0 we wrote Bm = B (am, rm). This implies that rm < rn and |am −am| < rn, 0 so that Bm is contained in Bn. Let us now prove that every absolute value | · | on k(T ) that extends the absolute value on k arises from a family B. We choose a sequence (an)n∈N of elements in k such that |T − an| is non-increasing and converges to r = inf |T − a|. a∈k

Note that r > 0 since otherwise the sequence (an) would converge to a point a ∈ k by completeness and we would have |T − a| = 0. + We set rn = |T − an| and Bn = B (an, rn) for every n. Then the ultrametric inequality implies that |an+1 − an| ≤ rn for every n, so that the sequence of balls B = (Bn)n∈N is decreasing. We claim that | · | = | · |B. To prove this, it suffices to test the equality on functions of the form T − b with b ∈ k. We have either |T − b| > |T − an| = rn for sufficiently large n, or |T − b| = r. In the former case, the ultrametric inequality implies that rn < |T − b| = |b − an| = |T − b|an,rn when n is large enough and the 16 JOHANNES NICAISE result follows. In the latter case, we find that |b − an| ≤ rn for all n, so that

|T − b|an,rn = rn and the result follows again.  1,an Corollary 2.5.9. Every point of Pk is of type 1, 2, 3 or 4 (and these types are disjoint).

1,an (2.5.10) The division of the points of Pk into four types is also reflected 1,an in their residue fields. Let x be a point of Pk . If x is of type 1 then its residue field is k. If x is of type 2, then its residue field H (x) is a a valued extension of k with value group |H (x)∗| = |k∗| and with residue ∼ ∗ field Hf(x) = ek(u). If x = ηa,r is of type 3, then |H (x) | is the subgroup of ∗ (R>0, ·) generated by |k | and r, and Hf(x) = ek. Finally, if x is of type 4, then H (x) is an immediate extension of k. Exercise 2.5.11. Prove these claims about the residue fields of the points 1,an of Pk . Type 4 requires some thought; have another look at the proof of Proposition 2.5.2.

1,an (2.5.12) To conclude our example, we study the topology on Pk . It is best described in terms of a natural tree structure, where branches of the tree correspond to continuously growing or shrinking families of balls. 1,an Alternatively, we can also construct Pk as a projective limit of finite graphs (this viewpoint will be generalized below). Writing down a formal description of the topology is a bit cumbersome; for now, it is better to illustrate it by means of a picture (cf. lecture).

2.6. An application to dynamical systems.

1,an (2.6.1) The analytification of the projective line Pk already gives rise to some deep applications, in particular to the study of dynamical properties of rational maps (that is, the behaviour of the sequence of iterates of a rational map). It turns out that one can develop a potential theory on 1,an Pk in analogy with classical potential theory over the complex numbers (the study of harmonic functions). The use of Berkovich spaces is essential for such applications: one aspect of potential theory is the construction of suitable measures, and these measures tend to be supported entirely on the 1,an 1 part of Pk that lies over the generic point of Pk (that is, the complement of the locus of type I points). A nice introduction to this topic is [Baker & Rumely, Potential theory and dynamics on the Berkovich projective line].

(2.6.2) Here is an example of a problem that was solved by these methods. We fix an integer d ≥ 2. For every complex number c we consider the polynomial map d fc : C → C : x 7→ z + c. A complex number a is called a preperiodic point of fc if the set of iterates (n) {fc (a) | a ∈ N} NON-ARCHIMEDEAN GEOMETRY 17

(n) is finite; this means that the value fc (a) becomes periodic in n for sufficiently large n. Now let a and b be two distinct complex numbers. d d It is obvious that, when a = b , the point a is preperiodic for fc if and only if b is preperiodic for fc. Baker and Demarco (Duke 2011) proved a converse to this property, answering a question by Zannier: they showed that, if there are infinitely many complex numbers c such that a and b are d d both preperiodic for fc, then a = b . This innocent-looking statement is surprisingly difficult to prove. The proof of Baker and Demarco uses 1,an potential theory on the projective line Pk both for k = C and for k = Cp (a completed algebraic closure of Qp).

2.7. Higher genus curves.

(2.7.1) Let k be an algebraically closed non-archimedean field and let C be a connected, smooth and projective algebraic curve over k. If k is trivially an 1,an valued then the structure of C is quite similar to that of Pk : when k is algebraically closed, the points of C(k) are joined to the Gauss point ηG (the trivial absolute value on the function field k(C)) by means of segments of absolute values | · |a,ε that measure the orders of vanishing of rational functions at the points a of C(k). The main difference is the residue field at ηG, which is now the function field k(C) with the trivial absolute value.

(2.7.2) The picture becomes more intricate if k is non-trivially valued. We still have a cloud of closed points of C at the boundary of Can, but Can is not always contractible. Describing the precise structure of Can requires some tools from arithmetic geometry (specifically, the stable reduction theorem) that will be described in the next section.

3. Curves over discrete valuation rings

3.1. Models of curves.

(3.1.1) Let k be a non-archimedean field with non-trivial absolute value. Let C be a smooth, projective, geometrically connected curve over k. To study the geometric and arithmetic properties of C, it is often useful to consider models of C over the valuation ring ko of k. Such a model is a projective ko-scheme C endowed with an isomorphism of k-schemes

C ×ko k → C. We can think of such a model as the choice of a set of homogeneous equations for C inside some projective space such that the equations have coefficients in ko. It is clear that a model always exists, as we can multiply any system of equations for C by a suitable element in ko to force the coefficients to lie o in k . We call = × o k the generic fiber of the model, and = × o k Ck C k Cek C k e the special fiber. 18 JOHANNES NICAISE

(3.1.2) Our hope is that is easier to study than C but still gives us Cek valuable information about C. Why should be easier to understand? Cek For one thing, the residue field ek usually has a lower arithmetic complexity (a simpler Galois group) than k; in many applications, it is algebraically closed, or finite. Moreover, often breaks up into multiple irreducible Cek components which have a simpler geometry (lower genus) than C; then a part of the geometry of C is translated into the combinatorial structure of (the ways these components intersect). Of course, if we want to be able Cek to deduce anything about C from the study of , then we need to impose Cek some conditions on our model C . The most basic condition is that C is flat over ko. Since C is itself integral, this is equivalent to requiring C to be integral. In particular, C does not have any irreducible components that are concentrated in the special fiber (which would have no relation whatsoever with C). Thus we can think of C as a “continuous” family of curves over Spec ko. From now on, we will always include flatness in the definition of a model. Definition 3.1.3. Let k be a non-archimedean field with non-trivial absolute value. Let C be a smooth, projective, geometrically connected curve over k.A model of C is a flat projective ko-scheme C endowed with an isomorphism of k-schemes

C ×ko k → C. If C 0 is another such model, a morphism of models C 0 → C is a morphism of ko-schemes whose restriction to the generic fibers commutes with the isomorphisms to C. If C 0 and C are fixed, then there exists at most one morphism of models C 0 → C (since it is completely determined on the generic fibers, C 0 is ko-flat and C is separated). If it exists, we say that C 0 dominates C .

(3.1.4) Another useful condition is that the singularities of the model C are as mild as possible. What this means depends on the nature of the field k. The most obvious condition is regularity of the scheme C , but this only makes sense for noetherian schemes, and ko is noetherian if and only if k ∗ is discretely valued (that is, the value group |k | is isomorphic to Z). We can also consider the singularities of the special fiber . This leads to the Cek notion of (semi)stable model. These issues will be discussed in the following sections.

3.2. Resolution of singularities.

(3.2.1) Throughout Section 3.2 we will assume that k is discretely valued; thus ko is noetherian. Then the maximal ideal koo is principal, and any generator π is called a uniformizer of k. We fix such a uniformizer π for the remainder of the section. Good examples to keep in mind are • k = F ((π)), where F is any field; • k = Qp, in which case p is a uniformizer. NON-ARCHIMEDEAN GEOMETRY 19

(3.2.2) Let C be a regular model for C over ko, and let x be a point of . We say that has strict normal crossings at x if we can find a unit Cek Cek u and a regular system of local parameters z1, . . . , zr in the local ring OC ,x such that r Y Ni π = u (zi) i=1 in OC ,x, for some nonnegative integers N1,...,Nr. The set of points where has strict normal crossings is open in . We say that is a strict normal Cek Cek Cek crossings divisor on , or that is an snc-model, if has strict normal C C Cek crossings at every point. This is equivalent to saying that the irreducible components of are regular and intersect each other transversally. A Cek slightly weaker notion is that of normal crossings: we say that has Cek normal crossings at x if there exists an ´etalemorphism U → C whose image contains x such that has normal crossings at any point above x. Uek We say that is a normal crossings divisor on , or that is an nc- Cek C C model, if has normal crossings at every point. The main difference with Cek the strict normal crossings case is that we allow transversal self-intersections of components in the special fiber. We can always turn an nc-model into an snc-model by blowing it up at each of these self-intersection points.

(3.2.3) A regular, resp. nc, resp. snc-model C is called relatively minimal if every morphism to a model with the same property is an isomorphism. It is called minimal if every model with the same property admits a morphism of models to C . It is clear from the definition that minimal models are unique up to canonical isomorphism. More precisely, if a minimal model exists, then every relatively minimal model is canonically isomorphic to the minimal model. The first important result about the existence of “nice” models for curves is the following. Theorem 3.2.4. Assume that k is discretely valued, and let C be a smooth, projective, geometrically connected curve over k. (1) Let C be a normal model for C. Then there exists a morphism of models C 0 → C , obtained as a composition of normalized point blow- ups, such that C 0 is an snc-model. (2) Any morphism of regular models C 0 → C can be written as a composition of point blow-ups. (3) The curve C has relatively minimal regular, nc and snc-models. If the genus of C is positive, then C has minimal regular, nc and snc- models. Proof. See [Lipman, Desingularization of two-dimensional schemes] and [Lichtenbaum, Curves over discrete valuation rings]. Note that we can apply resolution of singularities here because ko is excellent. The rough idea of the proof is that we start from any model, prove that it becomes snc after a finite number of normalized point blow-ups, and then make it relatively minimal by contracting suitable rational curves, using a variant of Castelnuovo’s criterion.  1 Example 3.2.5. We will show that C = Pko is a relatively minimal regular 1 0 1 model of Pk. Let C be a regular model of Pk such that there exists a 20 JOHANNES NICAISE morphism of models h : C → C 0. Then h is an isomorphism on the generic fibers, by the definition of a morphism of models. The image of h is closed by properness, so that h is surjective; then h is finite, because h cannot contract the special fiber. But C 0 is normal, so that h is an isomorphism. 1 However, C is not a minimal regular model of Pk, because the 1 1 automorphism (x : y) 7→ (x : πy) of Pk does not extend to Pko . Thus 1 Pk has no minimal regular model. Example 3.2.6. The minimal regular models of elliptic curves C over k have been classified by N´eroninto ten types (Kodaira had earlier given a similar classification for complex elliptic fibrations over a curve). See for instance [Silverman, Advanced topics in the arithmetic of elliptic curves]. From this classification, one can easily compute the minimal snc-models by repeatedly blowing up the points where the special fiber is not snc. In order to determine the type of a smooth plane cubic, one can apply Tate’s algorithm to its equation. For instance, if C is the plane cubic over k defined by the affine equation y2 = x3 + π then the minimal regular model of C is the closed subscheme C 2 of Pko defined by the same affine equation. It is not an nc-model, because the special fiber has a cuspidal singularity at the origin (it is defined Cek by y2 = x3). We need three point blow-ups to turn C into an snc-model. In the Kodaira-N´eronclassification, the curve C has reduction type II (in Kodaira’s notation). If ek has zero there is a similar classification for genus 2 curves, but the number of types if already much larger.

3.3. The stable reduction theorem.

(3.3.1) We drop the assumption that k is discretely valued. Let C be a smooth, projective, geometrically connected curve over k. We fix an algebraic closure eka of ek. A model C of C is called semi-stable if := × ka Ceka Cek e is reduced and has at most ordinary double points as singularities. We say that the model is strictly semi-stable if, moreover, the irreducible components of are regular. A semi-stable model is called stable if Ceka C the automorphism group of is finite; this is equivalent to saying that Ceka the arithmetic genus of is at least 2 and every non-singular rational Ceka component of meets the other components in at least three points. We Ceka say that C has semi-stable, resp. strictly semi-stable, resp. stable reduction if there exists a semi-stable, resp. strictly semi-stable, resp. stable model.

(3.3.2) If k is not algebraically closed, it is not reasonable to expect that C always has a semi-stable model. For if C is a semi-stable model, its ko- smooth locus is non-empty, and every ek-rational smooth point on Ck lifts to a k-rational point on C because ko is henselian. However, it may very well happen that C(k) is empty and ek is algebraically closed, in which case C cannot have a semi-stable model over ko. The following theorem states that semi-stable models do exist after a finite extension of the base field k. NON-ARCHIMEDEAN GEOMETRY 21

Theorem 3.3.3. Let C be a smooth, projective, geometrically connected curve over k, of genus g. 0 0 • There exists a finite separable extension k of k such that C ×k k has strictly semi-stable reduction. • If g ≥ 2 then there exists a finite separable extension k0 of k such 0 that C ×k k has stable reduction. • If k is discretely valued, then there exists a finite separable extension k0 of k such that every relatively minimal snc-model of C is strictly semi-stable. Proof. The discretely valued case was already proven by Deligne and Mumford in their famous paper on moduli spaces of curves. Here the advantage is that we already have a candidate to work with, namely, a relatively minimal snc-model. Thus we need a criterion that guarantees that this model is semi-stable, and then check that this criterion is satisfied after base change to a suitable finite separable extension of k. Several approaches have appeared in the literature, but the most natural one is probably the one due to T. Saito: if k has algebraically closed residue field and g ≥ 2 (or g = 1 and C(k) is non-empty) then the minimal nc-model C of C is semi-stable if and only if every element of the Galois group of k acts unipotently on the degree one `-adic cohomology of C. This is always true after base change to a finite separable extension k0 by Grothendieck’s Mondromy Theorem. If C is not snc, then blowing up the self-intersection points, performing an additional degree 2 base change and resolving the singularities, we can also find a strictly semi-stable model. The proof of the general case is quite different and uses non-archimedean geometry. The first proof was given by Bosch and L¨utkebohmert in the framework of rigid geometry. Proofs using Berkovich spaces were later given by M. Temkin and Ducros. One first reduced to the case where k is algebraically closed by an approximation/descent argument. The main idea is then that we can produce a model by constructing a suitable cover of Can by so-called affinoid domains. If these affinoid domains have the right shape, the model will be semi-stable. The key point is thus to prove the existence of such a cover. This can be done by choosing a finite morphism 1 from C to Pk and making a careful study of the ramification locus. 

Imperial College London E-mail address: [email protected]