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 absolute value 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 real number 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 nonarchimedean 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):
k o = {x ∈ k | |x| ≤ 1}
k oo = {x ∈ k | |x| < 1}
e
k = k o /k oo
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Exercise 1.1.6. Show that k o is a valuation ring 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 k o .
Show that k oo is the unique maximal ideal in k o ; thus e
k is a field.
We call k o the valuation ring of k and e
k the residue field of k. The set
of absolute values of non-zero elements in k is a subgroup of (R>0 , ·),
called the value group of k.
|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 nonarchimedean field, with k = k o = e
k and k oo = (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
and |x| = εordt (x) 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 k o = F [[t]], k oo = (t) and e
k = 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
|0| = 0 and |x| = εordp (x) 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
are different as absolute values. We have k o = Zp , k oo = (p) and
e
k = 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 k 0 of k (here we use the
assumption that k is complete). The field k 0 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 k 0 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
[
ka =
C((t1/n ))
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 k a
and determine the completion of k a . If k = Qp and k a is an algebraic
ca is denoted by Cp and called the field of
closure, the completion k
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).
• Archimedean: | · |r∞ 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? Nonarchimedean 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
polynomial 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-toglobal 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
f : X → A1C = Spec C[t]
such that f −1 (1) is isomorphic to the curve X and X0 = f −1 (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).
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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 X an (we
will recall the construction below). The important points about this process
are the following:
(1) We can apply new tools to X an 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 X an .
This is guaranteed by Serre’s GAGA principle (GAGA=géométrie
algébrique et géométrie analytique). 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 X an , 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 naı̈ve definition of analytic manifold over a non-archimedean field.
(1.3.2) Let us briefly recall the construction of X an . 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
a closed subvariety into some affine space AnC . Then U is described by
polynomial equations inside AnC and the same equations define a complex
manifold U an inside Cn . Gluing these together we get X an .
(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
f an : ι−1 (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 OX an on X(C). One can show that the locally ringed
space X an = (X(C), OX an ) 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 k n , we
call a function V → k analytic if it can locally be written as a converging
power series in the coordinates on k n 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
pair of the form (V, OV ) with V an open in k n 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 welldefined, 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.
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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 naı̈ve
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 nonarchimedean 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 nonempty 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 k oo in k o ; 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 Lütkebohmert 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 naı̈ve definition. Adding these points gives a more solid
structure to the manifold and gives rise to a satisfactory theory of nonarchimedean 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.
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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
Berkovich analytification X an 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 X an 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 X an .
(2.2.2) We can endow X an 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
ι : X an → 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
|f | : ι−1 (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 X an 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 X an as the
weakest topology such that the function |f | : X an → 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 X an 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 X an , we define the residue field H (x) of X an
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
f(x) (notice the clash of terminology here:
ideal H (x)oo and residue field H
we consider the residue field of the residue field at x).
(2.2.6) Finally, we can define a sheaf of analytic functions on X an . An
analytic function f on an open V of X an 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 X an that is called the
structure sheaf of X an and denoted by OX an .
(2.2.7) If g : Y → X is a morphism of k-schemes of finite type then it
induces a natural continuous map g an : Y an → X an and a local morphism
of sheaves in k-algebras
OX an → g∗an 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 g an 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 X an 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
ι−1 (ξ) is empty except when ξ is closed. Hence, X an = 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 X an . A fundamental
complication (which also arises in complex analytic geometry) is that the
algebras of analytic functions on open subsets of X an 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 OX an by also
considering analytic functions on certain closed subsets of X an (for instance,
the closed unit disc in A1,an
k ). Doing this in a systematic way requires the
introduction of a Grothendieck topology, called the G-topology, where such
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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 X an is
locally compact and locally path-connected. Moreover, we have the following
equivalences:
(1) X is connected if and only if X an is connected;
(2) X is separated if and only if X an is Hausdorff;
(3) X is proper if and only if X an 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 X an is
locally contractible.
If X is smooth over k, this had previously been proven by Berkovich using
de Jong’s theory of alterations.
(2.3.3) As a first example, we will describe the topological space Pk1,an ,
the Berkovich analytification of the projective line. One can show that,
for any k-scheme X of finite type and every algebraic closure k a of k, the
ca )an and the space X is
absolute Galois group Gal(k a /k) acts on (X ×k k
canonically homeomorphic to the quotient space of this action. This reduces
the problem to the case where k = k a ; 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
ι : P1,an
→ P1k
k
over any closed point of P1k is a singleton, consisting of the closed point
together with the given absolute value on its residue field k. Thus P1k (k) =
k ∪ {∞} injects naturally into Pk1,an . It remains to classify the absolute
values on the function field k(T ) that extend the absolute value on k, and
to describe the topology on the space Pk1,an . Note that we can identify Ak1,an
with P1,an
\ {∞}.
k
2.4. The projective line – trivially valued case.
(2.4.1) We first consider the case where the absolute value on k is trivial.
Then P1,an
has a canonical point (invariant under all automorphisms of P1k ),
k
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 rational function at a. More
precisely, if a ∈ k then we can write every non-zero element R(T ) of k(T )
as (T − a)v Q(T ) where Q(T ) is a rational function without zero or pole at
a; then orda R(T ) = v. If a = ∞ then we set orda R(T ) = −degR(T ). This
valuations defines a family of absolute values | · |a,ε on k(T ), with ε ∈ (0, 1),
by setting
|R(T )|a,ε = εorda R(T )
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 polynomials 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 .
(2.4.3) We denote by ηa,ε the point of Pk1,an 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
of k ∪ {∞}, sit in the topological space Pk1,an . 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
γa : [0, 1) → X an : ε 7→ ηa,ε
is a homeomorphism onto its image, and this image is an open subset of
X an . A subset S of P1,an
is open if and only if it satisfies the following
k
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 ).
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JOHANNES NICAISE
Proof. For notational convenience, we assume that a 6= ∞; this can always
be arranged by means of an automorphism of P1k . The map γa is obviously
injective. It is very easy to check that ι ◦ γa : [0, 1) → P1k is continuous, and
that
|f | ◦ γa : γa−1 (U ) → R
is continuous for every Zariski-open U in P1k and every regular function f
on U . Thus γa is continuous. It is also open, because for all real numbers
δ1 < δ2 ≤ 1, γa maps (δ1 , δ2 ) ∩ [0, 1) to the open subset of A1,an
defined by
k
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
V = {x ∈ Ak1,an | 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
on P1,an
k . The induced topology on k ∪ {∞} coincides with the metric
topology on P1k (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
\ {∞} is not compact; this reflects the
= P1,an
exercise). Observe that A1,an
k
k
lack of properness of A1k .
Exercise 2.4.6. Describe the k-algebras of analytic functions on the opens
of P1,an
k .
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),
and the picture is quite similar to the one for Pk1,an . 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
complicated. The points of P1,an
have been classified by Berkovich into four
k
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
f (T ) =
ci (T − a)i
i=0
where the coefficients ci belong to k, and we set
|f (T )|a,r = max |ci |ri .
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
ci (1/T )i |∞,r = max |ci |ri .
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 )
as c0 + . . . + cd T d , this means that maxi |ci | = 1. We must show that there
exists an element z of k o = 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 k oo is
non-zero. Let z be any element of k o such that its reduction modulo k oo is
not a root of fe. Then f (z) does not lie in k oo , and thus |f (z)| = 1.
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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
|z − a0 | = |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
only if r = r0 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
denote by ηa,r the point of P1,an
corresponding to |·|a,r . We also set ηa,0 = a.
k
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 nonarchimedean 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 ∩n Bn
gives rise to a new kind of absolute value, characterized by
|f (T )|B = inf |f (T )|an ,rn
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
in Bn . We denote the corresponding point of P1,an
by ηB , and we call ηB a
k
point of type 4. We are now ready to state Berkovich’s classification theorem
for points on P1,an
k .
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 )|B = inf |f (T )|an ,rn
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
is of this form. If B 0 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 ⊂ B and B ⊂ B 0 . We will express
can find m ≥ 0 such that Bm
n
m
n
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 ∈ ∩n Bn , 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
B 0 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 B 0 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 ⊂ B .
element of N. By symmetry, it suffices to find m ∈ N such that Bm
n
Choose an element a in k such that |T − a|B < rn . Then when m is large
0 < rn , where
enough, we must also have |T − a|am ,rm < rn and |T − a|a0m ,rm
0
+
0
0
0
we wrote Bm = B (am , rm ). This implies that rm < rn and |a0m − am | < rn ,
0 is contained in B .
so that Bm
n
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.
Corollary 2.5.9. Every point of Pk1,an is of type 1, 2, 3 or 4 (and these
types are disjoint).
(2.5.10) The division of the points of P1,an
into four types is also reflected
k
in their residue fields. Let x be a point of P1,an
k . 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
f(x) ∼
field H
k(u). If x = ηa,r is of type 3, then |H (x)∗ | is the subgroup of
=e
f(x) = e
(R>0 , ·) generated by |k ∗ | and r, and H
k. 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
of P1,an
k . Type 4 requires some thought; have another look at the proof of
Proposition 2.5.2.
(2.5.12) To conclude our example, we study the topology on P1,an
k . 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.
as a projective limit of finite
Alternatively, we can also construct P1,an
k
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.
(2.6.1) The analytification of the projective line Pk1,an 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
in analogy with classical potential theory over the complex numbers
P1,an
k
(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
part of P1,an
that lies over the generic point of P1k (that is, the complement
k
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
fc : C → C : x 7→ z d + c.
A complex number a is called a preperiodic point of fc if the set of iterates
{fc(n) (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.
It is obvious that, when ad = bd , 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
both preperiodic for fc , then ad = bd . This innocent-looking statement
is surprisingly difficult to prove. The proof of Baker and Demarco uses
potential theory on the projective line Pk1,an 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
valued then the structure of C an is quite similar to that of Pk1,an : 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 C an , but C an is not
always contractible. Describing the precise structure of C an 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 k o of k. Such a model is a
projective k o -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 k o . It is clear that a model always exists, as we can multiply any system
of equations for C by a suitable element in k o to force the coefficients to lie
in k o . We call Ck = C ×ko k the generic fiber of the model, and Cek = C ×ko e
k
the special fiber.
18
JOHANNES NICAISE
(3.1.2) Our hope is that Cek is easier to study than C but still gives us
valuable information about C. Why should Cek be easier to understand?
For one thing, the residue field e
k usually has a lower arithmetic complexity
(a simpler Galois group) than k; in many applications, it is algebraically
closed, or finite. Moreover, Cek often breaks up into multiple irreducible
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
Cek (the ways these components intersect). Of course, if we want to be able
to deduce anything about C from the study of Cek , then we need to impose
some conditions on our model C . The most basic condition is that C is
flat over k o . 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 k o . 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 k o -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 k o -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 k o -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 k o 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 Cek . This leads to the
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 k o is noetherian. Then the maximal ideal k oo 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 k o , and let x be a point of
Cek . We say that Cek has strict normal crossings at x if we can find a unit
u and a regular system of local parameters z1 , . . . , zr in the local ring OC ,x
such that
r
Y
π = u (zi )Ni
i=1
in OC ,x , for some nonnegative integers N1 , . . . , Nr . The set of points where
Cek has strict normal crossings is open in Cek . We say that Cek is a strict normal
crossings divisor on C , or that C is an snc-model, if Cek has strict normal
crossings at every point. This is equivalent to saying that the irreducible
components of Cek are regular and intersect each other transversally. A
slightly weaker notion is that of normal crossings: we say that Cek has
normal crossings at x if there exists an étale morphism U → C whose
image contains x such that Uek has normal crossings at any point above x.
We say that Cek is a normal crossings divisor on C , or that C is an ncmodel, if Cek has normal crossings at every point. The main difference with
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 blowups, 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 sncmodels.
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 k o 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.
Example 3.2.5. We will show that C = P1ko is a relatively minimal regular
model of P1k . Let C 0 be a regular model of P1k 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.
However, C is not a minimal regular model of P1k , because the
automorphism (x : y) 7→ (x : πy) of P1k does not extend to P1ko . Thus
P1k has no minimal regular model.
Example 3.2.6. The minimal regular models of elliptic curves C over k
have been classified by Néron into 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
y 2 = x3 + π then the minimal regular model of C is the closed subscheme C
of P2ko defined by the same affine equation. It is not an nc-model, because
the special fiber Cek has a cuspidal singularity at the origin (it is defined
by y 2 = x3 ). We need three point blow-ups to turn C into an snc-model.
In the Kodaira-Néron classification, the curve C has reduction type II (in
Kodaira’s notation).
If e
k has characteristic 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 e
k a of e
k. A model C of C is called semi-stable if
Ceka := Cek × e
ka
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 Ceka are regular. A semi-stable model C is called stable if
the automorphism group of Ceka is finite; this is equivalent to saying that
the arithmetic genus of Ceka is at least 2 and every non-singular rational
component of Ceka meets the other components in at least three points. We
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 k o smooth locus is non-empty, and every e
k-rational smooth point on Ck lifts to
a k-rational point on C because k o is henselian. However, it may very well
happen that C(k) is empty and e
k is algebraically closed, in which case C
cannot have a semi-stable model over k o . 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.
• There exists a finite separable extension k 0 of k such that C ×k k 0
has strictly semi-stable reduction.
• If g ≥ 2 then there exists a finite separable extension k 0 of k such
that C ×k k 0 has stable reduction.
• If k is discretely valued, then there exists a finite separable extension
k 0 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 k 0 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 Lütkebohmert 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
C an 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
from C to P1k and making a careful study of the ramification locus.
Imperial College London
E-mail address: j.nicaise@imperial.ac.uk