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A non-archimedean Ax-Lindemann theorem
Antoine Chambert-Loir
Univ. Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive
Gauche, UMR 7586, F-75013, Paris, France
E-mail: [email protected]
François Loeser
Sorbonne Universités, UPMC Univ Paris 06, UMR 7586 CNRS, Institut Mathématique de
Jussieu-Paris Rive Gauche, F-75005, Paris, France
E-mail: [email protected]
À Daniel Bertrand, en témoignage d’amitié
1. Introduction
1.1. — The classical Lindemann–Weierstrass theorem states that if algebraic numbers α1 , . . . , αn are Q-linearly independent, then their exponentials
exp(α1 ), . . . , exp(αn ) are algebraically independent over Q. More generally, if
α1 , . . . , αn are complex numbers which are no longer assumed to be algebraic,
Schanuel’s conjecture predicts that the field Q(α1 , . . . , αn , exp(α1 ), . . . , exp(αn ))
has transcendence degree at least n over Q. In [2], Ax established power series and
differential field versions of Schanuel’s conjecture. In particular the part of Ax’s
results corresponding to the Lindemann–Weierstrass theorem can be recasted into
geometrical terms as follows:
Theorem 1.2 (Exponential Ax-Lindemann). — Let exp : Cn → (C× )n be the
morphism (z1 , . . . , zn ) 7→ (exp(z1 ), . . . , exp(zn )). Let V be an irreducible algebraic
subvariety of (C× )n and let W be an irreducible component of a maximal algebraic
subvariety of exp−1 (V ). Then W is geodesic, that is, W is defined by a finite family
P
of equations of the form ni=1 ai zi = b with ai ∈ Q and b ∈ C.
In the breakthrough paper [23], Pila succeeded in providing an unconditional
proof of the André-Oort conjecture for products of modular curves. One of his main
ingredients was to prove an hyperbolic version of the above Ax-Lindemann theorem,
which we now state in a simplified version.
Let H denote the complex upper half-plane and j : H → C the elliptic modular
function. By an algebraic subvariety of Hn we shall mean the trace in Hn of an
algebraic subvariety of Cn . An algebraic subvariety of Hn if said to be geodesic
if it is defined by equations of the form zi = ci and zk = gk` z` , with ci ∈ C and
gk` ∈ GL+
2 (Q).
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ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
Theorem 1.3 (Hyperbolic Ax-Lindemann). — Let j : Hn → Cn be the morphism (z1 , . . . , zn ) 7→ (j(z1 ), . . . , j(zn )). Let V be an irreducible algebraic subvariety
of Cn and let W be an irreducible component of a maximal algebraic subvariety
of j −1 (V ). Then W is geodesic.
Pila’s method to prove this Ax-Lindemann theorem is quite different from the
differential approach of Ax. It follows a strategy initiated by Pila and Zannier in their
new proof of the Manin-Mumford conjecture for abelian varieties [27]; that approach
makes crucial use of the bound on the number of rational points of bounded height
in the transcendental part of sets definable in an o-minimal structure obtained by
Pila and Wilkie in [26]. Recently, still using the Pila and Zannier strategy, Klingler,
Ullmo and Yafaev have succeeded in proving a very general form of the hyperbolic
Ax-Lindemann theorem valid for any arithmetic variety ([16], see also [30] for the
compact case).
1.4. — In the recent paper [8], Cluckers, Comte and Loeser established a nonarchimedean analogue of the Pila-Wilkie theorem of [26] in its block version of [22].
The purpose of this paper is to use this result to prove a version of Ax-Lindemann for
products of algebraic curves admitting a non-archimedean uniformization and whose
corresponding Schottky group is “arithmetic” and has rank at least 2 (theorem 2.7).
In particular, this theorem applies for products of Shimura curves admitting a p-adic
uniformization à la Cherednik-Drinfeld (see section 3).
The basic strategy we use is strongly inspired by that of Pila [23] (see also [24]),
though some new ideas are required in order to adapt it to the non-archimedean
setting. Similarly as in Pila’s approach one starts by working on some neighborhood of the boundary of our space (which, instead of a product of Poincaré upper
half-planes, is a product of open subsets of the Berkovich projective line). Analytic
continuation and monodromy arguments are replaced by more algebraic ones and
explicit matrix computations by group theory considerations. We also take advantage of the fact that Schottky groups are free and of the geometric description of
their fundamental domains.
To conclude, let us note that there are cases where p-adic analogues of theorems
in transcendental number theory seem to require other methods than those used to
prove their complex counterparts. For instance, it is still an open problem to prove
a p-adic analogue, for values of the p-adic exponential function, of the classical
Lindemann-Weierstrass theorem.
Since his first works (see, for example, [4]), Daniel Bertrand has shown deep
insight into p-adic transcendental number theory, and disseminated his vision within
the mathematical community. We are glad that the Schwarzian derivative, which is
so dear to his heart, plays a role here, and we are pleased to dedicate this paper to
him.
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
3
Acknowledgements. — The research leading to this paper was initiated during
the 2014 MSRI program “Model Theory, Arithmetic Geometry and Number Theory”. We would like to thank the MSRI for its congenial atmosphere and hospitality. This research was partially supported by ANR-13-BS01-0006 (Valcomo) and by
ANR-15-CE40-0008 (Défigéo). The second author was partially supported by the
European Research Council under the European Community’s Seventh Framework
Programme (FP7/2007-2013)/ERC Grant Agreement nr. 246903 NMNAG and the
Institut Universitaire de France.
During the early stage of this project, the second author benefited from stimulating discussions with Barry Mazur at MSRI, to whom we express our heartfelt thanks.
We are also grateful to Daniel Bertrand for continuous support and encouragement.
The comments of Yves André, Jean-François Boutot, Zoé Chatzidakis, Antoine
Ducros, Florent Martin and Jonathan Pila helped us to improve this paper; we
thank them warmly.
2. Statement of the theorem
2.1. Non-archimedean analytic spaces. — Given a complete non-archimedean
valued field F , we shall consider in this paper F -analytic spaces in the sense of
Berkovich [3]. However, the statements, and essentially the proofs, can be carried
on mutatis mutandis in the rigid analytic setting. In this context, there is a notion
of irreducible component (see [14], or [11] for the rigid analytic version).
If V is an algebraic variety over F , we denote by V an the corresponding F -analytic
space. It canonically contains V (F ) as a closed subset.
2.2. Schottky groups. — Let p be a prime number and let F be a finite extension
of Qp . The group PGL(2, F ) acts by homographies on the F -analytic projective
line (P1 )an . In the next paragraphs, we recall from [15] a few definitions concerning
Schottky groups in PGL(2, F ) and their limit sets and the associated uniformizations
of algebraic curves.
One says that a discrete subgroup Γ of PGL(2, F ) is a Schottky group if it is
finitely generated, and if no element (6= id) has finite order [15, I, (1.6)]. If Γ is a
Schottky group, then Γ is free; moreover, any discrete finitely generated subgroup
possesses a normal subgroup of finite index which is a Schottky group, [15, I, (3.1)].
We say that Γ is arithmetic if its elements can be represented by matrices whose
coefficients lie in a number field. Since Γ is finitely is generated, there exists a
number field K ⊂ F such that Γ ⊂ PGL(2, K).
2.3. Limit sets. — Let Γ be a Schottky subgroup of PGL(2, F ). Its limit set is
the set LΓ of all points in (P1 )an of the form limn (γn · x), where (γn ) is a sequence
of distinct elements of Γ and x ∈ (P1 )an , [15, I, (1.3)]. By [15, I, (1.6)], the limit
set LΓ is a compact subset of P1 (F ). If the rank of Γ is at least 2, then LΓ is a
perfect subset of P1 (F ), see [15, I, (1.6.3) and (1.7.2)].
Let ΩΓ = (P1 )an \ LΓ ; it is a Γ-invariant open set of (P1 )an .
4
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
2.4. Quotients. — Let us assume that Γ is a Schottky group and let g be its
rank. The group Γ acts freely on ΩΓ , and the quotient space ΩΓ /Γ is naturally
an F -analytic space so that the projection pΓ : ΩΓ → ΩΓ /Γ is topologically étale.
Moreover, ΩΓ /Γ is the F -analytic space associated with a smooth, geometrically
connected, projective F -curve XΓ of genus g [15, III, (2.2)].
2.5. — Let us now consider a finite family (Γi )16i6n of Schottky subgroups of
Q
Q
PGL(2, F ) of rank > 2. Let us set Ω = ni=1 ΩΓi and X = ni=1 XΓi , and let
p : Ω → X an be the morphism deduced from the morphisms pΓi : ΩΓi → XΓani .
2.6. Flat subvarieties. — Let W be a closed analytic subspace of Ω.
We say that W is irreducible algebraic if there exists an F -algebraic subvariety Y
of (P1 )n such that W is an irreducible component of the analytic space Ω ∩ Y an .
We say that W is flat if it can be defined by equations of the following form:
(1) zi = c, for some i ∈ {1, . . . , n} and c ∈ Ω;
(2) zj = g · zi , for some pair (i, j) of elements of {1, . . . , n} and g ∈ PGL(2, F ).
Then, W is irreducible algebraic.
We say that W is geodesic if, moreover, the elements g in (2) can be taken such
that gΓi g −1 and Γj are commensurable (ie, their intersection has finite index in both
of them).
Here is the main result of this paper.
Theorem 2.7 (Non-archimedean Ax-Lindemann theorem)
Let F be a finite extension of Qp and let (Γi )16i6n be a finite family of
arithmetic
Q
Schottky subgroups of PGL(2, F ) of rank > 2. As above, let us set Ω = ni=1 ΩΓi and
Q
X = ni=1 XΓi , and let p : Ω → X an be the morphism deduced from the morphisms
pΓi : ΩΓi → XΓani .
Let V be an irreducible algebraic subvariety of X and let W ⊂ Ω be an irreducible
component of a maximal algebraic subvariety of p−1 (V an ). Then W is flat.
The proof of this theorem is given in section 7; it follows the strategy of Pila–
Zannier. In the archimedean setting, this strategy relies crucially on a theorem of
Pila–Wilkie about rational points on definable sets; we recall in section 4 the nonarchimedean analogue of this theorem, due to Cluckers, Comte and Loeser (see [8]),
which is used here. In section 6, we recall the reader a few more facts on p-adic
Schottky groups and p-adic uniformization, essentially borrowed from the book [15].
3. The example of Shimura curves
3.1. Complex Shimura curves. — Let B be a quaternion division algebra with
center Q; we assume that it is indefinite, namely B ⊗Q R ' M2 (R). Let H
be the algebraic group of units of B, modulo center, defined by H(R) = (B ⊗Q
R)× /Z((B ⊗Q R)× ) for every Q-algebra R. In particular, the group H(R) is isomorphic to PGL(2, R); and we fix such an isomorphism. Then the group G(R) acts
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
5
by homographies on the double Poincaré upper half-plane
h± = C \ R.
Let then OB be a maximal order of B, that is a maximal sub-algebra of B which is
isomorphic to Z4 as a Z-module. Let also ∆ be a congruence subgroup of H(OB ),
small enough so that the stabilizer of every point of h± is trivial. The quotient
h± /∆ has a natural structure of a compact Riemann surface and the projection
p : h± → h± /∆ is an étale covering.
This curve parameterizes triples (V, ι, ν), where V is a complex two dimensional
abelian variety, ι : OB → End(V ) is a faithful action of OB on V and ν is a level
structure “of type ∆” on A. When ∆ is the kernel of H(OB ) to H(OB /N ), for some
integer N > 1, such a level structure corresponds to an equivariant isomorphism
of VN , the subgroup of N -torsion of V , with OB /N .
It admits a canonical structure of an algebraic curve S which can be defined over
a number field E in C.
3.2. p-adic uniformization of Shimura curves. — Let p be a prime number at
which B ramifies, which means that B ⊗Q Qp is a division algebra. Let also F be a
completion of the field E at a place dividing p; we denote by Cp a p-adic completion
of an algebraic closure of F
Let Ω = (P1 )an
F \ P1 (Qp ) be the extension of scalars to F of Drinfeld’s upper half
plane. According to the theorem of Cherednik-Drinfeld ([7, 13]; see also [5] for a
detailed exposition), and up to replacing F by a finite unramified extension, the
F -analytic curve S an admits a “p-adic uniformization” which takes the form of a
surjective analytic morphism
j : Ω → S an ,
which identifies S an with the quotient of Ω by the action of a subgroup Γ
of PGL(2, Qp ). Up to replacing ∆ by a smaller congruence subgroup, which
replaces S by a finite (possibly ramified) covering, we may also assume that Γ is a
p-adic Schottky subgroup acting freely on Ω, and that j is topologically étale.
Let us describe these subgroups. Let A be the quaternion division algebra over Q
with the same invariants that B, except for those invariants at p and ∞ which are
switched. In particular, A ⊗Q R is Hamilton’s quaternion algebra, while A ⊗Q Qp '
M2 (Qp ). Let G be the algebraic group of units of A, modulo center, defined by
G(R) = (A ⊗Q R)× /Z((A ⊗Q R)× )
for every Q-algebra R. In particular, G(Qp ) ' PGL(2, Qp ) and the discrete subgroup Γ is the intersection of G(Q) with a compact open subgroup of G(Af ), the
adelic group associated with G where the place at ∞ is omitted.
Lemma 3.3. — The group Γ is conjugate to an arithmetic Schottky subgroup
in PGL(2, Qp ), its rank is at least 2, and its limit set is equal to P1 (Qp ).
Proof. — The group Γ is a discrete subgroup of PGL(2, Qp ) hence its limit set LΓ
is a Γ-invariant subset of P1 (Qp ). In other words, the Drinfeld upper half-plane Ω =
6
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
an
Pan
1 \ P1 (Qp ) is an open subset of ΩΓ = P1 \ LΓ . By the theory of Mumford curves,
an
the analytic curve (P1 \LΓ )/Γ is algebraic, and admits the analytic curve S an = Ω/Γ
as an open subset. According to the Cherednik-Drinfeld theorem, the curve S an is
projective. This implies that Ω = Pan
1 \ LΓ , hence LΓ = P1 (Qp ).
After base change to Qp , the algebraic Q-group G becomes isomorphic to
PGL(2)Qp . Consequently, there exists a finite algebraic extension K of Q, contained in Qp , such that GK ' PGL(2)K . By such an isomorphism, G(Q) is mapped
into PGL(2, K); this implies that the group Γ is conjugate to an arithmetic group.
Since Γ is a Schottky group, it is free. Since it is non-abelian, its rank is at
least 2.
By this lemma, the following result is a special case of our main theorem 2.7.
Theorem 3.4. — Let F be a finite extension of Qp , let Ω = Pan
1 \ P1 (Qp ) and
let j : Ωn → S an be the Cherednik–Drinfeld uniformization of a product of Shimura
curves. Let V be an irreducible algebraic subvariety of S and let W ⊂ Ωn be an
irreducible component of a maximal algebraic subvariety of j −1 (V an ). Then W is
flat.
3.5. — By the same arguments, one can show that our main theorem 2.7 also
applies to the uniformizations of Shimura curves associated with quaternion division
algebras over totally real fields, as considered by Cherednik [7] and Boutot–Zink [6].
3.6. — As suggested by J. Pila and explained to us by Y. André, theorem 3.4 can
also be deduced from its complex analogue, which is a particular case of [30]. The
crucial ingredient is a deep theorem of André ([1], III, 4.7.4) stating that the p-adic
uniformization and the complex uniformization of Shimura curves satisfy the same
non-linear differential equation. His proof relies on a delicate description of the
Gauss-Manin equation in terms of convergent crystals and on the tempered fundamental group introduced by him. From that point on, one can apply Seidenberg’s
embedding theorem [29] in differential algebra to prove that both the complex and
non-archimedean Ax-Lindemann theorems are equivalent to a single statement in
differential algebra, in the original spirit of Ax’s paper [2].
4. Definability — A p-adic Pila-Wilkie theorem
4.1. — There are two distinct notions of p-adic analytic geometry, one is “naïve”,
and the other one is rigid analytic or Berkovich geometry. These two notions give
rise to two notions of subanalytic sets, and we shall use both in this paper.
a) Semialgebraic and subanalytic subsets of Qnp are defined by Denef and van den
Dries in [12]; see also [8, p. 26].
b) In [17], Lipshitz defined a notion of rigid subanalytic subset of Cnp . We shall
use in this paper the variant ([18], definition 2.1.1) where the coefficients of all
polynomials and power series involved belong to a fixed finite extension F of Qp .
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
7
Considering affine charts, all three classes of sets can be defined in algebraic
varieties. They are stable under boolean operations and projections (corollary 4.3
of [19]), admit cell decompositions (theorem 7.4 of [9]) and a natural notion of
dimension (in fact, they are b-minimal in the sense of [10]).
We shall use the fact that if Z is a rigid subanalytic subset of Cnp , then Z(F ) =
Z ∩ Qnp is a subanalytic subset of Qnp . Indeed, Z can be defined by a quantifierfree formula of the above-mentioned variant of Lipshitz’s analytic language, and our
claim follows from the very definition of this language.
4.2. — Let F be a finite extension of Qp . A block in Qnp is either empty, or a
singleton, or a smooth subanalytic subset of pure dimension d > 0 which is contained
in a smooth semialgebraic subset of dimension d.
A family of blocks in Qnp × Qsp is a subanalytic subset W such that there exists
an integer t > 0 and a semialgebraic set Z ⊂ Qnp × Qtp such that for every σ ∈ Qsp ,
thre exists τ ∈ Qtp such that Wσ and Zτ are smooth of the same dimension, and
Wσ ⊂ Zτ . (In particular, the sets Wσ , for σ ∈ Qsp , are blocks in Qnp .)
4.3. Weil restriction. — Let F be a finite extension of Qp , of degree d. The Weil
restriction functor maps the affine line A1F over F to AdQp , identifying F n with Qdn
p .
This gives rise to a notion of semialgebraic or subanalytic subsets of F n . Observe
that if Z is a rigid subanalytic subset of Cnp , then Z(F ) is a subanalytic subset of F n .
This also leads to a notion of blocks in F n , or to a family of blocks in F n × Qsp .
4.4. — Let H be the standard height function on Q; for x ∈ Q, written as a
fraction a/b in lowest terms, one has H(x) = max(|a| , |b|). We also write H for the
n
height function on Q defined by H(x1 , . . . , xn ) = maxi (H(xi )). Viewing GL(d, Q)
d2
as a subspace of Q , it defines a height function on GL(d, Q). There exists a strictly
positive real number c such that H(gg 0 ) 6 cH(g)H(g 0 ) for every g, g 0 ∈ GL(d, Q),
and H(g −1 ) H(g)c for every g ∈ SL(d, Q). When d = 2, one even has H(g −1 ) =
H(g).
By abuse of language, if G is a linear algebraic Q-group, we implicitely choose an
embedding in some linear group, which furnishes a height function on G(Q). The
actual choice of this height function depends on the chosen embedding, any other
height function H 0 is equivalent, in the sense that there is a strictly positive real
number c such that H(x)1/c H 0 (x) H(x)c for every x ∈ G(Q).
4.5. — Let Z be a subset of F n and let K be finite extension of Q contained in F .
We write Z(K) = Z ∩ K n (K-rational points of Z). For every real number T ,
we define Z(K; T ) = {x ∈ Z(K) ; H(x) 6 T }; for every integer D, we also define
Z(D; T ) to be the set of points x ∈ Z(F ) such that [Q(xi ) : Q] 6 D for every
i ∈ {1, . . . , n} and H(x) 6 T . These are finite sets.
We say that Z has many K-rational points if there exist strictly positive real
numbers c, α such that
Card Z(K; T ) > cT α
8
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
for all T > 1. This notion only depends on the equivalence class of the height.
4.6. — In [8], Cluckers, Comte and Loeser established a p-adic analogue of a theorem of Pila-Wilkie [26] concerning the rational points of a definable set. We will
use the following variant of [8, Theorem 4.2.3].
Theorem 4.7. — Let F be a finite extension of Qp and let K be a finite extension
of Q, contained in F . Let Z ⊂ F n be a subanalytic subset. Let ε > 0. There exists
s ∈ N, c ∈ R and a family of blocks W ⊂ F n × Qsp , such that for any T > 1, there
S
exists a subset S ⊂ Qsp of cardinality < cT ε such that Z(K; T ) ⊂ σ∈S Wσ .
Proof. — Let d = [F : Qp ]. By Krasner’s lemma, algebraic numbers are dense in F ,
so that there exists a Qp -basis (e1 , . . . , ed ) of F which consists of algebraic numbers.
This furnishes a Qp -linear bijection ϕ : F ' Qdp , and Z 0 = ϕ(Z) is a subanalytic
subset of Qnd
p .
If x ∈ Z(K), then the coordinates of ϕ(x) in Qnd
p are algebraic numbers, of
degrees 6 [Q(e1 , . . . , ed ) : Q] [K : Q] = D. Since ϕ is linear, as well as its inverse,
there exists a positive real numbers a > 0 such that a−1 H(x) 6 H(ϕ(x)) 6 aH(x)
for every x ∈ Z(E).
The definition of a family of blocks that we have adopted here is slightly stronger
than the one used in Theorem 4.2.3 of [8]. However, all proofs go over without
s
any modification, so that there exists a family of blocks W 0 ⊂ Qnd
p × Qp such
ε
s
that for any T > 1, there exists a subset ST ⊂ Qp of cardinality < cT such that
S
s
Z 0 (D; T ) ⊂ σ∈ST Wσ0 . Let ψ : F n × Qsp → Qnd
p × Qp be the map (x, y) 7→ (ϕ(x), y)
and let W = ψ −1 (W 0 ) ⊂ F n × Qsp . By definition, W is a family of blocks in F n .
Moreover, for any T > 1, one has
Z(F ; T ) ⊂ ψ −1 (Z 0 (D; aT )) ⊂
[
σ∈SaT
ϕ−1 (Wσ0 ) =
[
Wσ .
σ∈SaT
Since Card(SaT ) 6 caε T ε , the family of blocks W satisfies the requirements of the
theorem.
5. Zariski closures and analytic functions
5.1. — Let F be a complete non-archimedean valued field. Let V be an F -scheme
of finite type. One says that a subset K of V an is sparse if there exist a set T and a
subset Z of V an × T such that for every t ∈ T , Zt = {x ∈ V ; (x, y) ∈ Z} is a closed
S
Zariski subset of V with empty interior, and K = t∈T Ztan . Let us remark that a
sparse set has empty interior; indeed, Abhyankar points are dense in V an , and each
of them is Zariski dense if V is irreducible.
Lemma 5.2. — Let us assume that K is sparse, and let C ⊂ V be a geometrically
irreducible curve such that C an 6⊂ K. Then C an \ K is connected.
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
9
Proof. — Let K = t∈T Ztan be a description of K as above. By assumption, for
every t ∈ T , C 6⊂ Ztan ; consequently, Ztan ∩ C an consists of rigid points of C an ,
hence K ∩ C an consists of rigid points of C an . In the topological description of
(geometrically irreducible) analytic curves as real graphs ([3], chapter 4), their rigid
points are endpoints, so that C an \ (K ∩ C an ) is connected as well.
S
Lemma 5.3. — Let F be a complete non-archimedean valued field. Let V be
an F -scheme of finite type which is geometrically connected (resp. geometrically
irreducible) and let K be a sparse subset of V an . Then V an \ K is a geometrically
connected (resp. geometrically irreducible) analytic space.
Proof. — We may assume that F is algebraically closed. Let us assume that V is
connected and let us prove that V an \ K is connected as well. Let x, y ∈ V (F ) \ K.
By [20, p. 56], there exists an irreducible curve C ⊂ V which passes through x
and y. Then C an is connected. One has C 6⊂ K, hence it follows from lemma 5.2
that C an \ (K ∩ C an ) is connected. Consequently, any two points of V (F ) \ K are in
the same connected components of V an \ K. Since V (F ) is dense in V an , V (F ) \ K
is dense in V an \ K, hence V an \ K is connected.
Let us now assume that V is irreducible. The normalization morphism p : W → V
is finite, and W is connected. Since p−1 (K) is a sparse subset of W , it follows from
the first part of the lemma that W an \ p−1 (K) is connected. Since W an is the
normalization of V an , then W an \ p−1 (K) = p−1 (V an \ K) is the normalization of
V an \ K. By theorem 5.17 of [14], this implies that V an is irreducible.
Corollary 5.4. — Let F be a complete valued field, let V be an F -scheme of finite
type and let K be a sparse subset of V an . The set of irreducible components of V an \K
is finite.
Proof. — Let Ω = V an \ K. Let E be the completion of the algebraic closure of F .
By lemma 5.3, ΩE ∩ Z an is irreducible, for every irreducible component Z of YE , and
the family of these intersections is the family of irreducible components of WE . The
corollary then follows from [14, lemme 4.25].
Proposition 5.5. — Let F be a finite extension of Qp . Let A be an affine scheme
of finite type over F and let Ω ⊂ Aan be the complement of a sparse subset. Let
X be a closed analytic subspace of Ω. Let V be a semi-algebraic subset of A(F ),
contained in X(F ), and let W be its Zariski-closure in A. If W is irreducible, then
W an ∩ Ω ⊂ X.
Proof. — The following proof is inspired by that of lemma 4.1 in [25]. Up to repeatedly replacing X by its singular locus Xsing , we may assume that V (F ) 6⊂ Xsing .
Let o ∈ V (F ) be a smooth point of X.
The rational points of a non-empty and non-geometrically irreducible variety
are not Zariski dense. Since W is irreducible, it is geometrically irreducible; by
lemma 5.3, W an ∩ Ω is an irreducible analytic space.
10
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
Let R the Weil restriction functor from F to Qp ; it is a right adjoint to the
functor of extensions of scalars from Qp to F . Recall that R(AnF ) ' And
Qp , where
d = [F : Qp ].
Let Z be the Zariski closure of V inside R(A). By Weil restriction, the inclusion
Z → R(A) gives rise to a morphism j : ZF → A; one has W = j(ZF ). Let ZF0 be
the set of points of ZF at which j is open; it contains a Zariski dense open subset
of ZF . Let also Y = j −1 (X) ⊂ (ZF )an ; it is a closed analytic subset of (ZF )an .
Let us consider a semi-algebraic cellular decomposition of R(A) which is adapted
to V and to Z(Qp ). Let C be a cell of dimension m = dim(V ) = dim(Z(F ))
which meets V . Let (u1 , . . . , um ) be a family of regular functions on R(A) which
parameterizes C.
Every point x ∈ C is a smooth point of V (in the sense of Qp -analytic geometry)
and of Z (in the sense of algebraic geometry). The tangent space Tx Y at x is
an F -vector subspace of Tx ZF ' F m which contains Tx V = Qm
p ; consequently,
Tx Y = Tx ZF for every x ∈ C. This implies that Y (F ) is a neighborhood of x
in ZF (F ), hence X(F ) contains a semi-algebraic subset of W of maximal dimension.
Since W an ∩Ω is irreducible, such a subset is dense for the Zariski topology of W an ∩Ω,
hence W an ∩ Ω ⊂ X.
6. Complements on p-adic Schottky groups and uniformization
Let F be a finite extension of Qp .
6.1. — We endow P1 (Cp ) with the distance given by
d(x, y) =
|x − y|
max(1, |x|) max(1, |y|)
for x, y ∈ Cp — it is invariant under the action of PGL(2, OCp ). Moreover, an
elementary calculation shows that every element g ∈ PGL(2, Cp ) is Lipschitz for
this distance (see also Thm 1.1.1 of [28]).
6.2. — Let Γ be a Schottky group in PGL(2, F ), let LΓ be its limit set and ΩΓ =
Pan
1 \ L . For x ∈ ΩΓ , let δΓ (x) be the distance of x to LΓ ; if |x| 6 1, this is the
largest real number r such that the disk D(x, r) of radius r centered at x is contained
in ΩΓ . For every γ ∈ PGL(2, F ), there exists a positive real number c such that
δΓ (γ · z) 6 cδΓ (z) for every z ∈ ΩΓ .
6.3. — By [15, I, (4.3)], every Schottky group in PGL(2, F ) admits a good fundamental domain FΓ , in the following sense:
(1) There exists a finite family (B1 , . . . , Bg , C1 , . . . , Cg ) of open disks in Pan
1 , with
S
S
centers in k, such that FΓ = Pan
\
B
∪
C
;
i
i
1
(2) The closed disks B1+ , . . . , Bg+ , C1+ , . . . , Cg+ are pairwise disjoint; let then F◦Γ =
S + S +
Pan
Bi ∪ Ci ;
1 \
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
11
(3) The group Γ is generated by elements γ1 , . . . , γg such that γi (P1 \ Bi ) = Ci+
and γi (P1 \ Bi+ ) = Ci for every i ∈ {1, . . . , g}.
Moreover, the following properties are satisfied:
(4) One has γ∈Γ γ · FΓ = P1 \ LΓ ;
(5) For γ ∈ FΓ , one has FΓ ∩ γ · FΓ 6= ∅ if and only if γ ∈ {1, γ1±1 , . . . , γg±1 };
(6) For every γ ∈ Γ \ {1}, one has F◦Γ ∩ γ · FΓ = ∅.
S
In this context, we denote by `Γ (g) the length of an element g ∈ Γ with respect
to the generating set {γ1 , . . . , γg }.
Lemma 6.4. — Let Γ be an arithmetic Schottky group in PGL(2, F ) and let H
be a height function on PGL(2, Q). There exists a positive real number c such that
H(γ) 6 c`Γ (γ) , for every γ ∈ Γ.
Proof. — As above, let (γ1 , . . . , γg ) be a generating family of minimal cardinality
of Γ. Let c1 be a positive real number such that H(hh0 ) 6 c1 H(h)H(h0 ) for every
h, h0 ∈ PGL(2, Q). For every γ ∈ Γ, one proves by induction on `Γ (γ) that
` (γ)−1
H(γ) 6 c1Γ
sup(H(γ1 ), . . . , H(γt ))`Γ (γ) .
Let c = sup(1, c1 , H(γ1 ), . . . , H(γt )); then H(γ) 6 c`Γ (γ) for every γ ∈ Γ.
Lemma 6.5. — Let Γ be a Schottky group in PGL(2, F ).
(1) For every point ξ ∈ LΓ and every open neighborhood Ui of ξ in Pan
1 , there
exists an affinoid domain FΓ contained in U which is a good fundamental domain
for the action of the group Γ.
(2) There exists positive real numbers a, b such that for every x ∈ ΩΓ , there exists
γ ∈ Γ such that γx ∈ FΓ and `Γ (γ) 6 a − b log(δΓ (x)).
Proof. — If FΓ is a good fundamental domain for Γ, then so is γ ·FΓ for every γ ∈ Γ.
Moreover, it is proven in [15, I, §4, p. 29] that the domain γ ·FΓ is contained in a ball
(denoted there by B(γ)) whose radius has an upper bound of the form c−`Γ (γ) ,
for some positive real number c > 1. This implies the lemma.
7. Proof of the theorem
7.1. — Let us recall the notation: for i ∈ {1, . . . , n}, we are given an arithmetic
Schottky subgroup Γi of PGL(2, F ) of rank > 2, let LΓi be its limit set, ΩΓi =
(P1 )an \ LΓi , and XΓi be the projective algebraic curve over F such that ΩΓi /Γi '
XΓani ; let also pΓi : ΩΓi → XΓani be the canonical morphism. Let X = X1 × · · · × Xn ,
Ω = ΩΓ1 × · · · × ΩΓn and p : Ω → X an be the associated morphism. Let V be an
algebraic subvariety of X and W ⊂ p−1 (V ) ⊂ Ω be an irreducible component of a
maximal algebraic subvariety of p−1 (V ). We need to prove that W is flat. The proof
requires intermediate steps and will be concluded in proposition 7.8.
12
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
7.2. — Let Y be the closure of W in (P1 )n and let m be its dimension. We may
assume that m > 0. We also assume that there does not exist j ∈ {1, . . . , n} such
that the jth projection qj : (P1 )n → P1 is constant on Y .
One then has q1 (Y ) = P1 . Since LΓ1 is a perfect set, it is dense in P1 for the
Zariski topology, and there exists a smooth point ξ ∈ Y such that q1 (ξ) ∈ LΓ1 .
Then there exists a finite subset J of {1, . . . , n} containing 1 and of cardinality m
such that the projection qJ : Pn1 → PJ1 induces a generically étale map from Y to PJ1 .
Up to reordering the coordinates, we assume that J = {1, . . . , m}. There exists a
smooth rigid point ξ 0 ∈ W such that q1 (ξ 0 ) ∈ LΓ1 and such that qJ |Y is étale at ξ 0 ;
we replacing ξ by ξ 0 , we henceforth assume that qJ |W is étale at ξ. Then there exists
an open neighborhood U of ξ in Y , of the form U1 × · · · × Un such that qJ |Y is étale
on U and finite to its image. Let ϕ = (ϕ1 , . . . , ϕn ) : V → Y be an analytic section,
defined around qJ (ξ). If ϕj is not constant, then it takes some value which does
not belong to LΓj ; shrinking V , we then assume that for each j > 2, either ϕj is
constant, or ϕj avoids LΓj .
7.3. — For every i, let Fi be a good fundamental domain for the action of Γi ; we
also assume that F1 is contained in U1 . Let F = F1 ×. . . Fn . Let G be the Q-algebraic
group PGL(2)n , and let G0 be the algebraic subgroup of G defined by
(7.3.1)
(g1 , . . . , gn ) ∈ G0
⇔
g2 = · · · = gm = 1
and let R be the subset of G0 (F ) defined by
(7.3.2)
g∈R
⇔
dim(gW ∩ F ∩ p−1 (V )) = m.
Lemma 7.4. — The set R is a subanalytic subset of G0 (F ).
Proof. — The sets V and W are algebraic, hence rigid subanalytic. Since F is
affinoid, the function p|F is rigid subanalytic, so that F ∩ p−1 (V ) is rigid subanalytic
as well. Consequently, (gW ∩ F ∩ p−1 (V ))g is a rigid subanalytic family of rigid
subanalytic subsets of Ω, parameterized by G0 (Cp ). By b-minimality, the set of
points g ∈ G0 (Cp ) such that dim(gW ∩ F ∩ p−1 (V )) = m is a rigid subanalytic
subset of G0 (Cp ). It then follows from the remark at the end of §4.1 that R is a
subanalytic subset of G0 (F ).
Lemma 7.5. — Let r be a positive real number and let f ∈ Cp [[z]] be a power series
which converges on the closed disk D(0, r). Let L1 and L2 be closed subsets of Cp
such that f −1 (L2 ) ⊂ L1 ; for every x ∈ Cp , let δL1 (x) and δL2 (x) be the distances
of x to L1 and L2 respectively. Then there exists real numbers m > 0, c > 0, and s
such that 0 < s < r and such that δL2 (f (x)) > cδL1 (x)m for every x ∈ D(0, s).
Proof. — Write f (z) = cn z n ; for simplicity of notation, we assume that r = 1
and that |cn | 6 1 for all n; write D = D(0, 1).
Let us first treat the case where f (0) 6∈ L2 . Then there exists a real number s > 0
such that D(f (0), s) ∩ L2 = ∅. For every x ∈ D such that |x| < s, one has
|f (x) − f (0)| < s, hence δL2 (f (x)) > s. It suffices to set m = 0 and c = s.
P
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
13
We now
assume that f (0) ∈ L2 , hence 0 ∈ L1 . Let m = ord0 (f − f (0)). Since
P
f 0 (z) = n>m ncn z n−1 , there exists a real number s such that 0 < s 6 1 and such
that |f 0 (z)| = |mcm | |z|m−1 provided |z| 6 s. Moreover, f (n) (z)/n! 6 1 for every
n > 0 and any z ∈ D. Considering the Taylor expansion
X 1
f (y) =
f (n) (x)(y − x)n ,
n>0 n!
we then see that there exists a real number s0 such that
f (D(x, u)) = D(f (x), |f 0 (x)| u)
for every real number u such that 0 < u 6 s0 and every x ∈ D such that 0 < |x| 6 s.
If u < δL1 (x), then D(x, u)∩L1 = ∅, hence D(f (x), |f 0 (x)| u)∩L2 = ∅; consequently,
δL2 (f (x)) > |f 0 (x)| δL1 (x). Since 0 ∈ L1 , one has |x| > δL1 (x). Consequently,
δL2 (f (x)) > |mcm | δL (x)m−1 δL1 (x) > |mcm | δL1 (x)m .
This concludes the proof.
Lemma 7.6. — There exists a real number c > 0 such that for every large enough
real number T , R ∩ Γ contains at least T c points of height 6 T .
Proof. — Let q be the genus of XΓ1 ; at the end of the proof, it will be fundamental
that q > 2. We recall that the fundamental domain F1 is the complement of a union
0
0
of 2q open disks in Pan
1 , with disjoint closures, say D1 , . . . , Dq , D1 , . . . , Dq , such that
ξ1 ∈ D1 ⊂ U1 ⊂ P1 \ D10 . Moreover, there exist independent elements α1 , . . . , αq
0
of Γ1 such that αi · (Pan
1 \ D1 ) ⊂ D1 .
For every i ∈ {1, . . . , m}, let ai ∈ ΩΓi ∩ Ui ; let a = (a1 , . . . , am ).
For a word γ1 of large length in α1 , . . . , αq , we will consider the point a(γ1 ) =
(γ1 · a1 , a2 , . . . , am ) of U1 × · · · × Um and its image ϕ(a(γ1 )) under the section ϕ.
We set γ2 = · · · = γm = 1. Let j > m. If ϕj is constant, then we set γj = 1 ∈ Γj .
Otherwise, we observe that ϕj (a(γ1 )) takes only values in ΩΓj , hence the preimage of LΓj by ϕj is contained in LΓ1 . By lemma 7.5, one has an inequality
d(ϕj (a(γ1 )), LΓj ) d(γ1 · a1 , LΓ1 )k , for some integer k > 0. By §6.2, one also
has an inequality d(αj · a1 , LΓ1 ) d(a1 , LΓ1 ), uniformly in a1 .
By lemma 6.5, (2), and lemma 6.4, there exists γj ∈ Γj such that γj ·ϕj (a(γ1 )) ∈ Fj
and
H(γj ) d(ϕj (a(γ1 )), LΓj )−κ ,
where κ is a positive real number, independent of γ1 .
If `(γ1 ) denotes the length of γ1 as a word in α1 , . . . , αq , this implies that
H(γj ) d(γ1 · a1 , LΓj )−mκ c`(γ1 ) ,
where c is a positive real number, independent of γ1 .
Let us fix such a real number c which works for all j. Enlarging c, we also assume
that H(γ1 ) 6 c`(γ1 ) . Let γ = (γ1 , γ2 , . . . , γn ). By construction, γ · W ∩ F ∩ p−1 (V ) =
γ · W ∩ F is of dimension m, so that γ ∈ R, and H(γ) 6 c`(γ1 ) . The number of words
14
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
of length ` in α1 , . . . , αq is q ` . Consequently, R(F ) ∩ Γ contains at least q ` points of
height 6 c` . This implies the lemma.
Lemma 7.7. — The stabilizer of W inside G0 ∩ Γ is infinite.
Proof. — Let K be a number field contained in F such that all groups Γj are
contained in PGL(2, K). The points of R ∩ Γ are K-rational points. By lemma 7.6,
the subset R of PGL(2, F )n has many K-rational points, in the sense of section 4.5.
Since R is subanalytic, we may apply the p-adic Pila-Wilkie theorem of [8], as stated
in theorem 4.7. Let thus s ∈ N, c ∈ R, ε > 0, and B ∈ PGL(2, F )n × Qsp be a
family of blocks such that for every T > 1, there exists a subset Σ ∈ Qsp of cardinality
S
< cT ε such that R(K; T ) ⊂ σ∈Σ Bσ . Let also t ∈ N and Z ⊂ PGL(2, F )n × Qtp be
a semi-algebraic subset such that for every σ ∈ Qsp , there exists τ ∈ Qtp such that
Bσ ⊂ Zτ and dim(Bσ ) = dim(Zτ ). Let c0 be an upper bound for the number of
irreducible components of the Zariski closure of the sets Zτ , for τ ∈ Qtp .
By the pigeonhole principle, there exists σ ∈ Σ such that
1
Card(R(K; T ) ∩ Γ ∩ Bσ ) > T c1 −ε .
c
Moreover, the Zariski closure of Bσ in PGL(2, F )n has at most c0 irreducible components. Consequently, we may choose such an irreducible component whose trace M
on Bσ satisfies
1
Card(R(K; T ) ∩ Γ ∩ M ) > 0 T c1 −ε .
cc
Observe that M is the chosen irreducible component of Bσ .
Let g ∈ M . Let W 0 be the Zariski closure of g −1 M · W in PGL(2)F . It is
irreducible.
There exists a finite extension F 0 of F such that W (F 0 ) is Zariski dense in WF 0 .
The Zariski closure of g −1 M · W (F 0 ) in PGL(2)F 0 is an irreducible component W00
of WF0 0 . By proposition 5.5, one has (W00 )an ∩ ΩF 0 ⊂ p−1
F 0 (VF 0 ). Since p is defined
0 an
−1
over F , this implies that (W ) ∩ Ω ⊂ p (V ).
The inclusions W ⊂ W 0 and (W 0 )an ∩ Ω ⊂ p−1 (V ), the irreducibility of W 0
and the maximality hypothesis on W imply that W 0 = W . Consequently, g −1 M
stabilizes W .
We have shown that the stabilizer of W inside G0 ∩ Γ contains subsets of the form
R(K; T ) ∩ Γ ∩ M , whose cardinality can be prescribed to be larger than any given
integer. This concludes the proof of the lemma.
Proposition 7.8. — The subvariety W is flat.
Proof. — We show that all components of the section ϕ are either constant or given
by homographies. Fix an integer j such that m < j 6 n and we consider ϕj as a
function of the first variable, all other variables being fixed.
For 2 6 i 6 m, fix an element ai ∈ ΩΓi ∩ Ui . Let us assume that the function
z 7→ ϕj (z, a2 , . . . , am ) is not constant.
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
15
Recall that Y is the Zariski closure of W in (P1 )n . By lemma 7.7, the number
of irreducible components of Y ∩ F is smaller than the cardinality of the stabilizer
of W inside G0 ∩ Γ. This furnishes functional equations of the form
(7.8.1)
h1 ϕj (g1 z, a2 , . . . , am ) = h2 ϕj (g2 z, a2 , . . . , am )
where g1 6= g2 ∈ Γ1 , and h1 , h2 ∈ Γj .
Let g = g2 g1−1 and h = h−1
2 h1 ; then ϕj (gu, a2 , . . . , am ) = hϕj (u, a2 , . . . , am ).
Consequently, the Schwarzian derivative Sϕj of ϕj satisfies Sϕj (gu, a2 , . . . , am ) =
Sϕj (u, a2 , . . . , am ), a relation which is valid for u ∈ g1−1 U1 .
Since ϕj is algebraic, so is Sϕj . By construction, g is a non-trivial element of the
Schottky group Γ1 , hence g is not of finite order. Consequently, Sϕj is constant.
We now apply a remark of Nevanlinna ([21], p. 344–345) according to which the
function ψ = (ϕ0j )−1/2 satisfies the differential linear equation ψ 00 + 21 (Sϕj )ψ = 0. If
Sϕj were a non-zero constant, then the function ψ = 0 would be the only algebraic
solution of this equation. This shows that Sϕj = 0.
This proves that for every a2 , . . . , am , the function z1 7→ ϕj (z1 , a2 , . . . , am ) is a
homography. By construction, it is defined over F . If we represent it as a matrix,
its coefficients will be analytic functions of a2 , . . . , am which belong to F ; necessarily
they are constant hence z1 7→ ϕj (z1 , a2 , . . . , am ) does not depend on a2 , . . . , am .
Let now identify ϕm+1 , . . . , ϕn with these homographies. The image of the map
from (P1 )m to (P1 )n given by (z1 , . . . , zm ) 7→ (z1 , . . . , zm , ϕm+1 (z1 ), . . . , ϕn (z1 )) is
then contained in Y . Since Y is irreducible and m-dimensional, this concludes the
proof.
8. Remarks on geodesic subvarieties
8.1. — Let F be a finite extension of Qp , let Γ and Γ0 be two Schottky subgroups
of PGL(2, F ). If Γ and Γ0 are commensurable, then LΓ = LΓ0 .
8.2. — In the rest of this section, we fix the following notation. Let F be a finite extension of Qp and let (Γi )16i6n be a finite family of Schottky subgroups of
Q
Q
PGL(2, F ). Let us set Ω = ni=1 ΩΓi , X = ni=1 XΓi , and let p : Ω → X an be the
morphism deduced from the morphisms pΓi : ΩΓi → XΓani .
Proposition 8.3. — If W is a geodesic subvariety of Ω, then p(W ) is algebraic.
Proof. — By the definition of a geodesic subvariety, the proof immediately reduces
to the particular case where
all the subgroups Γi are commensurable, and W is the
T
diagonal of Ω. Let Γ0 = i Γi and X0 be the algebraic curve associated with ΩΓ0 /Γ0 .
Then, for every i, the morphism fi : W → Xian deduced from f = p|W factors as
the composition of the uniformization p0 : ΩΓ0 → X0an and with a finite morphism
X0an → Xian . By gaga ([3], corollary 3.5.2), a finite analytic morphism of algebraic
curves is algebraic; consequently, there exists a finite morphism qi : X0 → Xi such
that fi = qian ◦ p0 . Then p(W ) is the image of X0 by the finite morphism q =
(q1 , . . . , qn ) : X0 → X, hence is algebraic.
16
ANTOINE CHAMBERT-LOIR & FRANÇOIS LOESER
Proposition 8.4. — Let W be an irreducible algebraic subvariety of Ω. Assume
that the Zariski closure of p(W ) in X has dimension dim(W ). If, moreover, n = 2,
then W is geodesic.
We expect that proposition 8.4 holds in general, without assuming that n = 2.
Proof. — Let V ⊂ X be the smallest algebraic subvariety of X such that V an
contains p(W ). By assumption, one has dim(V ) = dim(W ). By construction, W is
a maximal irreducible algebraic subvariety of p−1 (V an ). By theorem 2.7, W is flat;
let us prove that W is geodesic. This is obvious if dim(W ) = 0 or dim(W ) = 2,
hence we may assume that dim(W ) = 1. Let then g ∈ PGL(2, F ) be such that
W = {(z, g · z)} ∩ Ω and let us prove that Γ2 and gΓ1 g −1 are commensurable,
a property which is equivalent to the finiteness of both orbit sets Γ2 \Γ2 gΓ1 and
Γ1 \Γ1 g −1 Γ2 .
Let us argue by contradiction and assume that Γ2 \Γ2 gΓ1 is infinite. (The other
case is analogous, or follows by symmetry.) Fix a rigid point z ∈ ΩΓ1 . Let A ⊂ Γ1
be a set such that gA is a set of representatives of Γ2 \Γ2 gΓ1 ; by assumption, A is
infinite. Since Γ\W ⊂ V an , the algebraic variety V contains the infinite set of points
p(a · z, g · az) = (p1 (z), p2 (ga · z)), for a ∈ A, hence it contains its Zariski closure
{p1 (z)} × X2 . Since this holds for every z ∈ W , we deduce that V contains X1 × X2 ,
contradicting the assumption that dim(W ) = 1.
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