Arithmetic Differential Equations
Alexandru Buium
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
E-mail address: buium@math.unm.edu

2000 Mathematics Subject Classification. Primary 14G20, 14G35, 30F10, 14H52 ;Secondary 53A55, 37F05, 14L24
Key words and phrases. p−adic numbers, correspondences, uniformization
Abstract. This research monograph develops an arithmetic analogue of thetheory of ordinary differential equations in which derivations are replaced by
Fermat quotient operators. The theory is then applied to the construction andstudy of certain quotient spaces of algebraic curves with respect to correspondences having infinite orbits.

Preface
The main purpose of this research monograph is to develop an arithmetic analogue of the theory of ordinary differential equations. In our arithmetic theory the“time variable” t is replaced by a fixed prime integer p. Smooth real functions,t 7→ x(t), are replaced by integer numbers a ∈ Z or, more generally, by integers invarious (completions of) number fields. The derivative operator on functions,x(t) 7→

dx
(t),
dt

is replaced by a “Fermat quotient operator” δ which, on integer numbers, acts as
δ : Z → Z,
a 7→ δa :=a − ap.
p

Smooth manifolds (configuration spaces) are replaced by algebraic varieties definedover number fields. Jet spaces (higher order phase spaces) of manifolds are replacedby what can be called “arithmetic jet spaces” which we construct using δ in placeof d/dt. Usual differential equations (viewed as functions on usual jet spaces) are
replaced by “arithmetic differential equations” (defined as functions on our “arithmetic jet spaces”). Differential equations (Lagrangians) that are invariant undercertain group actions on the configuration space are replaced by “arithmetic differential equations” that are invariant under the action of various correspondences on
our varieties.
As our main application we will use the above invariant “arithmetic differential equations” to construct new quotient spaces that “do not exist” in algebraic
geometry. To explain this we start with the remark that (categorical) quotients ofalgebraic curves by correspondences that possess infinite orbits reduce to a point
in algebraic geometry. In order to address the above basic pathology we propose
to “enlarge” algebraic geometry by replacing its algebraic equations with our moregeneral arithmetic differential equations. The resulting new geometry is referred toas δ−geometry. It then turns out that certain quotients that reduce to a point inalgebraic geometry become interesting objects in δ−geometry; this is because there
are more invariant “arithmetic differential equations” than invariant algebraic equations. Here are 3 classes of examples for which this strategy works:
1) Spherical case. Quotients of the projective line P1 by actions of certain
finitely generated groups (such as SL2 (Z));2) Flat case. Quotients of P1 by actions of postcritically finite maps P1 → P1
with (orbifold) Euler characteristic zero;3) Hyperbolic case. Quotients of modular or Shimura curves (e.g. of P1 ) byactions of Hecke correspondences.
v

vi

PREFACE

Our results will suggest a general conjecture according to which the quotient of acurve (defined over a number field) by a correspondence is non-trivial in δ−geometry
for almost all primes p if and only if the correspondence has an “analytic uniformization” over the complex numbers. Then the 3 classes of examples above correspondto spherical, flat, and hyperbolic uniformization respectively.
Material included. The present book follows, in the initial stages of its
analysis, a series of papers written by the author [17]-[28]. A substantial part ofthis book consists, however, of material that has never been published before; thisincludes our Main Theorems stated at the end of Chapter 2 and proved in theremaining Chapters of the book. The realization that the series of papers [17]-[28]consists of pieces of one and the same puzzle came relatively late in the story andthe unity of the various parts of the theory is not easily grasped from reading thepapers themselves; this book is an attempt at providing, among other things, alinear, unitary account of this work. Discussed are also some of the contributions
to the theory due to C. Hurlburt [71], M. Barcau [2], and K. Zimmerman [29].
Material omitted. A problem that was left untouched in this book is that ofputting together, in an adelic picture, the various δ−geometric pictures, as p varies.
This was addressed in our paper [26] where such an adelic theory was developed
and then applied to providing an arithmetic differential frameworkfor functions ofP
the form (p, a) 7→ c(p, a), p prime, a ∈ Z, where L(a, s) = n c(n, a)n−s are various
families of L−functions parameterized by a. Another problem not discussed in thisbook is that of generalizing the theory to higher dimensions. A glimpse into what
the theory might look like for higher dimensional varieties can be found in [17] and[3]. Finally, we have left aside, in this book, some of the Diophantine applicationsof our theory such as the new proof in [18] of the Manin-Mumford conjecture about
torsion points on curves and the results in [21], [2] on congruences between classicalmodular forms.
Prerequisites. For most of the book, the only prerequisites are the basic
facts of algebraic geometry (as found, for instance, in R. Hartshorne’s textbook[66]) and algebraic number theory (as found, for instance, in Part I of S. Lang’s
textbook [87]). In later Chapters more background will be assumed and appropriate
references will be given. In particular the last Chapter will assume some familiarity
with the p−adic theory of modular and Shimura curves. From a technical point ofview the book mainly addresses graduate students and researchers with an interestin algebraic geometry and / or number theory. However, the general theme of thebook, its strategy, and its conclusions should appeal to a general mathematicalaudience.
Plan of the book. We will organize our presentation around the motivating“quotient space” theme. So quotient spaces will take center stage while “arithmetic
jet spaces” and the corresponding analogies with the theory of ordinary differential
equations will appear as mere tools in our proofs of δ−geometric theorems. Accordingly, the Introduction starts with a general discussion of strategies to constructquotient spaces and continues with a brief outline of our δ−geometric theory. We
also include, in our Introduction, a discussion of links, analogies, and / or discrepancies between our theory and a number of other theories such as: differential
equations on smooth manifolds [114], the Ritt-Kolchin differential algebra [117],[84], [32], [13], the difference algebraic work of Hrushovski and Chatzidakis [69],PREFACEvii

the theory of dynamical systems [109], Connes’ non-commutative geometry [36],the theory of Drinfeld modules [52], Dwork’s theory [53], Mochizuchi’s p−adicTeichmüller theory [111], Ihara’s theory of congruence relations [73] [74], and thework of Kurokawa, Soulé, Deninger, Manin, and others on the “field with one element” [86], [128], [98], [46]. In Chapter 1 we discuss some algebro-geometricpreliminaries; in particular we discuss analytic uniformization of correspondenceson algebraic curves. In Chapter 2 we discuss our δ−geometric strategy in detail
and we state our main conjectures and a sample of our main results. In Chapters
3, 4, 5 we develop the general theory of arithmetic jet spaces. The corresponding 3 Chapters deal with the global, local, and birational theory respectively. In
Chapters 6, 7, 8 we are concerned with our applications of δ−geometry to quotient
spaces: the corresponding 3 Chapters are concerned with correspondences admitting a spherical, flat, or hyperbolic analytic uniformization respectively. Details
as to the contents of the individual Chapters are given at the beginning of each
Chapter. All the definitions of new concepts introduced in the book are numbered
and an index of them is included after the bibliography. A list of references tothe main results is given at the end of the book. Internal references of the form
Theorem x.y, Equation x.y, etc. refer to Theorems, Equations, etc. belonging to
Chapter x (if x 6= 0) or the Introduction (if x = 0). Theorems, Propositions, Lemmas, Corollaries, Definitions, and Examples are numbered in the same sequence;
Equations are numbered in a separate sequence. Here are a few words about thedependence between the various Chapters. The impatient reader can merely skimthrough Chapter 1; he/she will need to read at least the (numbered) “Definitions”(some of which are not standard). Chapters 2-5 should be read in a sequence.
Chapters 6-8 are largely (although not entirely) independent of one another but
they depend upon Chapters 2-5.
Acknowledgments. The author wishes to thank D. Bertrand, P. Deligne,
E. Hrushovski, Y. Ihara, M. Kim, Y. I. Manin, B. Mazur, S. Lang, A. Pillay, B.
Poonen, F. Pop, T. Scanlon, J. Tate, D. Thakur, D. Ulmer, and J. F. Voloch
for encouragement and discussions at various stages of development of these ideas.
While writing this book the author was supported in part by NSF Grant #0096946.
Alexandru Buium
Albuquerque, May 2005

Introduction
We start, in Section 1, by explaining our main motivation which comes fromthe fact that categorical quotients of correspondences on curves tend to be trivial in algebraic geometry. In Section 2 we explain how one can fix this problem
for a remarkable class of correspondences; this is done by developing a geometry,called δ−geometry, which is obtained from usual algebraic geometry by adjoining
a “Fermat quotient operator” δ. If one views δ as an analogue of a derivation withrespect to a prime number then δ−geometry can be viewed as obtained from usual
algebraic geometry by replacing algebraic equations with “arithmetic differential
equations”. In Section 3 we discuss relations between our theory and some othertheories.
The present Introduction is written in an informal style; a formal presentationof this material will be made in the body of the book.
0.1. Motivation and strategy
0.1.1. Correspondences and their categorical quotients. It is convenient to start in complete generality by considering an arbitrary category C. (Morally
C should be viewed as a category of “spaces” in some geometry.) A correspondence in C is a tuple X = (X, X̃, σ1 , σ2 ) where X and X̃ are objects of C and
σ1 , σ2 : X̃ → X are morphisms in C. We sometimes write X = (X, σ), whereσ := (X̃, σ1 , σ2 ). Following the standard terminology of geometric invariant theory [113] we define a categorical quotient for X to be a pair (Y, π) where Y is anobject of C and π : X → Y is a morphism in C satisfying the following properties: 1) π ◦ σ1 = π ◦ σ2 ; 2) For any pair (Y 0 , π 0 ) where Y 0 is an object of C andπ 0 : X → Y 0 is a morphism such that π 0 ◦ σ1 = π 0 ◦ σ2 there exists a uniquemorphism γ : Y → Y 0 such that γ ◦ π = π 0 . Categorical quotients are sometimes
referred to as co-equalizers. Of course, if a categorical quotient (Y, π) exists then it
is unique up to isomorphism and we shall write Y = X/σ. Correspondences form,
in a natural way, a category: a morphism X → X0 between two correspondences
X = (X, X̃, σ1 , σ2 ) and X0 = (X 0 , X̃ 0 , σ10 , σ20 ) is, by definition, a pair of morphisms(π, π̃), π : X → X 0 , π̃ : X̃ → X̃ 0 , such that π ◦ σi = σi0 ◦ π̃, i = 1, 2. We will assume,for each category C we shall be considering, that a class of objects in C is given
which we refer to as trivial objects. (Morally trivial objects should be viewed as
spaces that reduce to a point.)
An important example of correspondences is provided by (discrete) dynamicalsystems (i.e. self maps). Indeed if X ∗ is an object in our category C and s :X ∗ → X ∗ is a morphism then one can attach to these data a correspondenceX∗ = (X ∗ , X ∗ , idX ∗ , s). More generally if we assume C possesses fiber products
then one can consider, in a natural way, the pull-back X := f ∗ X∗ of X∗ via anyix

x

INTRODUCTION

morphism f : X → X ∗ ; cf. Equation ??. Categorical quotients of correspondencesof the form X∗ , or more generally of the form X, should be viewed as (categorical)
spaces of orbits of the “dynamical system defined by s”.
Another example of correspondences appearing in nature are groupoids. To
review this we shall tacitly assume that all products intervening (explicitly or implicitly) in the discussion below exist in C. A groupoid in C is a tuple(X, X̃, σ1 , σ2 , ν, ι, )
where X and X̃ are objects in C andσ1 : X̃ → X,

σ2 : X̃ → X,

ν : X̃ ×σ2 ,X,σ1 X̃ → X̃,ι : X̃ → X̃, : X → X̃are morphisms such that ν satisfies the usual associativity axiom,  satisfies σ1 ◦  =σ2 ◦  = idX plus the usual unit axiom, and ι satisfies σ1 ◦ ι = σ2 , σ2 ◦ ι = σ1
plus the usual inverse axiom. The datum (X, X̃, σ1 , σ2 ) can then be viewed as acorrespondence.
In their turn groupoids often arise from group actions. Again we tacitly assume
that all products intervening (explicitly or implicitly) in the discussion below exist
in C. Let G be a group object in the category C and µ : G × X → X an action on
an object X in C. Then one can consider the groupoid(X, G × X, pr2 , µ, ν, ι, )
where pr2 is the second projection, and ν, ι,  are naturally induced by the group
operations and the action. So in particular group actions µ : G × X → X definecorrespondences(X, G × X, pr2 , µ).
Many of the examples of correspondences we shall be interested in will notcome, however, from dynamical systems or groupoids.
0.1.2. Basic pathology. Typically it turns out that, in many classical geometric situations, one has interesting correspondences X = (X, σ) = (X, X̃, σ1 , σ2 )whose categorical quotient X/σ is trivial. This “basic pathology” manifests itself,
for instance, in the case when C is the category of algebraic varieties over an algebraically closed field and the trivial objects are the points: if X and X̃ above are
algebraic varieties and the smallest equivalence relation hσi ⊂ X × X containing
the image of σ1 × σ2 : X̃ → X × X is Zariski dense in X × X then the categoricalquotient X/σ is trivial. Examples of this kind are very common. For instanceassume that dim X = dim X̃ = 1, and σ has an infinite orbit by which we meanthat there exists an infinite sequence of points P1 , P2 , P3 , ... ∈ X and a sequence ofpoints P̃1 , P̃2 , P̃3 , ... ∈ X̃ such that σ1 (P̃i ) = Pi , σ2 (P̃i ) = Pi+1 for all i ≥ 1. Thenhσi ⊂ X × X is dense in X × X and hence X/σ reduces to a point in the categoryof varieties. Note, by the way, that in the example above it is impossible to embed
X into a correspondence X0 = (X 0 , X̃ 0 , σ10 , σ20 ) in the category of schemes of finite
type such that X0 admits a groupoid structure and dim X 0 = dim X̃ 0 = 1; here,by an embedding we mean a morphism (ι, ι̃) : X → X0 with ι and ι̃ embeddings.
Intuitively, X does not “generate” a groupoid “of finite type” and, in some sense,
this is what prevents usual algebraic geometry from “controlling” the quotient.
With the above “basic pathology” in mind one is tempted to “enlarge” usual
algebraic geometry so as to make X/σ non-trivial; one can then pursue the study

0.1. MOTIVATION AND STRATEGYxi

of X/σ in this enlarged geometry. The present book is an attempt to develop suchan enlarged geometry and to apply it to the study of categorical quotients.
Before proceeding to explain our approach it might be useful to put things inperspective by discussing some general aspects of quotient theory.
0.1.3. Two viewpoints in quotient theory. Assume we are given a correspondence X = (X, σ) in some category of “spaces” (e.g. smooth manifolds,
varieties, schemes, etc.) There are (at least) two possible paths towards the ideaof “quotient” of X by σ which we shall call here, for convenience, invariant theory
and groupoid theory.
In invariant theory one seeks to construct a categorical quotient as a “genuine
space” X/σ such that the (locally defined) functions on it identify with the (locallydefined) “σ−invariant functions” on X. (Here a locally defined function ϕ on X
is called σ−invariant if ϕ ◦ σ1 = ϕ ◦ σ2 on their common domain of definition).
This viewpoint is actually quite successful in algebraic geometry, in case (X, σ)comes from a reductive algebraic group action µ : G × X → X; geometric invariant
theory codifies the situation in this case. On the other hand the invariant theoretic
approach in algebraic geometry utterly fails for correspondences with Zariski dense
equivalence relation hσi; these will be, by the way, the correspondences that will
be at the heart of this book.
In groupoid theory one assumes that the correspondence X = (X, X̃, σ1 , σ2 )has a groupoid structure (ν, ι, ) and one defines the “space X/σ” to be the groupoid(X, X̃, σ1 , σ2 , ν, ι, ) itself regarded up to an appropriate equivalence on the classof groupoids. (This equivalence should allow roughly speaking to transfer modules
from one groupoid to another in a sense that depends on the particular context
we are in.) Non-commutative geometry and stack theory adopt this viewpoint.
By the way, one of Connes’ original motivations for developing non-commutative
geometry [36] was to address the “basic pathology” above as it manifests itself in
topology and differential geometry; this has been enormously successful and canbe adapted to algebro-geometric situations [127], [99]. On the other hand passing
from algebraic varieties to algebraic stacks, as one sometimes does in moduli space
theory, is not sufficiently drastic to make the “basic pathology” go away !
0.1.4. Strategy of the present approach. Our approach towards the “basic pathology” above will use the viewpoint of invariant theory (as opposed to thatof groupoid theory); but in order that invariant theory be non-trivial we will haveto enlarge algebraic geometry by adjoining to it some new functions. Having “morefunctions” will increase our chances to find, as we will, interesting σ−invariant functions. Now there is a general recipe to enlarge algebraic geometry by adjoining to
its functions an extra function, δ, satisfying “polynomial compatibility conditions”
with respect to addition and multiplication. Cf. [20]; see also the last Section ofChapter 2 in our book. Under a certain genericity condition it turns out, as weshall see, that “locally” there are exactly 4 types of such δ’s which can be referred
to as derivation operators, difference operators, p−derivation operators (= “Fermat
quotient operators”), and p−difference operators. If δ is a derivation one is led tothe differential algebra (and corresponding geometry) of Ritt [117] and Kolchin[84]; cf. also [32], [12], [14], [15], [16]. If δ is a difference operator one is led tothe difference algebra (and corresponding geometry) of Ritt-Cohn [35]; cf. also thework of Chatzidakis-Hrushovski [33] and Hrushovski [70]. If δ are p−derivation

xii

INTRODUCTION

operators attached to various prime numbers p one is led to what can be calledarithmetic differential algebra (and its corresponding geometry). The case when δ
is a p−difference operator seems to lead to a less interesting theory. Of the above
4 types of δ, p−derivations are the only ones that do not vanish identically on the
integers. This makes p−derivations especially suited for arithmetic applications; so
we shall exclusively be interested, in this book, in arithmetic differential algebra.
The study of arithmetic differential algebra and its associated geometry was begun
in our paper [17] and developed in a series of papers [18]-[28], [2], [3], [71]. Earlier
papers (cf. especially papers by Joyal [79] and Ihara [75]) contain indications that
Fermat quotients should be viewed as analogues of derivations. However note that
there is an important difference between our approach and the one proposed by
Ihara; cf. the remarks surrounding Equation 0.31 below. Arithmetic differential
algebra can be viewed as an “arithmetic” analogue of the Ritt-Kolchin differential
algebra and also as a “regularized” analogue of the Cohn difference algebra; indeed
arithmetic differential algebra can be viewed as obtained from difference algebra byadjoining to it certain divergent series (cf. Remark ?? below).
As a matter of terminology note that, in the Ritt-Kolchin differential algebra
one uses the symbol δ as an abbreviation for “differential”. By analogy with thetradition in the Ritt-Kolchin theory we will use the symbol δ as an abbreviation for“arithmetic differential”; in particular arithmetic differential algebra will be referred
to as δ−algebra while its “associated” geometry, which can be called arithmetic
differential geometry, will be referred to as δ−geometry. There will be no danger ofconfusion with differential algebra and difference algebra terminology because nouse of the latter two types of algebra will be made in this book.
The main purpose of this book is to first develop some of the basic elements of
δ−geometry and then to construct (and study) interesting categorical quotients, inδ−geometry, for correspondences whose categorical quotient in algebraic geometryis trivial. A conjecture will emerge to the effect that the δ−geometric picture isinteresting if and (essentially) only if our correspondences admit a complex analytic
uniformization. Below we give a rough outline of our theory; the details of thetheory will be explained in Chapter 2.
0.2. Rough outline of the theory
0.2.1. Background and notation. We start with a number field F (always
assumed of finite degree over Q). Fix a finite place ℘ of F which is unramified
over Q; later we will vary ℘. We identify ℘ with a maximal ideal in the ring ofintegers OF of F . Let p = char O/℘. Let O℘ be the localization of OF at ℘,consider its completion Ô℘ , and let Ô℘ur be the maximum unramified extension ofÔ℘ (obtained by adjoining to Ô℘ all the roots of unity of order prime to p in an
algebraic closure of its fraction field). Then Ô℘ur is a discrete valuation ring withmaximal ideal generated by p. Consider the completion of Ô℘ur which we denote byR℘ := (Ô℘ur )ˆ.
The ring R℘ has a simple
P∞ well known structure: any element of it can be represented
uniquely as a series i=0 ζi pi where ζi are roots of unity of order prime to p or 0.
The ring R℘ has a unique automorphism φ inducing P
the p−power P
Frobenius map∞∞on the residue field k℘ := R℘ /pR℘ ; it is given by φ( i=0 ζi pi ) = i=0 ζip pi . For

0.2. ROUGH OUTLINE OF THE THEORY

xiii

F = Q, ℘ = (p), we have Ẑ(p) = Zp , the ring of p−adic numbers, so we simply
urwrite Rp = Ẑurp in place of (Ẑ(p) )ˆ. By the way, for a general F and ℘ the naturalmorphism Rp → R℘ is an isomorphism so the ring R℘ depends only on p and not
on F and ℘.
0.2.2. δ−geometry. With the notation above we let δ = δp : R℘ → R℘ be
the “Fermat quotient operator” defined by δx = (φ(x) − xp )/p. We consider a“geometric category”, Cδ , which morally underlies δ−geometry at ℘. The objects ofCδ will be called δ−ringed sets; a δ−ringed set Xδ consists of an underlying set Xset
equipped with a family (Xs ) of subsets and a family of “structure rings” (Os ) ofR℘ −valued functions on the Xs ’s such that f ∈ Os implies δ ◦ f ∈ Os . Here s runs
through a monoid S and one assumes that Xst = Xs ∩ Xt for s, t ∈ S. One also
assumes compatibility with restrictions i.e. f ∈ Os implies f|Xst ∈ Ost . A δ−ringed
set is called trivial if Os = R℘ for all s. There is a natural concept of morphismof δ−ringed sets; with these morphisms the δ−ringed sets form a category Cδ . Any
correspondence in Cδ has a categorical quotient. So the existence of categoricalquotients is not here the issue; the issue will always be their non-triviality.
0.2.3. Passing from algebraic geometry to δ−geometry. Let X℘ = X
be a smooth scheme of finite type over R℘ with irreducible geometric fibers. We
will attach to the scheme X a δ−ringed set Xδ ; the underlying set Xset of Xδ willbe the set X(R℘ ) of R℘ −points of X and the structure rings Os will consist ofdcertain functions P 7→ ϕ(P ) that locally, in coordinates x ∈ R℘
, look likeϕ(x) =F (x, δx, ..., δ r x)
G(x, δx, ..., δ r x)

where F, G are restricted power series with R℘ −coefficients. Let us make this
precise. First, a function f : X(R℘ ) → R℘ is called a δ−function of order ≤ r if
for any P ∈ X(R℘ ) there is a Zariski open set U ⊂ X, P ∈ U (R℘ ), and a closeddbyembedding U ⊂ Ad such that f|U (R℘ ) is given in coordinates x ∈ R℘
f (x) = F (x, δx, ..., δ r x)
where F is a restricted power series with coefficients in R℘ . Recall that a power
series is called restricted if its coefficients converge p−adically to 0. Denote by
Or (V ) the rings of δ−functions of order ≤ r on a Zariski open set V ⊂ X. Then
V 7→ Or (V ) defines a sheaf Or of rings on X for the Zariski topology. Define aδ−line bundle to be a locally free sheaf of Or − modules of rank P
one. Consider the
ring W := Z[φ] ⊂ End(R℘ ) and let W+ be the set of all w Q=
ai φi ∈ W with×wiai ≥ 0. The multiplicative monoid W acts on R by λ = (φ (λ))ai , λ ∈ R× .
For any δ−line bundle L one can define a δ−line bundle Lw by acting with w upon
the defining cocycle of L. Let L = K −1 be now the anticanonical bundle on X,
viewed as a δ−line bundle. We consider the graded ringM
H 0 (X, Lw ).
w∈W+

Its homogeneous elements will be called δ−sections. We let S be the monoid ofδ−sections of weight 6= 0, not divisible by p, in the above ring. For any s ∈ S ofdegree w0 we let
Xs := {P ∈ X(R℘ ) | s(P ) 6≡ 0 mod p},

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INTRODUCTION

Os := {P 7→ ϕ(P ) := t(P )/sw (P ) | t ∈ H 0 (X, Lww0 )}.
The data (Xs ), (Os ) define a δ−ringed set Xδ with underlying set X(R℘ ).
Let now X℘ = X = (X, σ) = (X, X̃, σ1 , σ2 ) be a correspondence in the categoryof schemes over R℘ . Assume X and X̃ are smooth of finite type over R℘ and haveirreducible geometric fibers. Then by the discussion above one can attach to theschemes X and X̃ two δ−ringed sets Xδ and X̃δ . Assume moreover that σ1 andσ2 are étale. Then the morphisms σ1 , σ2 : X̃ → X induce morphisms between thecorresponding δ−ringed sets. So we end up with a correspondence Xδ = (Xδ , σδ ) in
the category Cδ . One can easily describe the categorical quotient Xδ /σδ in Cδ . The
underlying set of the categorical quotient Xδ /σδ is the set of equivalence classes ofpoints in X(R℘ ) with respect to the smallest equivalence relation containing theimage of the map
σ1 × σ2 : X̃(R℘ ) → X(R℘ ) × X(R℘ ).
Let L̃ be the anticanonical bundle on X̃. Let us say that a δ−section s ∈ H 0 (X, Lw )is σ−invariant if its pull-backs to H 0 (X̃, L̃w ) via σ1∗ and σ2∗ coincide; σ− invariant
δ−sections will simply be called δ− invariants. Then the monoid parameterizing
the structure rings of Xδ /σδ is the monoid of all δ−invariants s ∈ H 0 (X, Lw ) which
are not divisible by p. For s of degree w0 the corresponding structure ring of Xδ /σδis{[P ] 7→ ϕ(P ) := t(P )/sw (P ) | t ∈ H 0 (X, Lww0 ) a δ−invariant}.
So morally Xδ /σσ is non-trivial if one can find at least two R℘ −linearly independent
δ−invariants in one of the spaces H 0 (X, Lw ), where 0 6= w ∈ W+ . This agrees, ofcourse, with the ideology of classical invariant theory.
0.2.4. Conjectural picture. At this point we consider a global situation.
Let S be a finite set of finite places of the number field F , containing all the places
ramified over Q, assume (for simplicity) that S consists of all the places containingsome integer m ∈ Z, and let O := OF [1/m] be the ring of S−integers of F . Consider
a correspondence
X = XO = (X, σ),σ = (X̃, σ1 , σ2 ), in the category CO of schemes over O. We will assume X andX̃ are of finite type and smooth over O, have irreducible geometric fibers of dimension one (so they are “curves” over O), and σ1 , σ2 are étale. We have induced
correspondencesX℘ = (X℘ , σ℘ ), XC = (XC , σC )in the category of schemes over R℘ and C respectively. We will always assume
that the smallest equivalence relation in XC × XC containing the image of X̃C →XC × XC is Zariski dense in XC × XC so that the categorical quotient XC /σC istrivial in the category of schemes over C. As we already saw, for all ℘ 6∈ S, we may
consider a correspondenceXδ = (Xδ , σδ )in the category Cδ . Our general guiding conjecture will (roughly speaking) assert
that:
The categorical quotient Xδ /σδ is non-trivial for almost all places ℘ if and(essentially) only if the correspondence XC admits an “analytic uniformization”.

0.2. ROUGH OUTLINE OF THE THEORY

xv

Here, by XC admitting an analytic uniformization, we understand that there
exist morphisms of correspondences in the category of (open) Riemann surfaces (i.e.
in the category of complex analytic manifolds of dimension one), (ι, ι̃) : XC → YC ,0(π, π̃) : YC
→ YC , where ι, ι̃ are open immersions, π, π̃ are (possibly infinite)
0
Galois covers unramified above X, X̃, and YC= (S, S, τ1 , τ2 ), with S a simply
connected Riemann surface and τ1 , τ2 automorphisms of S. We always assume herethat the Galois groups of π and π̃ are admissible in the sense that they act properly discontinuously on S and are either finite (in case S = P1 ) or infinite (in case
S = C) or of finite covolume (if S = H, the upper half plane) respectively. According to whether S is P1 , C, or H there are three classes of correspondences admitting
analytic uniformizations which will be called spherical, flat, and hyperbolic correspondences. Analytically uniformizable correspondences can be completely classified in algebraic terms; cf. Chapter 1. Up to an appropriate equivalence relation
we have, very roughly speaking, the following description. (N.B. The actual situation is slightly more complicated.) The spherical uniformizable correspondences
are, essentially, of the form (P1 , P1 , π, π ◦ τ ) where π : P1 → P1 is a finite Galois
cover and τ : P1 → P1 is an isomorphism; the categorical quotient in the categoryof sets of such a correspondence is the same as the quotient of P1 by the action of
the group hΓ, τ i generated by τ and the Galois group, Γ, of π. Of course the grouphΓ, τ i is generally infinite. Flat uniformizable correspondences are, essentially, ofthe form (P1 , P1 , σ1 , σ2 ) where σi are either multiplicative functions t 7→ tdi or
Chebyshev polynomials, or Lattès functions [109] (the latter being induced by endomorphisms of elliptic curves E via isomorphisms E/hγi ' P1 where γ : E → E
is an automorphism). The hyperbolic uniformizable correspondences (with infinite
orbits) are, essentially, Hecke correspondences on modular or Shimura curves (thatare classically described in terms of quaternion algebras over totally real fields). So
the “if” part of our conjecture essentially says that our δ−geometric theory givesa rich picture in all the above examples; the “only if” part of the conjecture saysthat the above examples are, essentially, the only ones for which our δ−geometrictheory gives a rich picture.
0.2.5. Results. Our main results show that the “if” part of the conjecture
above holds (under some mild assumptions) in the spherical case, in the flat case,and in the “rational hyperbolic” case. (Here the “rational hyperbolic” case refers
to the case when S = H and the quaternion algebra describing the situation has
center Q.) For correspondences X admitting such a uniformization we will study the“geometry” and “cohomology” of the quotients Xδ /σδ . In particular we will showthat the quotients Xδ /σδ tend to behave like “rational varieties” and the quotient
maps Xδ → Xδ /σδ tend to look like “pro-finite covers” whose Galois properties weshall study. The “only if” part of the conjecture is much more mysterious. We will
be able to prove a local analogue of the “only if” part of our conjecture. Also we will
prove a global result along the “only if” direction saying (roughly speaking) that
if Xδ /σδ is “sufficiently non-trivial” for almost all places ℘ then XC is criticallyfinite (in the sense of complex dynamics) and X℘ ⊗ k℘ has a generically trivial
pluricanonical bundle. The latter property will allow us to prove a version of the“only if” part of the conjecture in the “dynamical system case”.
0.2.6. Proofs. Here are a few words about the strategy of our proofs. The
first move is to attach to any smooth scheme of finite type X℘ over R℘ = Ẑur
p a

xvi

INTRODUCTION

projective system of formal schemes(0.1)... → J r (X℘ ) → J r−1 (X℘ ) → ... → J 0 (X℘ ) = X̂℘such that for each r the ring Or (X℘ ) of δ−functions of order ≤ r on X℘ (R℘ )identifies with the ring O(J r (X℘ )) of global functions on J r (X℘ ):(0.2)

Or (X℘ ) ' O(J r (X℘ )).

The formal scheme J r (X℘ ) should be viewed intuitively as an “arithmetic jet space”
and will be referred to as the p−jet space of order r of X℘ . It is an arithmetic analogue of the jet space of a manifold (relative to a submersion) in differential geometry (cf. Equation 0.6) and, in particular, it carries certain structures reminiscentof structures in classical mechanics. The elements of the rings in Equation 0.2 canthen be viewed intuitively as “arithmetic differential equations” in the same way inwhich functions on jet spaces in differential geometry are interpreted as “differential
equations”. The above construction reduces the study of “δ−geometry” of X℘ tothe study of (usual) algebraic geometry of J r (X℘ ). Actually the construction ofp−jet spaces is quite easy to explain. Assume, for simplicity, that X℘ is affine given
by
(0.3)

X℘ = Spec R℘ [T ]/(F ),where T is a tuple of indeterminates and F is a tuple of elements in R℘ [T ]. Assume
furthermore, for simplicity, that the components of F have coefficients in Zp . ThenJ r (X℘ ) is, by definition, the formal spectrum(0.4)

J r (X℘ ) := Spf R℘ [T, T 0 , T 00 , ..., T (r) ]ˆ/(F, δF, δ 2 F, ..., δ r F )where ˆ means “p−adic completion”, T 0 , T 00 , ... are new tuples of variables andδF, δ 2 F, ... are defined by the formulae
F (T p + pT 0 ) − F (T )p,(0.5)(δF )(T, T 0 ) :=p
(δF )(T p + pT 0 , (T 0 )p + pT 00 ) − ((δF )(T, T 0 ))p,(δ 2 F )(T, T 0 , T 00 ) :=p
etc. The polynomials δF, δ 2 F, ... should be viewed as arithmetic analogues of “iterated total derivatives” of F and the construction of our “arithmetic jet spaces”
is then analogous to that of “differential algebraic jet spaces” to be discussed
presently; cf. Equation 0.20 below. Also, like in the case of the latter, and as
in the case of differential geometry, the fibers of the maps J r (X℘ ) → J r−1 (X℘ ) are(p−adic completions of) affine spaces.
Having reduced our problems to problems about p−jet spaces the next step isto construct, in each of the spherical, flat, and hyperbolic cases, some remarkableδ−invariants. In the spherical case this will be elementary. In the flat case theconstruction will be essentially based on our theory of δ−characters [17]; the latterare arithmetic analogues of the Manin maps in [97]. In the hyperbolic case theconstruction will be based on our theory of isogeny covariant δ−modular forms[21], [23], [24]. We will develop the theories of δ−characters and δ−modular forms
ab initio in this book; we will partly follow the above cited papers and then we
will further develop these theories up to a point where we can use them for ourapplications.
Finally we will need methods to prove that all δ−invariants essentially occur
via the constructions mentioned above. There will be a number of methods used

0.3. COMPARISON WITH OTHER THEORIES

xvii

for this purpose. There is, for instance, a Galois theoretic method that is sufficientto handle, in particular, the spherical case. Another method, especially useful inthe flat and hyperbolic case, is close in spirit to that used by Hilbert in his original
approach to invariant theory and is based on the existence of a certain system of(usual) partial differential operators acting on appropriate rings of invariants. In
our case the partial differential operators in question will live on arithmetic jet
spaces and will be derived by analogy with classical mechanics.
0.3. Comparison with other theories
0.3.1. Differential equations on smooth manifolds. Many of the δ− geometric concepts to be introduced and studied in this book are arithmetic analoguesof concepts related to differential equations on smooth manifolds derived from classical mechanics. Let us quickly review here some of these smooth manifold concepts.
Let M be a smooth manifold (i.e. a C ∞ real manifold) equipped with a submersion π : M → R. For each t ∈ R we think of π −1 (t) as the “configuration
space” at time t. (The map M → R should be viewed as the object whose arithmetic analogues are the morphisms of schemes, X → Spec F , defining varieties X
over number fields F .) For any smooth map f : R → M we denote by J0r (f ) ther−jet of f at 0. We define the jet space of M (relative to π) by(0.6)

J r (M ) := {J0r (f ) |f : R → M smooth, π ◦ f : R → R a translation}.

Then J r (M ) has a natural structure of smooth manifold. Any smooth sections : R → M of π : M → R lifts to a smooth map pr s : R → J r (M ) defined by(pr s)(t) := J0r (s ◦ τt ),where τt : R → R is the translation τt (u) := u + t. The smooth functions L :J r (M ) → R can be thought of as (time dependent) Lagrangians. Alternatively
we may think of such smooth functions L as differential equations on M . The
natural projections J r (M ) → J r−1 (M ) are fiber bundles with fiber Rn , where
dim M = n + 1.
Next, in order to simplify our discussion, we will work in coordinates. So we
let M := R × Rn with coordinates (t, x) = (t, x1 , ..., xn ) and we let π : M → R be
the first projection. Then we have a natural identification J r (M ) = R × Rn(r+1)
sending the r−jet J0r (f ) of a map f : R → M , f (t) = (t + t0 , x(t)), into the tuple


dr x
dx
t0 , x(0), (0), ..., r (0) .
dt
dt
If s : R → R × Rn , s(t) = (t, x(t)), is a section of π then pr s : R → R × Rn(r+1)
is given by


dx
dr x
(0.7)
(pr s)(t) = t, x(t), (t), ..., r (t) .
dt
dt
Cf. [114], p. 96. We denote by (t, x, x0 , ..., x(r) ) the coordinates on R × Rn(r+1) ,where x0 , ..., x(r) are n−tuples of variables; for each index i the variables x0i , x00i , ...
are the variables conjugate to xi . A (time dependent) Lagrangian is simply a smoothfunction(0.8)

L(t, x, x0 , ..., x(r) )on R × Rn(r+1) . Below is a list of key concepts related to the above context.

xviiiINTRODUCTION

A (vertical) vector field on M is a derivation operator (on smooth functions oft, x) of the form
X
∂
(0.9)
V :=
ai (t, x)∂xiiwhere ai are smooth functions. More generally a (vertical) generalized vector field(sometimes called Bäcklund vector field) is a derivation operator (on smooth functions of t, x) of the form
X∂;
(0.10)V :=ai (t, x, x0 , ..., x(r) )∂xii
cf. [114], p. 289.
The total derivative operator is the derivation operator (on smooth functionsof t, x, x0 , ..., x(r) ) defined by
X (j+1)
∂
∂
+
xi
·
;
(0.11)
Dt :=
(j)
∂t
∂xi
ij
cf. [114], p. 109. It is characterized by the property thatd
(L ◦ pr s)
dt
for all Lagrangians L and all sections s of π. One should view Dt as the “generatorof the Cartan distribution”.
The prolongation of a (vertical) vector field as in Equation 0.9 is defined by
X j∂(0.12)
pr V :=(Dt ai ) ·
;
(j)
∂xi
ij
(Dt L) ◦ (pr s) =cf. [114], p. 110; it is the unique (smooth) vector field on J r (M ) that commuteswith Dt and coincides with V on functions of t and x.
A vector field as in Equation 0.9 is an infinitesimal symmetry for an R−linear
space L of Lagrangians if(0.13)(pr V )(L) ⊂ L.

If (Li ) is a family of Lagrangians and L is the ideal generated by the family (Dtj Li )in the ring of all Lagrangians then any infinitesimal symmetry of L in the sense
above is an infinitesimal symmetry of the “system of differential equations Li = 0”in the sense of [114], p. 161.
The vector field V is a variational infinitesimal symmetry of the Lagrangian L
if(0.14)(pr V )(L) = 0;

cf. [114], p. 253.
The Fréchet derivative of a Lagrangian L is the operator on vector fields given
by
X j
X
∂L
∂
7→ (pr V )(L) = hpr V, dLi =(Dt ai ) ·;(0.15)
V =
ai
(j)
∂xi
∂xi
ij
i
cf. [114], p. 307.

0.3. COMPARISON WITH OTHER THEORIES

xix

The Euler-Lagrange form attached to a Lagrangian L is the 1−form

!
X X
∂L
 (−1)j Dtj
 dxi ;
(0.16)
EL(L) :=
(j)
∂x
i
j
i
cf. [114], p. 246.
Noether’s Theorem in this context is the (trivial) statement that for any Lagrangian L(t, x, x0 , ..., x(r) ) and any vector field V as in Equation 0.9 there exists
G = G(t, x, x0 , ..., x(2r−1) ) such that
(0.17)

hV, EL(L)i − (pr V )(L) = Dt G.

If V is a variational infinitesimal symmetry of L (i.e. Equation 0.14 holds) then G
is called the conservation law attached to V ; it is unique up to an additive constantand it has the property that if the function t 7→ x(t) satisfies the Euler-Lagrange
equations


d2r x
dx
∂
, EL(L)i t, x(t), (t), ..., 2r (t) = 0(0.18)
h∂xi
dt
dt
for i = 1, ..., n then the function
d2r−1 xdx
t 7→ G t, x(t), (t), ..., 2r−1 (t)dtdt
is constant.
Finally the Hamiltonian vector field attached to a function H = H(t, x, x0 ) andthe vector fields ∂/∂xi is the vector fieldX  ∂H ∂∂H ∂(0.19)−.
∂x0i ∂xi∂xi ∂x0ii
All the concepts above have arithmetic analogues some of which will play a keyrole in the proofs of our main theorems. The arithmetic analogues of jet spacesof manifolds, of total derivatives, and of Lagrangians were already encountered inour discussion of “arithmetic jet spaces”. The arithmetic analogue of prolongationof vector fields (and certain related operators) will play a key role in controllingδ−invariants in our theory. The arithmetic analogues of Fréchet derivatives willbe the tangent maps in δ−geometry. In some of our basic δ−geometric examples
we will prove the existence of “arithmetic (variational) infinitesimal symmetries”,
we will discover remarkable systems of commuting “arithmetic Hamiltonian vector
fields”, and we will compute “arithmetic Euler-Lagrange equations for free particles”. The way to the latter is as follows. The Lagrangian of a free particle
with one degree of freedom is (x0 )2 ; one should view this as attached to the group
Ga (R) = (R, +). If one formally sets y = ex then (x0 )2 = (y 0 /y)2 ; so (y 0 /y)2 should
be viewed as the Lagrangian for a free particle on the group Gm (R) = (R× , ·). Now
y 7→ y 0 /y is the basic differential character of Gm i.e. it is a group homomorphism(y1 y2 7→ (y10 /y1 )+(y20 /y2 )) defined by rational functions in the coordinates and theirderivatives. Manin maps (cf. the discussion below) provide differential charactersfor elliptic curves E. On the other hand our theory provides arithmetic analogues,
ψ, of differential characters for one dimensional algebraic groups Ga , Gm , E (which
we call δ−characters). So it is reasonable to define Lagrangians of free particles(with one degree of freedom) as squares, ψ 2 , of δ−characters for one dimensional
algebraic groups. Finally our arithmetic theory views δ−characters as “flat” andxx

INTRODUCTION

provides “spherical” and “hyperbolic” analogues of them called, in our book, f 1 .
Consequently we shall view the squares (f 1 )2 as spherical resp. hyperbolic analogues of Lagrangians of free particles (with one degree of freedom).
In order to clarify the analogy between the usual differential equations andarithmetic differential equations it will be useful to discuss an intermediate stepwhich is provided by the Ritt-Kolchin differential algebra [117], [84].
0.3.2. Differential algebra. The presentation here follows [13]. In differential algebra one usually starts with a field K equipped with a derivation δ : K → K;
recall that, by definition, δ is an additive map satisfying the usual Leibniz rule. We
shall assume, for simplicity, that K has characteristic zero and is algebraicallyclosed. In applications K usually contains the field C(t) of rational functions in
one variable and δ extends the derivation d/dt. (For the “most general” case of thistheory, in which K is replaced by any ring and δ is replaced by a “Hasse-Schmidt
derivation” we refer to Vojta’s preprint [131].) For any non-singular variety X over
K one can then define a projective system of non-singular varieties... → J r (X) → J r−1 (X) → ... → J 0 (X) = X
called the (differential algebraic) jet spaces of X. It is easy to review the construction of these varieties. Assume, for simplicity, that X is affine given by
X = Spec K[T ]/(F )where T is a tuple of indeterminates T1 , ..., TN and F is a tuple of elements in K[T ].
Then J r (X) is, by definition, the spectrum(0.20)J r (X) := Spec K[T, T 0 , T 00 , ..., T (r) ]/(F, δF, δ 2 F, ..., δ r F )where T 0 , T 00 , ... are new tuples of variables and δF, δ 2 F, ... are defined by the formulaeX ∂F(T ) · Tj0 ,(0.21)(δF )(T, T 0 ) := F δ (T ) +∂Tjj(δ 2 F )(T, T 0 , T 00 ) := (δF )δ +

X ∂δF
j

∂Tj

(T, T 0 ) · Tj0 +

X ∂δF
j

∂Tj0

(T, T 0 ) · Tj00 ,etc., where F δ , (δF )δ ,... are obtained from F, δF, ... by applying δ to the coefficients.
The polynomials δF, δ 2 F, ... are the iterated total derivatives of F . There is aclear analogy between the arithmetic jet spaces in Equation 0.4 and the differential
algebraic jet spaces J r (X) in Equation 0.20. On the other hand it can be shown([14], p.1443) that the K−points of the differential algebraic jet space J r (X) canbe identified with the morphisms of K−schemes,(0.22)

Spec K[[T ]]eT δ /(T r+1 ) → X,

where K[[T ]]eT δ is the ring of power series K[[T ]] viewed as a K−algebra via themap∞Xδn x nT .
eT δ : K → K[[T ]], eT δ (x) :=n!
n=0(The maps in Equation 0.22 should be viewed as a twisted version of arcs in X
where the twist is given by δ. Recall that usual arcs of order r in X are defined asmorphisms of K−schemes, Spec K[[T ]]/(T r+1 ) → X, where K[[T ]] is viewed as aK−algebra via the inclusion K ⊂ K[[T ]]; the set of all arcs of order r in X has a0.3. COMPARISON WITH OTHER THEORIES

xxi

natural structure of algebraic variety over K and is referred to as the arc space of
X of order r. Cf. [45]. Therefore the varieties J r (X) can be viewed as a twisted
analogue of arc spaces. They identify with arc spaces in case X is defined over thefield C = {a ∈ K|δa = 0} of constants of δ. But for X not descending to C arc
spaces are different from the varieties J r (X); cf. [14] for more on this.) Now there
is a clear analogy between the maps in Equation 0.22 and the jets belonging toJ r (M ) in Equation 0.6. This makes differential algebraic jet spaces an analogue ofjet spaces of smooth manifolds (relative to a submersion). The analogies describedabove can be summarized in the following diagram:(0.23)(1) − − − (2) − − − (3),(1)=jet spaces of smooth manifolds relative to a submersion; cf. Equation 0.6(2)=differential algebraic jet spaces; cf. Equation 0.20
(3)=arithmetic jet spaces; cf. Equation 0.4
Here are a few more remarks on differential algebraic jet spaces. As in the caseof smooth manifolds, the fibers of the maps J r (X) → J r−1 (X) are affine spaces.
Also there is a natural map at the level of K−points ∇ : X(K) → J r (X)(K)
naturally induced by the map ∇ : A1 (K) = K → J r (A1 )(K) = K r+1 defined by∇(a) = (a, δa, ..., δ r a). One defines the ring Or (X) of δ−polynomial maps on X
to be the ring of all functions f : X(K) → K that can be written as f = f˜ ◦ ∇where f˜ ∈ O(J r (X)) is a regular function on J r (X). One gets an isomorphism
Or (X) ' O(J r (X)) of which the isomorphism in Equation 0.2 is the arithmetic
analogue.
Next we address the following question: are the results of this book arithmetic
analogues of results involving (genuine) differential equations ? In a certain loose
sense this is indeed the case as explained below.
1) The spherical case of our theory here can be loosely viewed as an arithmetic analogue of some of the classical theory of differential invariants as found, forinstance, in [135], [114].
2) The flat case of our theory here can be loosely viewed as an arithmetic
analogue of the theory of the Manin map [97] and of the differential algebraic
theory we developed in [12], [14], [15].
3) The hyperbolic case of our theory here can be loosely viewed as an arithmetic
analogue of the classical theory of differential relations among modular forms andof the differential algebraic theory we developed in [16]; cf. also [8].
Let us provide some details.
0.3.2.1. Spherical case. We explain here a simple situation in the classical theory of differential invariants. Start with a group G acting by algebraic automorphisms on a variety X over K (which is usually affine). Then there is aninduced action of G on the jet spaces J r (X). The ring of G−invariant functions Or (X)G = O(J r (X))G corresponds to a special case of differential invariants in classical theory. A trivial example of this would be the natural actionof G := SL2 (C) on the affine plane A2 = Spec K[x, y] over K; the Wronskian
xy 0 − yx0 ∈ K[x, y, x0 , y 0 ] = O(J 1 (A1 )) is then G−invariant. The spherical case of
our arithmetic theory involves arithmetic analogues of computations of differential
invariants in the above sense.
More generally the classical differential invariant theory deals with groups G(or simply vector fields V ) acting directly on jet spaces J r (X) (in such a way that

xxii

INTRODUCTION

the action does not necessarily come “by functoriality” from an action on X itself).
The situation is described in detail in Olver’s book [114] in the setting of jet spaces
in differential geometry.
0.3.2.2. Flat case. As already mentioned our flat theory will involve an arithmetic analogue of the theory of Manin maps [97], [12], [14], [15]. Let us quickly
review the basics of the latter in the context of differential algebra following [12].
Assume X is an Abelian variety over K of dimension g. Then, by functoriality,each J r (X) has a natural structure of an algebraic group and is an extension of
X by a vector group (i.e. a sum of copies of the additive group Ga = Spec K[t]).
By the theory of extensions of Abelian varieties by vector groups there is a surjective homomorphism J r (X) → H onto a vector group H of dimension ≥ (r − 1)g.
Consequently if r ≥ 2 we have H 6= 0 hence Hom(J r (X), Ga ) 6= 0. Elements of
the latter composed with ∇ give rise to homomorphisms ψ : X(K) → K. These
homomorphisms are called Manin maps. They were discovered by Manin [97] (viaa construction different from the one above) and were used by him to provide thefirst proof of the Mordell conjecture over function fields. Now the strategy presented above works well in the arithmetic case. The p−jet spaces J r (X℘ ) of an
Abelian scheme X℘ over R℘ continue to be group objects and, although the kernelof the projection J r (X℘ ) → X̂℘ ceases to be a sum of copies of Ĝa := Spf R℘ [t]ˆ,one can still prove that Hom(J r (X℘ ), Ĝa ) 6= 0 provided r ≥ 2. As before elementsof the latter yield δ−functions ψ : X(R℘ ) → R℘ which are homomorphisms; theseshould be viewed as arithmetic analogues of Manin’s maps and play a key role inour theory in the flat case. Indeed maps of the form φ◦ψψ are invariant under themultiplication by integers N on our Abelian variety, [N ] : X℘ → X℘ ; this invarianceis the starting point for our use of ψ in our search for invariants in the flat case.
0.3.2.3. Hyperbolic case. We end our discussion of differential algebra by reviewing “differential algebraic invariants of isogenies” following [16]; this theory
has an analogue in the arithmetic case that plays a key role in our treatment of
the hyperbolic case. Let us consider the affine line A1 = Spec K[j] viewed as amoduli space for elliptic curves over K. For any elliptic curve E over K we letj(E) ∈ K be its j−invariant. On K = A1 (K) we have then an equivalence relation
called isogeny: two points a1 , a2 ∈ K are called isogenous if there exists an isogenyof elliptic curves E1 → E2 over K such that j(E1 ) = a1 and j(E2 ) = a2 . The
constant field C ⊂ K is saturated with respect to isogeny hence so is K\C. Then
one can show [16] that there exists a rational function U (j) ∈ Q(j) such that if(0.24)

χ(j, j 0 , j 00 , j 000 ) :=2j 0 j 000 − 3(j 00 )2+ (j 0 )2 U (j) ∈ K(j)[j 0 , (j 0 )−1 , j 00 , j 000 ]4(j 0 )2

then the function
K\C → K, a 7→ χ(a, δa, δ 2 a, δ 3 a)is constant on isogeny classes. (The fraction in Equation 0.24 is, of course, theclassical Schwarzian operator.) We refer to [16] for more on this and for a higher
dimensional generalization of this. It turns out that the theory behind the function
χ above has an arithmetic analogue which leads to what we call δ−modular forms;
cf. [21], [23], [24]. These will be used heavily in this book.
0.3.3. Difference algebra. In the Ritt-Cohn difference algebra [35] one starts
with a field K equipped with a ring endomorphism φ : K → K. For our discussion

0.3. COMPARISON WITH OTHER THEORIES

xxiii

here we will allow K to have arbitrary characteristic. For any variety X over K
one can then define a projective system of non-singular varieties... → Jφr (X) → Jφr−1 (X) → ... → Jφ0 (X) = X
which can be called the difference jet spaces of X. If X is affine given by
X = Spec K[T ]/(F )where T is a tuple of indeterminates and F is a tuple of elements in K[T ] thenJφr (X) is, by definition, the spectrum
2r

Jφr (X) := Spec K[T, T φ , T φ , ..., T φ ]/(F, φF, φ2 F, ..., φr F )(0.25)

2

where T φ , T φ , ... are new tuples of variables andφ : K[T, T φ , ..., T φ

r−1r] → K[T, T φ , ..., T φ ]is the unique ring endomorphism prolonging φ : K → K and sending φT = T φ ,2
φT φ = T φ ,... e.t.c. A basic difference between difference algebra and differential
algebra is that the fibers of Jφr (X) → Jφr−1 (X) are not affine spaces in general.
Actually it is plain that Jφr (X) splits as a product of varietiesr

Jφr (X) ' X × X φ × ... × X φ ,iwhere X φ is obtained from X by twisting the coefficients of the defining equations
by φi . In particular, if X is definable over the fixed field of φ then Jφr (X) is justthe (r + 1)−fold product of X with itself. In spite of this geometric difference
between difference algebra and differential algebra a lot of analogies between thetwo persist. Systems of linear equations, for instance, behave similarly [62], p. 20.
At the non-linear level the geometric model theory of the two is strikingly similar;
cf. especially the work of Hrushovski and Chatzidakis explained in [69].
It is interesting to compare difference algebra and arithmetic differential algebra. We refer to Remark ?? below for an argument suggesting that arithmetic
differential algebra could be viewed as obtained from difference algebra by adjoining certain divergent series. Here we close our discussion by indicating a possible
bridge at the level of jet spaces. Let us assume, for simplicity, that X is an affine
scheme given by the spectrum of R℘ [T ]/(F ) as in Equation 0.3. Let K℘ be the fieldof fractions of R℘ . Then one can consider a canonical algebraisation of the p−jet
space J r (X℘ ) given as the scheme(0.26)J r (X℘ ) := Spec R℘ [T, T 0 , T 00 , ..., T (r) ]/(F, δF, δ 2 F, ..., δ r F ).

Then clearly the p−adic completion of J r (X℘ ) is J r (X℘ ); in symbols:(0.27)J r (X℘ )ˆ = J r (X℘ ).

On the other hand it is trivial to see that the base change to K℘ of J r (X℘ ) isJφr (X℘ ⊗ K℘ ); in symbols,(0.28)J r (X℘ ) ⊗ K℘ = Jφr (X℘ ⊗ K℘ ).

By Equations 0.27 and 0.28 we see that J r (X℘ ) might be viewed as a “bridge”
between the jet spaces of difference algebra and arithmetic differential algebra.

xxiv

INTRODUCTION

Whether this bridge can actually be crossed is an open question. We can summarize
the above discussion by adjoining to the diagram in Equation 0.23 the diagram:(0.29)

(2)
|
|
(4)

−−−

−−−

(3)
|
|
(5)

(2)=differential algebraic jet spaces; cf. Equation 0.20
(3)=arithmetic jet spaces; cf. Equation 0.4(4)=difference jet spaces; cf. Equation 0.25(5)=canonical algebraisation of arithmetic jet spaces; cf. Equation 0.26.
0.3.4. Dynamical systems. Let us start by reviewing some aspects of theclassical iteration theory initiated by Fatou and Julia [109]. In this theory one
starts with a rational map s : P1 → P1 from the complex projective line into itself
and one studies the behavior of orbitsz, s(z), s2 (z), ...
as z moves in P1 . The z’s for which the orbit is “unstable” form a closed set called
the Julia set. The complement of the Julia set is called the Fatou set. Points thateventually return to themselves after a number of iterations are called periodic. Up
to finitely many points the Julia set turns out to coincide with the closure of the setof periodic points and usually looks like a “fractal”. A point z ∈ C is critical if thetangent map of s at z vanishes. The orbits of critical points hold the key to many ofthe dynamical properties of s. In particular the maps s for which all critical points
have finite orbits have been closely investigated by Thurston [51]; they are called
postcritically finite and one can attach orbifolds and Euler characteristics to them.
Postcritically finite maps with Euler characteristic 0 have been classified and they
are essentially those that admit an “analytic uniformization” by the complex plane
C. Let us call them here, for simplicity, flat maps. Flat maps can be explicitly
described as being either multiplicative maps, or Chebyshev polynomials or Lattèsmaps (the latter being maps induced by endomorphisms of elliptic curves). The
Julia sets of flat maps are “smooth” (i.e. not “fractals”). For multiplicative maps
the Julia set is a circle. For Chebyshev maps the Julia set is a segment. For Lattèsmaps the Julia set is the whole of P1 . (Note however that flat maps are not the
only maps that have smooth Julia sets !)Flat maps play a key role in our theory. Indeed let us attach to any rational map
s a correspondence X∗ = (P1 , P1 , id, s) in complex algebraic geometry. Assume,for simplicity, that s has rational coefficients. Fix a prime p. Then, as explained
earlier, we can (roughly speaking) attach to X a correspondence Xδ in δ−geometryat p. What we will prove in this book is, roughly speaking, that the categoricalquotient of Xδ in δ−geometry is non-trivial for almost all p if and only if s is a flat
map.
More generally the concept of postcritical finiteness (and a suitable generalization of it to the case of correspondences that do not come from dynamical systems)
plays a key role in the present book. This will be explained in detail in Chapters 1
and 2.
We close our discussion here by noting that complex dynamics has an interestingp−adic analogue. For instance, the Lubin-Tate theory of formal groups [92], [93]

0.3. COMPARISON WITH OTHER THEORIES

xxv

can be viewed as an early incarnation of a non-archimedian dynamics. For more
recent work see, for instance [93], [5], [6], [91], and the bibliographies therein.
Although our theory lives in the p−adic world and although we will use formalgroups at periodic points in our theory, we will not need to use, in our book, the“genuine” theory of p−adic dynamical systems in the above cited papers. Here, by“genuine” we mean “dealing with the p−adic Julia set, p−adic wandering Fatou
domains, etc.” (On the other hand, as already mentioned, we do need to use thetheory of complex dynamical systems through the orbifold characterization of flat
maps.) Needless to say it would be very interesting to make our theory interact
with the “genuine” theory of p−adic dynamical systems.
0.3.5. Connes’ non-commutative geometry. The approach in the present
book seems to be, in some sense, perpendicular to Connes’ non-commutative geometric approach to quotient spaces; cf. our previous remarks about invariant versus
groupoid ideology. Nevertheless, since some examples occur in both theories it is
not unreasonable to expect interactions between the two approaches. In what follows we shall discuss some general principles of non-commutative geometry and weshall examine some examples. Our discussion will inevitably be extremely superficial and not entirely precise; for precise, in-depth presentations we refer to [36],[99], [101].
Connes’ theory is formulated in the context of functional analysis and, althoughsome parts of it have a more algebro-geometric flavor [127] [99], our presentation
will follow Connes’ original approach. As already mentioned the starting point inthis theory is often a groupoid
G = (X, X̃, σ1 , σ2 , ν, ι, )in some “geometric” category, for instance the category of smooth manifolds. Writeγ1 γ2 = ν(γ1 , γ2 ). The first step is to attach to G a C ∗ −algebra, C ∗ (G), called theconvolution algebra of G. Naively C ∗ (G) might be thought of as (a norm completionof) some algebra of complex valued functions f on X̃ with multiplication given by“convolution”:
X(f1 ? f2 )(γ) =f1 (γ1 )f2 (γ2 ).
γ1 γ2 =γAs it is this cannot work because the sums involved are usually infinite. So one
replaces, according to the context, sums by integrals, functions by “densities”, etc.
The next step in Connes’ theory is the realization that “non-commutative
spaces” should not be defined as C ∗ −algebras but rather as Morita equivalenceclasses of C ∗ −algebras. Two C ∗ −algebras A and B are Morita equivalent if there
exists a bimodule A MB with certain extra data that allow one to “transfer” modules
between A and B. So, roughly speaking, one defines the non-commutative space
attached to the groupoid G as the C ∗ −algebra C ∗ (G) up to Morita equivalence.
At this point one is faced with the “topological and geometrical study of noncommutative spaces”; this constitutes the heart of the theory and, for this purpose,
a whole spectrum of techniques is brought into the picture such as: spectral theory,
index theory, K−theory, etc.
In what follows we examine a number of examples in non-commutative geometry and compare them with the spherical, flat, and hyperbolic examples we areable to treat in our theory.

xxvi

INTRODUCTION

0.3.5.1. Spherical case. Consider the action of P SL2 (Z) on P1 (C) by homographies. Note that
P1 (C) = H± ∪ P1 (R), H± := C\R = H+ ∪ H− ,H+ = H is the upper half plane in C, H− is the lower half plane, and P1 (R) is thereal projective line. The action of P SL2 (Z) on H± is properly discontinuous andthe quotient exists in the category of Riemann surfaces; it is the union of 2 copiesof the modular curve H/P SL2 (Z) parameterizing elliptic curves over the complex
numbers. However the action of P SL2 (Z) on the “stratum” P1 (R) is not properly
discontinuous and actually the categorical quotient in the category of manifolds
reduces to a point. What can be done in non-commutative geometry is to consider
the groupoid G attached to the action
P SL2 (Z) × P1 (R) → P1 (R)and consider the C ∗ −algebra C ∗ (G) modulo Morita equivalence. The latter is declared to be, by definition, the “quotient” P1 (R)/P SL2 (Z) and can be interpreted,in a natural way as a compactification of the modular curve H/P SL2 (Z); cf. theremarks in the next example. The non-commutative space P1 (R)/P SL2 (Z) has
been studied by Manin and Marcolli [100] who proved a number of deep results
about “limiting” behavior of objects from the usual modular curve H/P SL2 (Z)that tend to points on the boundary.
On the other hand the space of orbits of P1 under the action of P SL2 (Z) will
be one of the main examples of our theory in the spherical case; cf. the present
Introduction. It would be beautiful to understand if there is a connection between
the two approaches. Remark that from our perspective P1 modulo P SL2 (Z) is in a
“spherical situation” while in non-commutative geometry P1 (R) modulo P SL2 (Z)is at the boundary of a “hyperbolic commutative geometric object”.
0.3.5.2. Flat case. One of the first remarkable objects studied in non-commutative geometry were the non-commutative tori of Connes and Rieffel [36], [40].
There are analogies between these objects and the objects relevant in the flat caseof our theory. Let us briefly explain this in the case of “complex dimension” one. A
complex torus of dimension one is an elliptic curve E over C. One can represent E
as the quotient Eτ (in the category of Riemann surfaces) of the complex plane C by
the action (by translation) of the subgroup Z + τ Z, where τ ∈ H. Alternatively one
can describe Eτ as the quotient (in the category of Riemann surfaces) of C× by theaction of the subgroup hqτ i where qτ = e2πiτ . As τ approaches a point θ ∈ R\Q,qτ will approach qθ := e2πiθ which is on the unit circle S 1 ⊂ C, but not a root ofunity. In non-commutative geometry it is possible to define an object which canbe viewed as the limit of a family of elliptic curves Eτ when τ → θ. Here is theconstruction. One considers the actionµθ : Z × S 1 → S 1 , µθ (m, z) := qθm · z,
one attaches to this action a groupoid Gθ and one defines the C ∗ −algebra Aθ :=C ∗ (Gθ ); the Morita equivalence class of Aθ is interpreted as a non-commutative
elliptic curve which is the limit of Eτ as τ → θ. One of the crucial early discoveries
in non-commutative geometry was that for two irrational real numbers θ1 andθ2 the C ∗ −algebras Aθ1 and Aθ2 are Morita equivalent if and only if θ1 and θ2
are P SL2 (Z)−conjugate. This justifies the claim in the previous example that

0.3. COMPARISON WITH OTHER THEORIES

xxvii

P1 (R)/P SL2 (Z) can be interpreted as a compactification of the modular curve
H/P SL2 (Z). (Indeed P1 (Q)/P SL2 (Z) consists of one point, the classical cusp.)
Now let us note an analogy between the above example and what will be involved in the flat case of our theory. The quotient of S 1 by the action ρθ is a quotient
S 1 /hqθ i of the Lie group S 1 by a cyclic subgroup (which acts “wildly”). Accordingly
one can propose to study (categorical) quotients of the form G/hg1 , ..., gn i where G
is a commutative algebraic group and hg1 , ..., gn i is a finitely generated subgroup.
Such quotients have actually been studied in differential algebra [12], [13] in relation to the Mordell conjecture over function fields and their study involves theManin maps discussed before. As already mentioned Manin maps have arithmetic
analogues that play a key role in this book.
0.3.5.3. Hyperbolic case. Hecke correspondences play an important role in noncommutative geometry; cf. [39], [37], [38], [101]. On the other hand Hecke correspondences will be at the heart of the hyperbolic case of the present book. We
would like to compare the two theories in what follows. Let, as usual, Af denotethe finite adeles of Q i.e.


Z
⊗ Q.
Af := lim
← nZ
Then, following Shimura, one considers the space(0.30)

Sh = GL2 (Q)\(GL2 (Af ) × H± ).

For any congruence subgroup Γ ⊂ SL(2, Z) consider the modular curveShΓ := H/Γ,
where we wrote H/Γ instead of Γ\H in order to make notation agree with notationin the present book. One can consider then the inverse limit
Sh0 := lim ShΓ
←over all congruence subgroups Γ ⊂ SL2 (Z). The space Sh0 turns out to be aconnected component of Sh. All this is classical, i.e. commutative. Now, following[38] one can embed the set Sh into the “much larger” setSh(nc) := GL2 (Q)\(M2 (Af ) × H± ).
The set Sh(nc) does not have a nice “commutative space” structure but one canattach to it a C ∗ −algebra A2 starting from a certain (alternative !) presentationof Sh(nc) as a quotient by an action; cf. [101], p. 67. We refer to loc. cit. foran overview of the beautiful and deep theory of A2 and also for an overview of the
Bost-Connes A1 that was its GL1 prototype.
Now turning to our theory let us consider a prime number l, the matrix


l 0
τl :=
,
0 1
and the subgrouphΓ, τl i ⊂ GL2 (Q)(where Γ ⊂ SL2 (Z) is any congruence subgroup). Define the set
ShhΓ,τl i := H/hΓ, τl i.
This set should be viewed, roughly, as the quotient of ShΓ by a Hecke correspondence. Since the group hΓ, τl i does not act properly discontinuously on H the set

xxviii

INTRODUCTION

ShhΓ,τl i is not an object of analytic geometry anymore. Our theory can be viewed asa way to put on the set ShhΓ,τl i a geometric structure in δ−geometry. Summarizing
we have the following inclusions of sets
Sh0 ⊂ Sh ⊂ Sh(nc)and the following projections of sets
Sh0 → ShΓ → ShhΓ,τl i .
The set Sh(nc) is replaced, in non-commutative geometry, by the C ∗ −algebra A2
while the set ShhΓ,τl i is replaced, in our theory, by an object of δ−geometry. It isplain from the above discussion that the non-commutative theory and our theory“diverge” in this example. The set we are interested in, ShhΓ,τl i , is a “wild” quotientof the “commutative part” of Sh(nc) .
0.3.6. Drinfeld modules. The theory of Drinfeld modules [52], [62], [130]exhibits analogies with our theory that deserve being understood. Let us quickly
recall the definition of a Drinfeld module and make a few comments. One starts
with a smooth projective geometrically connected curve X over a finite field Fpm
equipped with a point ∞ ∈ X. Let A be the ring of regular functions on X\{∞}.
Let ι : A → A be a homomorphism into a field A. Let F : A → A be the pm −powerhomomorphism. Let A[F ] be the non-commutative ring generated by A and amvariable, called F , with relations F · a = F (a) · F = ap · F , a ∈ A. (Morally A[F ]is a convolution algebra attached
Pto the action of F on A.) Let e : A[F ] → A be thering homomorphism sending e( ai F i ) = a0 and let j : A → A[F ] be the inclusion.
Then a Drinfeld module is an Fpm −algebra homomorphism ψ : A → A[F ] suchthat e ◦ ψ = ι, ψ 6= j ◦ ι. The ring A[F ] acts on A (with the letter F acting on A as
the endomorphism F and A acting on A by multiplication). Hence A will act on A
via ψ. The theory of Drinfeld modules can be viewed therefore as a very non-trivial
example of “algebra / geometry with operators”; it entirely lives in characteristic
p. The ring A is viewed, in this theory, as an analogue of the ring Z of integers.
Its quotient field k plays the role of Q. The completion K of k at ∞ plays the roleof the real field, R. The completion C∞ of an algebraic closure of K plays the roleof the complex field C. The “first” examples of the theory are Drinfeld modules
ψ : A → C∞ [F ] attached to “lattices”. Drinfeld modules of ranks 1 and 2 play therole of one dimensional algebraic groups (multiplicative group and elliptic curves
respectively), viewed as Z−modules (or sometimes modules over rings of integers inimaginary quadratic fields). They can also be viewed as analogues of formal groups
viewed as Z−modules (or sometimes modules over rings of integers in p−adic fields).
There are analogies between the exponentials in the theory of Drinfeld modules andour arithmetic Manin maps; both classes of maps have, as kernels, objects that canbe viewed as “lattices”. It is a suggestion of Manin (cf. private communication
to the author) that maybe there exists a lifting to characteristic zero of Drinfeld
module theory which is similar in spirit to the arithmetic differential algebra of thepresent book.
0.3.7. Dwork’s theory. It is interesting to compare our theory of arithmetic
differential equations with what is usually understood by the arithmetic theory ofdifferential equations (as it appears in the work of Dwork, for instance; cf. [55]).
It turns out that the two theories are “perpendicular” in the very precise sensethat they deal with “differentiation” in two “perpendicular” directions. Indeed

0.3. COMPARISON WITH OTHER THEORIES

xxix

Dwork’s theory is about genuine differential equations (i.e. equations involvingd/dt) whereas our theory is about analogues of differential equations (involving,instead, the Fermat quotient operator δ). In a precise sense d/dt and δ point in
two different directions. Now although the two theories are perpendicular thisdoesn’t mean they don’t interact. Indeed part of Dwork’s theory can be rephrased
in terms of crystalline cohomology; cf. the report by Katz [82]. The genuine
differential equations can be read off the Gauss-Manin connection whereas ourarithmetic differential equations are closely related to the action of Frobenius. On
the other hand Gauss-Manin and Frobenius are, as well known, both present in the
crystalline picture. We will actually heavily rely in our proofs on this crystalline
interpretation when we get to discuss the hyperbolic case.
0.3.8. Mochizuchi’s p−adic Teichmüller theory. An interesting problem
is to find interactions between our theory here and Mochizuchi’s p−adic Teichmüller
theory [111]. Unlike the Tate and Mumford p−adic uniformizations of curves
with (totally) degenerate reduction mod p both our theory and Mochizuchi’s canbe viewed, in a certain sense, as “uniformization” theories in the case of good
reduction. As an example of what this might mean in the case of our theory wepropose to see the “inverse” of our arithmetic Manin map of an elliptic curve E (cf.
our discussion above) as a multivalued map from the affine line A1 into E. Thiscan be interpreted, in some bold sense, as a sort of uniformization of E.
0.3.9. Ihara’s congruence relations. There an interesting possibility thata profitable link can be established between our approach here and Ihara’s beautiful
work on congruence relations (i.e. liftings to characteristic zero of the correspondence Π ∪ Π0 on a curve of characteristic p, where Π is the graph of the Frobenius);
cf. [73], [74]. A hint as to such a link can be seen, for instance, in Lemma ??
of this book. On the other hand we would like to point out what we think is an
important difference between our viewpoint here and the viewpoint proposed by
Ihara in a related paper [75]. Our approach, in its simplest form, proposes to see
the operatora − ap,δ = δp : Z → Z, a 7→ δa =pwhere p is a fixed prime, as an analogue of a derivation with respect to p. In [75]Ihara proposed to see the mapYa − ap(0.31)
d:Z→
Z/pZ, a 7→
mod p
p
p
as an analogue of differentiation for integers and he proposed a series of very interesting conjectures concerning the “zeroes” of the differential of an integer; these conjectures are still completely open. The main difference between Ihara’s viewpoint
and ours is that we do not consider the reduction mod p of the Fermat quotients but
the Fermat quotients themselves. This allows the possibility of considering iterates
δ r of our δ which leads to the possibility of considering arithmetic analogues ofhigher order differential equations; and indeed most such equations relevant to ourtheory will have order ≥ 2. On the other hand it doesn’t make sense to consider
iterates of Ihara’s operator d. One way out of this dilemma would be
Q to find a map
D : Z → Z which, composed with the canonical projection Z → p Z/pZ, yields
Ihara’s map d. (Then one could consider iterates Dr of D.) Voloch [132] proved,

xxx

INTRODUCTION

however, that no such map D can exist at least if one assumes the conjecture that
there are only finitely many Mersenne primes.
0.3.10. Field with one element. The theory proposed in this book seems
to agree with certain aspects of the “myth of the field with one element” emerging
from the work of Kurokawa, Soulé, Deninger, Manin, and others; cf. [86], [128],urur[98], [46]. For instance the constants a ∈ Ẑurp , δa = 0, of the map δ : Ẑp → Ẑpurconsist of the roots of unity in Ẑp together with 0; on the other hand the roots ofunity and 0 are, according to the philosophy of the field with one element, precisely
the F̄1 − points of the affine line where F̄1 is the “algebraic closure of the field with
one element, F1 ”. Not all the predictions of the philosophy of the field with oneelement seem however to be in agreement with our theory. For instance there seems
to be an important ideological difference between our operatora − ap,δp : Z → Z, δp a =pand the absolute derivation introduced in [86],∂∂ e: Z → Z,(p m) := epe−1 m,∂p∂pwhere m ∈ Z is prime to p. Indeed the constants of δp are 0 and ±1 while theconstants of ∂/∂p are 0 and all the integers coprime to p. This suggests that,
according to the philosophy of F1 , there is no theory of F1 that involves one prime
only; the only theory about F1 would involve all the primes at the same time. On
the contrary, in our book, there is a theory for each individual prime.

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