Foundations of arithmetic differential geometry
Alexandru Buium
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
E-mail address: buium@math.unm.edu

2010 Mathematics Subject Classification. 11E57, 11F85, 14G20, 53C21
Key words and phrases. classical groups, p−adic numbers, curvaturePreface
Aim of the book. The aim of this book is to introduce and develop anarithmetic analogue of classical differential geometry; this analogue will be referred
to as arithmetic differential geometry. In this new geometry the ring of integers Z
will play the role of a ring of functions on an infinite dimensional manifold. The
role of coordinate functions on this manifold will be played by the prime numbersp ∈ Z. The role of partial derivatives of functions with respect to the coordinatesp∈ Z, of integers n ∈ Z withwill be played by the Fermat quotients, δp n := n−nprespect to the primes p. The role of metrics (respectively 2-forms) will be played bysymmetric (respectively anti-symmetric) matrices with coefficients in Z. The role ofconnections (respectively curvature) attached to metrics or 2-forms will be playedby certain adelic (respectively global) objects attached to matrices as above. Oneof the main conclusions of our theory will be that (the “manifold” correspondingto) Z is “intrinsically curved;” this curvature of Z (and higher versions of it) will
be encoded into a Q-Lie algebra holQ , which we refer to as the holonomy algebra,
and the study of this algebra is, essentially, the main task of the theory.
Needless to say, arithmetic differential geometry is still in its infancy. However,its foundations, which we present here, seem to form a solid platform upon which
one could further build. Indeed, the main differential geometric concepts of thistheory turn out to be related to classical number theoretic concepts (e.g., Christoffel
symbols are related to Legendre symbols); existence and uniqueness results for themain objects (such as the arithmetic analogues of Ehresmann, Chern, Levi-Cività,and Lax connections) are being proved; the problem of defining curvature (whichin arithmetic turns out to be non-trivial) is solved in some important cases (via
our method of analytic continuation between primes and, alternatively, via algebraization by correspondences); and some basic vanishing/non-vanishing theorems
are being proved for various types of curvature. It is hoped that all of the above
will create a momentum for further investigation and further discovery.
Immediate context. A starting point for this circle of ideas can be foundin our paper [23] where we showed how to construct arithmetic analogues of theclassical jet spaces of Lie and Cartan; these new spaces were referred to as arithmetic
jet spaces. The main idea, in this construction, was to replace classical derivation
operators acting on functions with Fermat quotient operators acting on numbers andto develop an arithmetic differential calculus that parallels classical calculus. There
were two directions of further development: one towards a theory of arithmetic
differential equations and another one towards an arithmetic differential geometry.
A theory of arithmetic differential equations was developed in a series of papers[23]-[42], [7] and was partly summarized in our monograph [35] (cf. also thesurvey papers [43, 102]); this theory led to a series of applications to invariantv

vi

PREFACE

theory [27, 7, 28, 35], congruences between modular forms [27, 37, 38], andDiophantine geometry of Abelian and Shimura varieties [24, 36]. On the other
hand an arithmetic differential geometry was developed in a series of papers [42][47], [8]; the present book follows, and further develops, the theory in this latter
series of papers.
We should note that our book [35] on arithmetic differential equations andthe present book on arithmetic differential geometry, although both based on thesame conceptual framework introduced in [23], are concerned with rather different
objects. In particular the two books are independent of each other and the overlap between them is minimal. Indeed the book [35] was mainly concerned witharithmetic differential calculus on Abelian and Shimura varieties. By contrast,
the present book is concerned with arithmetic differential calculus on the classicalgroupsGLn , SLn , SOn , Spn ,and their corresponding symmetric spaces.
Of course, the world of Abelian/Shimura varieties and the world of classicalgroups, although not directly related within abstract algebraic geometry, are closely
related through analytic concepts such as uniformization and representation theory.
The prototypical example of this relation is the identification
M1 (C) ' SL2 (Z)\SL2 (R)/SO2 (R)where M1 is the coarse moduli space of Abelian varieties
C/latticeof dimension one; the curve M1 is, of course, one of the simplest Shimura varieties. It is conceivable that the analytic relation between the two worlds referred
to above could be implemented via certain arithmetic differential correspondences(correspondences between arithmetic jet spaces); in particular it is conceivable thatthe subject matter of the present book and that of our previous book [35] might
be related in ways that go beyond what can be seen at this point. A suggestion forsuch a possible relation comes from the theory of δ-Hodge structures developed in[22] and from a possible arithmetic analogue of the latter that could emerge from[7, 35].
There are at least two major differential geometric themes that are missing
from the present book: geodesics and the Laplacian. It is unclear at this point
what the arithmetic analogues of geodesics could be. On the other hand, for firststeps towards an arithmetic Laplacian (and, in particular, for a concept of curvature
based on it), we refer to [34]. But note that the flavor of [34] is quite different fromthat of the present book. Again, it is conceivable that the arithmetic Laplacian
theory in [34] could be related, in ways not visible at this point, with our theory
here.
Larger context. By what has been said so far this book is devoted to unveiling a new type of “geometric” structures on Z. This is, of course, in line withthe classical effort of using the analogy between numbers and functions to theadvantage of number theory; this effort played a key role in the development of
number theory throughout the 20th century up to the present day, as reflectedin the work of Dedekind, Kronecker, Hilbert, Artin, Weil, Iwasawa, Grothendieck,PREFACE

vii

Manin, Parshin, Arakelov (to name just a few of the early contributors). However, as we shall explain in the Introduction, this classical theory (which views
number fields as analogous to Riemann surfaces) differs in essential ways from thetheory of the present book (which views number fields as analogous to infinite dimensional manifolds). Our theory should also be contrasted with more recently
proposed paradigms for “the discrete” such as: arithmetic topology [104]; the theory around the p-curvature conjecture of Grothendieck [87]; the Ihara differential[78]; the Fontaine-Colmez calculus [64]; discrete differential geometry [10]; and thegeometries over the field with one element, F1 , proposed by a number of people including: Tits [118], Kapranov-Smirnov [86], Soulé [115], Deitmar [56], Kurokawa
et.al. [93], Connes-Consani [55], Manin-Marcolli [117], Lorscheid [117], Haran[70], etc. The non-vanishing curvature in our theory also prevents our arithmetic
differential geometry from directly fitting into Borger’s λ-ring framework [13] for
F1 ; indeed, roughly speaking, λ-ring structure leads to zero curvature. For eachindividual prime, though, our theory is consistent with Borger’s philosophy of F1 ;cf. [13, 102] and the Introduction to [35]. In spite of the differences between these
theories and ours we expect interesting analogies and interactions.
This being said what is, after all, the position of our theory among more established mathematical theories? The answer we would like to suggest is that ourcurvature of Z, encoded in the Lie algebra holQ , could be viewed as an infinitesimalcounterpart of the absolute Galois group ΓQ := Gal(Qa /Q) of Q: our Lie algebra holQ should be to the absolute Galois group ΓQ what the identity component
Hol0 of the holonomy group Hol is to the monodromy group Hol/Hol0 in classicaldifferential geometry. As such our holQ could be viewed as an object of study in
its own right, as fundamental, perhaps, as the absolute Galois group ΓQ itself. A“unification” of holQ and ΓQ may be expected in the same way in which Hol0 and
Hol/Hol0 are “unified” by Hol. Such a unification of holQ with ΓQ might involve
interesting Galois representations in the same way in which the unification of ΓQ
with the geometric fundamental group via the arithmetic fundamental group in
Grothendieck’s theory yields interesting Galois representations [77, 57].

Organization of the book. The book starts with an Introduction in which
we give an outline of our theory and we briefly compare our theory with othertheories.
In Chapter 1 we briefly review the basic algebra and superalgebra concepts
that we are going to use throughout the book. Most of this material is standard so
the reader may choose to skip this chapter and consult it later as needed.
Chapter 2 will be devoted to revisiting classical differential geometry from analgebraic standpoint. We will be using the language of differential algebra [91, 112],
i.e., the language of rings with derivations. We call the attention upon the factthat some of the classical differential geometric concepts will be presented in asomewhat non-conventional way in order to facilitate and motivate the transition
to the arithmetic setting. This chapter plays no role in our book other than thatof a motivational and referential framework; so, again, the reader may choose to
skip this chapter and then go back to it as needed or just in order to compare thearithmetic theory with (the algebraic version of) the classical one.
Chapter 3 is where the exposition of our theory properly begins; here we present
the basic notions of arithmetic differential geometry and, in particular, we introduce

viii

PREFACE

our arithmetic analogues of connection and curvature. The theory will be presented
in the framework of arbitrary (group) schemes.
Chapter 4 specializes the theory in Chapter 3 to the case of the group scheme
GLn ; here we prove, in particular, our main existence results for Ehresmann, Chern,
Levi-Cività, and Lax connections respectively.
Chapters 5, 6, 7, 8 are devoted to the in-depth analysis of these connections; inparticular we prove here the existence of the analytic continuation between primes
necessary to define curvature for these connections and we give our vanishing/nonvanishing results for these curvatures. In Chapter 5, we also take first steps towards
a corresponding (arithmetic differential) Galois theory.
The last Chapter 9 lists some of the natural problems, both technical andconceptual, that one faces in the further development of the theory.
Chapters 1, 2, 3, 4 should be read in a sequence (with Chapters 1 and 2 possibly
skipped and consulted later as needed); Chapters 5, 6, 7, 8 depend on Chapters 1,
2, 3, 4 but are essentially independent of each other. Chapter 9 can be read rightafter the Introduction.
Cross references are organized in a series of sequences as follows. Sections arenumbered in one sequence and will be referred to as “section x.y.” Definitions, Theorems, Propositions, Lemmas, Remarks, and Examples are numbered in a separate
sequence and are referred to as “Theorem x.y, Example x.y,” etc. Finally equations
are numbered in yet another separate sequence and are referred to simply as “x.y.”
For all three sequences x denotes the number of the chapter.
Readership and prerequisites. The present book addresses graduate students and researchers interested in algebra, number theory, differential geometry,
and the analogies between these fields. The only prerequisites are some familiarity
with commutative algebra (cf., e.g., the Atiyah-MacDonald book [4] or, for more
specialized material, Matsumura’s book [103]) and with foundational scheme theoretic algebraic geometry (e.g., the first two chapters of Hartshorne’s book [72]).
The text also contains a series of remarks that assume familiarity with basic concepts of classical differential geometry (as presented in [89], for instance); but these
remarks are not essential for the understanding of the book and can actually beskipped.
Acknowledgments. Over the years the author has greatly profited from inspiring discussions and/or collaborations with a number of people including: M.
Barrett, D. Bertrand, J. Borger, J.-B. Bost, C. P. Boyer, P. Cartier, P. J. Cassidy,
P. Deligne, T. Dupuy, I. Fesenko, E. Hrushovski, Y. Ihara, M. Kim, E. R. Kolchin,S. Lang, Yu. I. Manin, H. Moscovici, A. Pillay, B. Poonen, F. Pop, E. Previato,
D. C. Rössler, A. Saha, T. Scanlon, S. R. Simanca, J. Tate, D. Thakur, D. Ulmer,
D. Vassilev, F. Voloch. Also, while writing this book, the author was partially
supported by the Simons Foundation through grant 311773. Finally the author is
grateful to the Max Planck Institut für Mathematik (Bonn), the Institut des HautesÉtudes Scientifiques (Bures sur Yvette), the Australian National University (Canberra), and the Institute of Mathematics of the Romanian Academy (Bucharest),
for hospitality and support during the writing of various parts of this book.
Alexandru Buium
Albuquerque, 2016

Introduction
By classical differential geometry we will understand, in this book, the theoryof differentiable manifolds equipped with various “geometric structures” such as
metrics, connections, curvature, etc. Our standard reference for this theory is [89].
However we shall only be interested in the algebraic aspects of this theory so wewill allow ourselves below to reintroduce some of its concepts using the language ofdifferential algebra [91, 112] (i.e., the language of rings equipped with derivations)and we will still refer to this “algebraized” version of differential geometry as classical differential geometry. On the other hand our purpose, in this book, is to develop
an arithmetic analogue of classical differential geometry, which we will refer to asarithmetic differential geometry; in this latter theory derivation operators acting onfunctions will be replaced by Fermat quotient type operators acting on numbers,in the spirit of [23, 35]. In this Introduction we present an outline of our theory
and we provide a quick comparison of our theory with other theories.

0.1. Outline of the theory
0.1.1. Classical versus arithmetic differentiation. In the classical theory of smooth (i.e., differentiable) manifolds one considers, for any m-dimensionalsmooth manifold M , the ring of smooth real valued functions C ∞ (M, R). For the
purposes of this Introduction it is enough to think of M as being the Euclidean
space M = Rm . In this book the arithmetic analogue of the manifold Rm will be
the scheme Spec Z. In order to stress our analogy between functions and numbers
it is convenient to further fix a subring(0.1)

A ⊂ C ∞ (Rm , R)

that is stable under partial differentiation and to frame the classical differential
geometric definitions in terms of this ring. The arithmetic analogue of A in 0.1
will then be the ring of integers Z or, more generally, rings of fractions A of ringsof integers in an abelian extension of Q. (The theory can be easily extended tothe case when Q is replaced by an arbitrary number field; we will not consider thismore general situation in the book.) A prototypical example of such a ring is the
ring
(0.2)

A = Z[1/M, ζN ],where M is an even integer, N is a positive integer, and ζN is a primitive N -throot of unity. Inverting M allows, as usual, to discard a set of “bad primes” (thedivisors of M ); adjoining ζN amounts to adjoining “new constants” to Z.
1

2

INTRODUCTION

Most of the times the ring A in 0.1 can be thought of as equal to C ∞ (Rm , R).
But it will be sometimes useful to think of the ring A in 0.1 as consisting of analytic functions; indeed we will sometimes view analytic/algebraic functions as corresponding, in our theory, to global arithmetic objects in which case C ∞ objects willcorrespond, in our theory, to adelic objects. We will adopt these various viewpoints
according to the various situations at hand.
The analogue of the set of coordinate functions(0.3)

U = {ξ1 , ξ2 , ..., ξm } ⊂ C ∞ (Rm , R)

on Rm will be a (possibly infinite) set of primes,
V = {p1 , p2 , p3 , ...} ⊂ Z(0.4)

not dividing M N . We denote by m = |V|≤ ∞ the cardinality of V. (In case Q is
replaced by an arbitrary number field the set V needs to be replaced by a set offinite places of that field.)
Next one considers the partial derivative operators on A ⊂ C ∞ (Rm , R),(0.5)

δi := δiA : A → A, δi f :=

∂f
,
∂ξi

i ∈ {1, ..., m}.

Following [23] we propose to take, as an analogue of the operators 0.5, the operators
δp on A = Z[1/M, ζN ] defined by(0.6)

δp := δpA : A → A, δp (a) =φp (a) − ap
, p ∈ V,ppwhere φp := φA
p : A → A is the unique ring automorphism sending ζN into ζN .
More generally recall that a derivation on a ring B is an additive map B → B
that satisfies the Leibniz rule. This concept has, as an arithmetic analogue, theconcept of p-derivation defined as follows; cf. [23, 35, 83].

Definition 0.1. Assume B is a ring and assume, for simplicity, that p is a
non-zero divisor in B; then a p-derivation on B is a set theoretic map(0.7)

δp := δpB : B → B

with the property that the map(0.8)

φp := φB
p :B →B

defined by(0.9)

φp (b) := bp + pδp b

is a ring homomorphism.
We will always denote by φp the ring homomorphism 0.8 attached to a pderivation δp as in 0.7 via the formula 0.9 and we shall refer to φp as the Frobenius
lift attached to δp ; note that φp induces the p-power Frobenius on B/pB.
0.1.2. Classical differential geometry revisited. Classical differential geometric concepts are often introduced by considering frame bundles P → M of rankn vector bundles over m-dimensional manifolds M . For our purposes we take here
M = Rm . Such a P is a principal homogeneous space for the group GLn (R); and if
the vector bundle is trivial (which, for simplicity, we assume in what follows) then
P is identified with M × GLn (R). Note that the rank n of the vector bundle andthe dimension m of M in this picture are unrelated.

0.1. OUTLINE OF THE THEORY

3

We want to review the classical concept of connection in P ; we shall do it
in a somewhat non-standard way so that the transition to arithmetic becomes
more transparent. Indeed consider an n × n matrix x = (xij ) of indeterminatesand consider the ring of polynomials in these n2 indeterminates over some A ⊂C ∞ (Rm , R) as in 0.1, with the determinant inverted,(0.10)

B = A[x, det(x)−1 ].

Note that B is naturally a subring of the ring C ∞ (P, R). Denote by End(B) the
Lie ring of Z-module endomorphisms of B. Also consider G = GLn = Spec B as ascheme over A.
Definition 0.2. A connection on P (or on G, or on B) is a tuple δ = (δi ) of
derivations(0.11)

δi := δiB : B → B, i ∈ {1, ..., m},extending the derivations 0.5. The curvature of δ is the matrix (ϕij ) of commutators
ϕij ∈ End(B),(0.12)

ϕij := [δi , δj ] : B → B,i, j ∈ {1, ..., m}.

The holonomy ring of δ is the Z-span hol in End(B) of the commutators(0.13)[δi1 , [δi2 , [...., [δis−1 , δis ]...]]],

where s ≥ 2; it is a Lie subring of End(B).
In particular one can consider the trivial connection δ0 = (δ0i ) defined by(0.14)

δ0i x = 0.

Here δ0i x = (δ0i xkl ) is the matrix with entries δ0i xkl ; this, and similar, notation
will be constantly used in the sequel. A connection is called flat if its curvaturevanishes: ϕij = 0 for all i, j = 1, ..., m. For instance δ0 is flat. For a flat δ theholonomy ring hol vanishes: hol = 0.
More generally let us define a connection on an arbitrary A-algebra B as a tupleof derivations 0.11 extending the derivations 0.5; one can then define the curvatureby the same formula 0.12.
Note that, in classical differential geometry, a prominent role is played by connections on vector bundles; the framework of vector bundles is, however, too “linear” to have a useful arithmetic analogue: our arithmetic theory will be essentiallya “non-linear” theory in which vector bundles (more generally modules) need to be
replaced by principal bundles (more generally by rings).
There are various types of connections that we shall be interested in and forwhich we will seek arithmetic analogues; they will be referred to as:(1) Ehresmann connections,(2) Chern connections,(3) Levi-Cività connections,(4) Fedosov connections,(5) Lax connections,(6) Hamiltonian connections,(7) Cartan connections,(8) Riccati connections,(9) Weierstrass connections,(10) Painlevé connections.

4

INTRODUCTION

In some cases our terminology above is not entirely standard. For instance what
we call Fedosov connection is usually called symplectic connection [96, 62, 66].
We changed the name by which we refer to these connections in order to avoid
confusion with the symplectic paradigm underlying the Hamiltonian case; our choiceof name is based on the fact that symplectic manifolds equipped with a symplectic
connection are usually called Fedosov manifolds [66]. Also there are a number ofobjects in classical differential geometry that are known under the name of Cartan
connection; what we call Cartan connection is an object corresponding to what
is usually called Cartan distribution [1], p. 133, and “lives” on jet bundles; ourCartan connection is unrelated, for instance, to the Maurer-Cartan connection.
The Maurer-Cartan connection, on the other hand, will play a role in our bookthrough the consideration of the logarithmic derivative map from an algebraic group
to its Lie algebra. Finally what we call connections in cases (5), (6), (8), (9), (10)are classically referred to as equations or flows; e.g. what we call Lax connection
is classically called Lax equation. Indeed connections classically refer to linear
equations whereas the equations involved in (5)-(10) are non-linear. So what wecall connection will not always refer to linear equations; this is in line with what
will happen later in the arithmetic case where connections will be “intrinsically
non-linear.”
In what follows we discuss the connections (1) through (7) in some detail. For
these connections we assume A as in 0.1; for the connections (1) through (5) we
assume, in addition, that B is as in 0.10; for connections (6) and (7) B will bedefined when we get to discuss these connections. Connections (8), (9), (10) will
not be discussed in this Introduction but will appear in the body of the book; these
are connections on curves (for (8), (9)) or surfaces (for (10)) appearing in relation
to the classical theory of differential equations with no movable singularities. The
Painlevé connections (10) fit into the Hamiltonian paradigm (6). Both (9) and (10)lead to elliptic curves.
Definition 0.3. A connection (δi ) on B in 0.10 is an Ehresmann connection
if it satisfies one of the following two equivalent conditions:
1a) There exist n × n matrices Ai with coefficients in A such that
(0.15)

δi x = Ai x1b) The following diagrams are commutative:(0.16)

B
δi ↓
B

µ

−→
µ

−→

B ⊗A B↓ δi ⊗ 1 + 1 ⊗ δ0i
B⊗B

Here µ is the comultiplication. Condition 1a can be referred to as saying that(δi ) is right linear. Condition 1b can be referred to as saying that δ is right invariant.
(As we shall see the arithmetic analogues of conditions 1a and 1b will cease to beequivalent.) If (δi ) is an Ehresmann connection the curvature satisfies ϕij (x) = Fij x
where Fij is the matrix given by the classical formula(0.17)

Fij := δi Aj − δj Ai − [Ai , Aj ].

Also there is a Galois theory attached to flat Ehresmann connections, the PicardVessiot theory. Indeed, for a flat Ehresmann connection δ = (δi ) consider the0.1. OUTLINE OF THE THEORY

5logarithmic derivative map, lδ : GLn (A) → gln (A)m , with coordinateslδi (u) = δi u · u−1 ,(0.18)

where gln is the Lie algebra of GLn . The fibers of the map lδ are solution sets ofsystems of linear equations
δi u = Ai · u.

(0.19)

And if one replaces A by a ring of complex analytic functions then Galois groups
can be classically attached to such systems; these groups are algebraic subgroups ofGLn (C) measuring the algebraic relations among the solutions to the corresponding
systems.
We next discuss Chern connections. We will introduce two types of Chern connections: real Chern connections and complex Chern connections. Complex Chern
connections will be modeled on the classical hermitian connections on hermitian
vector bundles [69, 88] which play a central role in complex geometry; real Chern
connections are a real analogue of the complex ones and do not seem to play animportant role in differential geometry (although see Duistermaat’s paper [61] fora special case of these connections). Both real and complex Chern connections will
have arithmetic analogues; the arithmetic analogue of real Chern connections will
be more subtle (and hence more interesting) than the arithmetic analogue of thecomplex Chern connection so we will mostly concentrate on the former than on thelatter.
To introduce these concepts let us consider again the rings A as in 0.1 and B
as in 0.10, and let q ∈ GLn (A) be an n × n invertible matrix with coefficients in A
which is either symmetric or anti-symmetric,
q t = ±q;(0.20)

so the t superscript means here transposition. Of course, a symmetric q as above isviewed as defining a (semi-Riemannian) metric on the trivial vector bundle
M × Rn → M,

(0.21)

on M = Rm while an anti-symmetric q is viewed as defining a 2-form on 0.21;
here metrics/2-forms on vector bundles mean smoothly varying non-degenerate
symmetric/anti-symmetric bilinear forms on the fibers of the bundle. With B be
as in 0.10 let G = GLn = Spec B be viewed as a scheme over A and consider themaps of schemes over A,(0.22)

Hq : G → G,

Bq : G × G → G,

Hq (x) = xt qx,

Bq (x, y) = xt qy.

defined on points by(0.23)

We continue to denote by the same letters the corresponding maps of rings B → B
and B → B ⊗A B. Consider the trivial connection δ0 = (δ0i ) on G defined by
δ0i x = 0. Then one has the following easy results:
theorem 0.4. There is a unique connection δ = (δi ) on G such that the following diagrams are commutative:(0.24)

δ

B
Hq ↑

i
←−

B

0i
←−

δ

B
↑ Hq
B

B
δi ⊗ 1 + 1 ⊗ δ0i ↑B ⊗A Bδ0i ⊗1+1⊗δi

←−
Bq

←−

B ⊗A B↑ Bq
B

6

INTRODUCTION

theorem 0.5. There is a unique connection δ = (δi ) on G such that the following diagrams are commutative:
δ0i ⊗1+1⊗δi

←−

B
Hq ↑

(0.25)

δ

0i
←−

B

B ⊗A B↑ Bq
B.

Definition 0.6. A connection δ is Hq -horizontal (respectively Bq -symmetric)
with respect to δ0 if the left (respectively right) diagrams in 0.24 are commutative
for all i. The unique connection that is Hq -horizontal and Bq -symmetric withrespect to δ0 (cf. Theorem 0.4) is called the real Chern connection attached to q.
Definition 0.7. A connection δ is (Hq , Bq )-horizontal with respect to δ0 if the
diagrams 0.25 are commutative for all i. The unique connection that is (Hq , Bq )horizontal with respect to δ0 (cf. Theorem 0.5) is called the complex Chern connection attached to q.
Real and complex Chern connections turn out to be automatically Ehresmannconnections. The arithmetic analogues of the real and complex Chern connections
will not be “Ehresmann” in general.
The definitions just given may look non-standard. To see the analogy betweenof our (real and complex) Chern connections and the classical hermitian Chern connections in [69, 88] we need to look at the explicit formulae for these connections;so we need to introduce more notation as follows.
For any connection (δi ) on B in 0.10 set
Ai=

Ai (x):= δi x · x−1 ∈ gln (B),

Γkij

=

−Aikj

=

(0.26)(k, j) entry of (−Ai ).

The quantities Γkij will be referred to as Christoffel symbols of the 2nd kind of theconnection. Also, for q as in 0.20, set(0.27)

Γijk := Γlij qlk ,which we refer to as the Christoffel symbols of the 1st kind. Here we use Einsteinsummation.
It is trivial to check that δ is Hq -horizontal if and only if the followng conditionis satisfied:(0.28)

δi qjk = Γijk ± Γikj .

Classically if 0.28 holds one says that q is parallel with respect to δ.
Similarly it is trivial to check that δ is Bq -symmetric if and only if the following
condition is satisfied:(0.29)

Γijk = ±Γikj .

This is, as we shall see, a condition different from the classical condition of symmetry
for a connection on the tangent bundle.
In 0.28 and 0.29 the upper (respectively lower) sign correspond to the upper(respectively lower) sign in 0.20.
Using 0.28 and 0.29 it is trivial to see that the real Chern connection attachedto q exists and is unique, being given by1(0.30)
Γijk = δi qjk .
2

0.1. OUTLINE OF THE THEORY

7

This proves Theorem 0.4. Note that, as promised earlier, the Chern connection is
an Ehresmann connection, i.e. Ai belong to gln (A) rather than gln (B).
By the way the real Chern connection has the following compatibility withthe special linear group SLn : if δi (det(q)) = 0 then the real Chern connection
δi : B → B attached to q sends the ideal of SLn into itself and hence induces
a “connection on SLn .” (This compatibility with SLn will fail in the arithmetic
case.)
Similarly it is easy to see that δ is (Hq , Bq )-horizontal with respect to δ0 if andonly if(0.31)

Γijk = δi qjk .

Hence the complex Chern connection exists and is unique, being given by 0.31,
which proves Theorem 0.5. Comparing 0.30 and 0.31 we see that the Christoffel
symbols of the real and complex Chern connections coincide up to a 21 factor andin this sense they are analogous to each other; on the other hand formula 0.31 isanalogous to the formula for the classical hermitian Chern connection in [69, 88].
We next discuss Levi-Cività connections. Assume A as in 0.1, B as in 0.10,
and assume that n = m. (Note we also implicitly assume here that a bijection
is given between the set indexing the derivations δi and the set indexing the rows
and columns of the n × n matrix x = (xij ); such a bijection plays the role of what
is classically called a soldering.) In the next definition we let z be an n-tuple ofvariables z1 , ..., zn .
Definition 0.8. The connection (δi ) is symmetric or torsion free if the following diagrams of A-algebras are commutative:s(0.32)

i
B
←−
sj ↑
ri
A[x] ←−

A[x]
↑ rj
A[z]

where si (x) = −Ati and rj (zk ) = xjk .
Trivially, the commutativity of 0.32 is equivalent to:(0.33)

Γijk = Γjik .

Note the difference between the condition 0.33 defining symmetry and the condition 0.29 defining Bq -symmetry: the two types of symmetry involve different pairsof indices. To avoid any confusion we will use the term torsion free rather thansymmetric in what follows. The fundamental theorem of Riemannian geometry is,in this setting, the following (completely elementary) statement:
theorem 0.9. Let q ∈ GLn (A), q t = q. Then there is a unique connection δthat is Hq -horizontal (i.e., satisfies equation 0.28 with the + sign) and is torsion
free (i.e., it satisfies equation 0.33); it is given by1(0.34)
Γkij = (δk qij + δi qjk − δj qki )2Definition 0.10. The connection in Theorem 0.9 is called the Levi-Cività
connection attached to q.
Note that, in particular, the Levi-Cività connection is an Ehresmann connection, i.e. Ai belong to gln (A) rather than gln (B). (The latter will fail in the

8

INTRODUCTION

arithmetic case.) The Levi-Cività connection is, in a precise sense to be discussed
later, “dual” to the real Chern connection and is generally different from the real
Chern connection; it coincides with the real Chern connection if and only if(0.35)

δi qjk = δj qik

in which case q is called Hessian (the “real” analogue of Kähler).
The curvature of the Levi-Cività connection is, of course, the main object ofRiemannian geometry. Let us recall some of the basic definitions related to it. First
some notation. For (Fij ) the curvature 0.17 of the Levi-Cività connection attached
to a metric q = (qij ) we set:
(0.36)kFij = (Fijkl ), Rlij:= −Fijkl ,mRijkl = qim Rjkl.

One refers to Rijkl as the covariant Riemann tensor; the latter has following symmetries:(0.37) Rijkl = −Rjikl , Rijkl = −Rijlk , Rijkl +Riklj +Rijlk = 0, Rijkl = Rklij .
Definition 0.11. A metric q is said to have constant sectional curvature if
there exists κ ∈ A with δi κ = 0 such that
(0.38)

Rijkl = κ · (qik qjl − qjk qil ).

We next discuss Fedosov connections partially following [96, 62, 66, 119]
where the term symplectic rather than Fedosov is being used. We consider, again,
A as in 0.1 and B as in 0.10.
Definition 0.12. Let q ∈ GLn (A), q t = −q, so n is even. A Fedosov connection
relative to q is a connection δ that is Hq -horizontal (i.e., satisfies equation 0.28 withthe − sign) and is torsion free (i.e. it satisfies 0.33).
Definition 0.13. q ∈ GLn (A), q t = −q, is symplectic if it satisfies(0.39)

δi qjk + δj qki + δk qij = 0.

One trivially checks that a Fedosov connection relative to q exists if and only
if q is symplectic.
Unlike in [62, 66, 119] our Fedosov connections here are not necessarily Ehresmann. And, for a given symplectic matrix q, Fedosov connections that are Ehresmann exist but are not unique; one Fedosov connection which is an Ehresmannconnection is given by(0.40)

Γijk =

1(δi qjk + δj qik ) .
3

We next discuss Lax connections. Let A be as in 0.1 and B as in 0.10.
Definition 0.14. A connection (δi ) on B is called a Lax connection if it satisfies(0.41)

δi x = [Ai (x), x] := Ai (x)x − xAi (x)for some n × n matrix Ai (x) with coefficients in B.

0.1. OUTLINE OF THE THEORY

9Note that, unlike Chern and Levi-Cività connections, Lax connections are nota subclass of the Ehresmann connections. For a Lax connection, the following
diagrams are commutative:
δ

i
B ←−
P↑

(0.42)

δ0i
A[z] ←−B↑P
A[z]where A[z] = A[z1 , ..., zn ] is a ring of polynomials in the variables zj , δ0i are theunique derivations extending the corresponding derivations on A with δ0i zj = 0,
and P is the A-algebra homomorphism with P(zj ) = Pj (x),(0.43)

det(s · 1 − x) =nX(−1)j Pj (x)sn−j .
j=0

The commutativity of 0.42 expresses the fact that the Lax connections describe“isospectral flows” on GLn .
By the way the theory summarized above has a “(1, 1)-analogue” for which werefer to the body of the book. Suffices to say here that, for the (1, 1) theory, one
may start with M = Cm and with a subringA ⊂ C ∞ (M, C)(0.44)of the ring of smooth complex valued functions on M which is stable under thederivations∂∂(0.45)δi :=, δi :=, i = 1, ..., m,∂zi∂z iwhere z1 , ..., zm are the complex coordinates on M . A connection on G = Spec B,
B = A[x, det(x)−1 ], is then an m-tuple of derivations δi : B → B extending thederivations δi : A → A. Consider the unique derivations δi : B → B, extending thederivations δi : A → A, such that δi x = 0. Then one defines the (1, 1)-curvature ofδ = (δi ) as the matrix (ϕij ) with entries the A-derivations(0.46)

ϕij := [δi , δj ] : B → B.

The theory proceeds from here.
So far we discussed connections on the ring B in 0.10. In what follows weinformally discuss Hamiltonian and Cartan connections; for a precise discussion werefer to the main text. These are connections on rings B other than 0.10.
To discuss Hamiltonian connections consider the ring
(0.47)

B = A[x],with A as in 0.1 and x = {x1 , ..., xd } a d-tuple of variables. Elements of the exterior
products of the module of Kähler differentials ΩB/A will be referred to as (vertical)∂
forms. Also consider the derivations ∂ j = ∂xon B.
jDefinition 0.15. A 2-form(0.48)

ω := ω ij · dxi ∧ dxj ,ω ij ∈ B, ωij + ωji = 0, is symplectic if det(ωij ) ∈ B × and(0.49)∂ k ω ij + ∂ j ω ki + ∂ i ω jk = 0.

10

INTRODUCTION

Note the contrast with the concept of symplectic matrices q: q are symplectic
in the “horizontal directions,”
δj = ∂/∂ξj ,cf. 0.39, while 2-forms ω in 0.48 are symplectic in the “vertical directions,”
∂j =

∂
.
∂xj

Definition 0.16. For any 2-form ω as in 0.48 the Lie derivative δk ω of ω withrespect to ξk is defined by:(0.50)

δk ω := (δk ω ij ) · dxi ∧ dxj + ω ij · d(δk xi ) ∧ dxj + ω ij · dxi ∧ d(δk xj ).

Definition 0.17. A Hamiltonian connection with respect to a symplectic 2form ω is a connection on a B as in 0.47 whose Lie derivatives vanish:δk ω = 0.
Hamiltonian connections with respect to symplectic forms naturally appear, by
the way, in the background of some of the basic differential equations of mathematical physics, in particular in the background of the Painlevé VI equations. This kindof Hamiltonian connections turn out to have an arithmetic analogue [42] which will
be discussed in the body of our book but not in this Introduction.
On the other hand one can consider Hamiltonian connections on a B as in 0.47
with respect to a Poisson structure. The Poisson story will be reviewed in themain body of the book. The two concepts of Hamiltonian connections (relative to
symplectic forms and relative to Poisson structures) are related in caseδk ω ij = 0.
An example of Poisson structure is provided by Lie-Poisson structures on Lie algebras in which case the corresponding Hamiltonian connections typically lead to Laxconnections. A classical example of equations arising from a Lie-Poisson structure
are the Euler equations for the rigid body which will be reviewed in the body ofthe book; the Euler equations are related to Lax equations in at least two different
ways (via 3 × 3 and 2 × 2 matrices respectively). Euler equations will have anarithmetic analogue, although this arithmetic analogue will not be a priori related
to our arithmetic analogues of Lax connections.
Finally we mention Cartan connections. In the same way we considered principal bundles and attached to them algebras B as in 0.10 one can consider “infinite
jet bundles” (which we do not define here) and attach to them polynomial algebras,(0.51)(α)

B = A[xj ; α ∈ Zm
≥0 , j = 1, ..., d],(α)

where A is as in 0.1 and xj are indeterminates; these algebras come equippedwith a natural flat connection
X X (α+e ) ∂∂(0.52)
δi :=
+
xj i
,
(α)
∂ξi
∂x
α
j
j

where ei is the canonical basis of Zm .
Definition 0.18. The flat connection (δi ) on the ring B in 0.51 defined by
0.52 is called the Cartan connection.

0.1. OUTLINE OF THE THEORY

11

Other names for our Cartan connection above are: the total derivative or the
Cartan distribution. The Cartan connections have an arithmetic analogue which
was thoroughly studied in [35, 34], will be reviewed in the main text of the present
book in case m = 1, and plays a central role in the whole theory.
0.1.3. Arithmetic differential geometry. Roughly speaking arithmetic differential geometry is obtained from classical differential geometry by replacingclassical differentiation (derivations) with arithmetic differentiation (p-derivations).
This is often a convoluted process that we now explain.
The first step is to consider the ring
B = A[x, det(x)−1 ]defined as in 0.10, with x = (xij ) an n × n matrix of indeterminates, but where A
is given now by
A = Z[1/M, ζN ]as in 0.2. We again consider the group scheme over A,
G = GLn := Spec B.
A first attempt to define arithmetic analogues of connections would be to consider
families of p-derivations
δp : B → B, p ∈ V,
extending the p-derivations 0.6; one would then proceed by considering their commutators on B (or, if necessary, expressions derived from these commutators). But
the point is that the examples of “arithmetic analogues of connections” we willencounter in practice will essentially never lead to p-derivations B → B! What weshall be led to is, rather, an adelic concept we next introduce. (Our guiding “principle” here is that, as mentioned before, C ∞ geometric objects should correspond toadelic objects in arithmetic while analytic/algebraic geometric objects correspondto global objects in arithmetic.)
To introduce our adelic concept let us consider, for each p ∈ V, the p-adic
completion of B:
B pb := lim B/pn B.

(0.53)←

Then we make the following:
Definition 0.19. An adelic connection on G = GLn is a family δ = (δp ) of
p-derivations(0.54)

δp := δpB : B pb → B pb, p ∈ V,

extending the p-derivations in 0.6.
If φp : B pb → B pb are the Frobenius lifts attached to δp and Gpb = Spf B pb is thep-adic completion of G = GLn = Spec B then we still denote by φp : Gpb → Gpb the
induced morphisms of p-adic formal schemes.
Next we will explore analogues of the various types of connections encountered in classical differential geometry: Ehresmann, Chern, Levi-Cività, Fedosov,
and Lax. (The stories of Hamiltonian connections and Cartan connections will be
discussed in the main body of the book and will be skipped here.)

12

INTRODUCTION

In what follows we need an analogue of the trivial connection in 0.14. It will
be given by the adelic connection (δ0p ) defined by(0.55)

δ0p x = 0.

The attached Frobenius lifts will be denoted by (φ0p ); they satisfy(0.56)

φ0p (x) = x(p)where x(p) is the matrix (xpij ). We call δ0 = (δ0p ) the trivial adelic connection.
To introduce arithmetic analogues of Ehresmann connections one starts bynoting that, for n ≥ 2, there are no adelic connections δ = (δp ) whose attached
Frobenius lifts (φp ) make the following diagrams commute:µ(0.57)

Gpb × Gpb −→φp × φ0p ↓µGpb × Gpb −→

Gpb
↓ φp
Gpb

Since 0.57 is an analogue of 0.16 one can view this as saying that there are no adelicconnections that are analogues of right invariant connections. This is an elementary
observation; we will in fact prove a less elementary result:theorem 0.20. For n ≥ 2 and p - n there are no adelic connections (δp ) and(δ1p ) whose attached Frobenius lifts (φp ) and (φ1p ) make the following diagrams
commute:µ(0.58)

Gpb × Gpb −→φp × φ1p ↓µ
Gpb × Gpb −→

Gpb
↓ φp
Gpb

There is a useful property, weaker than the commutativity of 0.57, which we now
introduce. Let T be the diagonal maximal torus of G and let N be the normalizer
of T in G. We will say that an adelic connection (δp ) with associated Frobenius lifts(φp ) is right invariant with respect to N if the following diagrams are commutative:µ(0.59)

Gpb × N pb −→φp × φ0p ↓µGpb × N pb −→

Gpb
↓ φp
Gpb

This latter property has its own merits but is too weak to function appropriately
as a defining property of Ehresmann connections in arithmetic. Instead, we willconsider an appropriate analogue of “linearity,” 0.15. What we will do will be toreplace the Lie algebra gln by an arithmetic analogue of it, gln,δp , and then we will
introduce an arithmetic analogue of the logarithmic derivative. This new framework
will naturally lead us to the following:
Definition 0.21. An adelic connection (δp ) is an Ehresmann connection if(0.60)

δp x = αp · x(p) ,where αp are matrices with coefficients in A.
By the way, clearly, Ehresmann connections are right invariant with respect to
N . We will attach Galois groups to such Ehresmann connections and develop thebasics of their theory. A natural expectation is that these Galois groups belong
to the group N (A)δ of all matrices in N (A) whose entries are roots of unity or 0.

0.1. OUTLINE OF THE THEORY

13

This expectation is not always realized but we will prove that something close to
it is realized for (αp ) “sufficiently general.” The above expectation is justified by
the fact that, according to the general philosophy of the field with one element F1 ,the union of the N (A)δ ’s, as A varies, plays the role of “GLn (Fa1 ),” where Fa1 is the“algebraic closure of F1 .”Next we explain our arithmetic analogue of Chern connections.
Let q ∈ GLn (A) with q t = ±q. Attached to q we have, again, maps
Hq : G → G,

Hq (x) = xt qx,

Bq : G × G → G;

Bq (x, y) = xt qy.

We continue to denote by Hq , Bq the maps induced on the p-adic completions Gpb
and Gpb × Gpb. Consider again the trivial adelic connection δ0 = (δ0p ) on G (soδ0p x = 0) and denote by (φ0p ) the attached Frobenius lifts (so φ0p (x) = x(p) ).
Then we will prove the following:
theorem 0.22. For any q ∈ GLn (A) with q t = ±q there exists a unique adelicconnection δ = (δp ) whose attached Frobenius lifts (φp ) make the following diagrams
commute:(0.61)

φp

Gpb
↓ Hq

Gpb
φp × φ0p ↓

φ0p

Gpb

Gpb × Gpb

Gpb −→Hq ↓Gpb −→

φ0p ×φp−→
Bq−→

Gpb × Gpb
↓ Bq
Gpb

Definition 0.23. An adelic connection δ = (δp ) is Hq -horizontal (respectively
Bq -symmetric) with respect to δ0 = (δ0p ) if the left (respectively right) diagrams
in 0.61 are commutative. The unique connection which is Hq -horizontal and Bq symmetric with respect to δ0 (cf. Theorem 0.22) is called the real Chern connection
attached to q.
The real Chern connection introduced above is an arithmetic analogue of thereal Chern connection in classical differential geometry. Unlike in the case of classical differential geometry our adelic Chern connections will not be special cases of
Ehresmann connections (although they will be right invariant with respect to N ).
On the other hand we will have the following easy:
theorem 0.24. For any q ∈ GLn (A) with q t = ±q there is a unique adelicconnection (δp ) on GLn whose attached Frobenius lifts (φp ) make the following
diagrams commute:(0.62)

Gpb
Hq ↓
Gpb

φ0p ×φp−→

Gpb × Gpb
↓ Bq

φ0p

Gpb

−→

Definition 0.25. An adelic connection δ = (δp ) is (Hq , Bq )-horizontal withrespect to δ0 = (δ0p ) if the diagrams 0.62 are commutative. The unique connection
which is (Hq , Bq )-horizontal with respect to δ0 (cf. Theorem 0.24) is called thecomplex Chern connection attached to q.
The concept of real Chern connection introduced above is, from an arithmetic
point of view, more subtle than the concept of complex Chern connection. Thiscan be seen already in the case of GL1 . Indeed let
q ∈ GL1 (A) = A× ,A = Z[1/M ].

14

INTRODUCTION

In this case it turns out that the “Christoffel symbols” defining the real Chern
connection are related to the Legendre symbol; explicitly the Frobenius lift φp :Gpb → Gpb corresponding to the real Chern connection attached to q is induced by
the homomorphism
φp : Zp [x, x−1 ]pb → Zp [x, x−1 ]pb(0.63)
satisfying(0.64)
where

φp (x) = q qp(p−1)/2

 
q
xp ,
p

is the Legendre symbol of q ∈ A× ⊂ Z(p) . On the other hand the Frobe-

nius lift φp : Gpb → Gpb corresponding to the complex Chern connection attached toq is simply defined by a homomorphism 0.63 satisfying
φp (x) = q p−1 xp .

(0.65)

In view of the above we will mainly concentrate, in this book, on the study of
the real Chern connection while leaving the simpler case of the complex Chern
connection to the reader. Note by the way that the quantity φ(x)/xp in 0.65 is
the square of the quantity φ(x)/xp in 0.64; this has an analogue for GLn with n
arbitrary and is analogous to the fact that the Christoffel symbols of the classicalcomplex Chern connection are twice the Christoffel symbols of the classical real
Chern connection; cf. 0.30 and 0.31.
We shall also introduce, in our book, adelic connections that are analogues ofLevi-Cività connections. They are already relevant in case V consists of one prime
only. So assume, in the theorem below, that V = {p} consists of one prime p. As
usual we write G = GLn and we set g = gln = Spec A[x].
theorem 0.26. For any q ∈ GLn (A) with q t = q there is a unique n-tuple(δ1p , ..., δnp ) of adelic connections on G with attached Frobenius lifts(φ1p , ..., φnp ),such that for all i, j = 1, ..., n the following diagrams are commutative(0.66)

φip

Gpb
↓ Hq

φ0p

Gpb

Gpb −→Hq ↓Gpb −→si
Gpb −→sj ↓ri
gpb −→gpb↓ rj(An )pb

where, for ∆i := δip x, the maps si , ri are defined on points a = (aij ), b = (bij ) by:si (a) = (a(p) )−1 ∆i (a),ri (b) = (b1i , ..., bni ).

Definition 0.27. The tuple (δ1p , ..., δnp ) in Theorem 0.26 is called the LeviCività connection attached to q,
The commutativity of the left diagrams in 0.66 says, of course, that, for each i,(δip ) is Hq -horizontal with respect to (δ0p ) and hence is analogous to the conditionof parallelism 0.28 (with the + sign). The commutativity of the right diagrams in
0.66 is equivalent to the condition(0.67)

δip xkj = δjp xki .

and is analogous to the condition of torsion freeness 0.33. This justifies our Definition 0.27 of the Levi-Cività connection. But note that, unlike in the case of0.1. OUTLINE OF THE THEORY

15

classical differential geometry, our adelic Levi-Cività connections are not special
cases of Ehresmann connections.
The adelic Levi-Cività connection and the adelic real Chern connection attached to q will be related by certain congruences mod p that are reminiscent of
the relation between the two connections in classical differential geometry.
The one prime paradigm of the Levi-Cività connection above can be viewed ascorresponding to the classical Levi-Cività connections attached to (translational)
cohomogeneity one metrics, by which we understand here metrics that, in appropriate coordinates, are invariant under a codimension one group of translations;
P equivalently metrics which in appropriate coordinates have the form g =gij dξi dξjwith(0.68)

δk gij = δl gij

for all i, j, k, l. As a rule we will omit, for simplicity, the adjective translationalfrom now on.
Continuing to assume V = {p} one can also attempt to develop an arithmetic analogue of Fedosov connections as follows. Consider an anti-symmetricq ∈ GLn (A), q t = −q. Let us say that an n-tuple of (δ1p , ..., δnp ) of adelic connections on G = GLn is a Fedosov connection relative to q if the diagrams 0.66
are commutative. We will prove that for n = 2 and any anti-symmetric q Fedosov
connections relative to q exist. However, in contrast with the Levi-Cività story, we
will prove that for n ≥ 4 there is no Fedosov connection relative to the split q, forinstance.
Finally there are adelic connections that are analogous to Lax connections.
In fact there are two such analogues which we call isospectral and isocharacteristic
Lax connections. They offer two rather different arithmetic analogues of isospectral
flows in the space of matrices. Indeed the isocharacteristic property essentially saysthat a certain characteristic polynomial has “δ-constant” coefficients whereas theisospectrality property essentially says that the characteristic polynomial has “δconstant” roots. (Here δ-constant means “killed by all δp ” which amounts to “beinga root of unity or 0”.) In usual calculus the two properties are equivalent but in ourarithmetic calculus these two properties are quite different. By the way isospectral
and isocharacteristic Lax connections will not be defined on the whole of G but
rather on certain Zariski open sets G∗ and G∗∗ of G respectively. Let us give somedetails of this in what follows. Let G∗ ⊂ G be the open set of regular matrices,consider the diagonal maximal torus T in G, and let T ∗ = G∗ ∩ T ; regular means
here “with distinct diagonal entries.” We will prove:
theorem 0.28. There exists a unique adelic connection δ = (δp ) on G∗ with
attached Frobenius lifts (φp ) such that the following diagrams commute:(0.69)(T ∗ )pb × Gpb
C↓(G∗ )pb

φ0p ×φ0p−→(T ∗ )pb × Gpb↓Cφp(G∗ )pb,−→

where C(t, x) := x−1 tx.
Definition 0.29. The adelic connection δ in Theorem 0.28 is called the canonical isospectral Lax connection.

16

INTRODUCTION

More generally let (δp ) be the canonical isospectral Lax connection, with attached Frobenius lifts (φp ), φp (x) = Φp , i.e., Φp = (Φpij (x)) where Φpij (x) =φp (xij ). Let αp (x) be n × n matrices with entries in O(G∗ )pb. Then, setting
p (x) = 1 + pαp (x), one can consider the isospectral Lax connection attached to()(αp ), defined by the family of Frobenius lifts (φp ),
()
−1
φ()
.
p (x) = Φp (x) := p (x) · Φp (x) · p (x)The latter has the following property that justifies the term isospectral. Let
up = a−1
p bp ap ,()

bp = diag(bp1 , ..., bpn ) ∈ T ∗ (Apb)ap ∈ GLn (Apb),()

be such that φp (u) = Φp (u). Thenδp bpi = 0,i = 1, ..., n.

So the eigenvalues bpi of the “solutions” up are δp -constant.
On the other we will prove:
theorem 0.30. There is an adelic connection δ = (δp ) on an open set G∗∗ ⊂ G,
(G∗∗ )pb 6= ∅, with attached Frobenius lifts (φp ), such that the following diagrams
commute:(0.70)(G∗∗ )pb −→P↓

φp(G∗∗ )pb↓P

φ0p(An )pb,(An )pb −→where An = Spec A[z].

The diagram 0.70 should be viewed as analogous to diagram 0.42. The adelicconnection δ is, of course, not unique.
Definition 0.31. An adelic connection δ as in Theorem 0.30 is called an
isocharacteristic Lax connection.
Among isocharacteristic Lax connections there will be a canonical one.
For any isocharacteristic Lax connection (δp ), if φp (x) = Φp (x) and up ∈G∗∗ (Apb) satisfies φp (u) = Φp (u) we have thatδp (Pj (u)) = 0,

j = 1, ..., n.

So the coefficients of the characteristic polynomials of the “solutions” up are δp constant.
Note that the canonical Frobenius lifts φp in the two diagrams 0.69 and 0.70 will
not coincide on the intersection G∗ ∩ G∗∗ ; so the isospectral and isocharacteristic
stories are really different.
Next we would like to explore the curvature of adelic connections.
Consider first the case of Ehresmann connections, 0.60. Since αp ∈ A for all p
our Frobenius lifts φp : B pb → B pb induce Frobenius lifts φp : A[x] → A[x] and hence
one can consider the “divided” commutators
1(0.71)
ϕpp0 := 0 [φp , φp0 ] : A[x] → A[x], p, p0 ∈ V.
pp
The family (ϕpp0 ) will be referred to as the curvature of the Ehresmann connection(δp ).

0.1. OUTLINE OF THE THEORY

17

The situation for general adelic connections (including the cases of real Chern
and Lax connections) will be quite different. Indeed, in defining curvature weface the following dilemma: our p-derivations δp in 0.54 do not act on the same
ring, so there is no a priori way of considering their commutators and, hence, itdoes not seem possible to define, in this way, the notion of curvature. It will turnout, however, that some of our adelic connections will satisfy a remarkable property
which we call globality along the identity (more generally along various subvarieties);
this property will allow us to define curvature via commutators.
Definition 0.32. Consider the matrix T = x − 1, where 1 is the identity
matrix. An adelic connection δ = (δp ) on GLn , with attached family of Frobenius
lifts (φp ), is global along 1 if, for all p, φp : B pb → B pb sends the ideal of 1 into itself
and, moreover, the induced homomorphism φp : Apb[[T ]] → Apb[[T ]] sends the ring
A[[T ]] into itself. If the above holds then the curvature of (δp ) is defined as thefamily of “divided” commutators (ϕpp0 ),(0.72)

ϕpp0 :=

1[φp , φp0 ] : A[[T ]] → A[[T ]],pp0

where p, p0 ∈ V. Let End(A[[T ]]) denote the Lie ring of Z-module endomorphisms of
A[[T ]]. Then define the holonomy ring hol of δ as the Z-linear span in End(A[[T ]])of all the Lie monomials[φp1 , [φp2 , ..., [φps−1 , φps ]...]] : A[[T ]] → A[[T ]]where s ≥ 2, pi ∈ V. Similarly define the holonomy Q-algebra holQ of δ as the
Q-linear span of hol in End(A[[T ]]) ⊗ Q. Finally define the completed holonomy
ring,c = lim holn ,hol←where holn is the image of the map
hol → End(A[[T ]]/(T )n ).

(0.73)The various maps referred to above can be traced on the following diagram:
B pb ⊂φp ↓B pb ⊂

Apb[[T ]]φp ↓Apb[[T ]]⊃A[[T ]]
φp ↓↓ φp0⊃
A[[T ]]⊂⊂b0

Ap [[T ]] ⊃↓ φ p0b0
Ap [[T ]] ⊃b0

Bp
↓ φ p0b0
Bp

The idea of comparing p-adic phenomena for different p’s by “moving along theidentity section” was introduced in [34] where it was referred to as analytic continuation between primes. Analytic continuation is taken here in the sense of Zariski[124], Preface, pp. xii-xiii, who was the first to use completions of varieties along
subvarieties as a substitute for classical analytic continuation over the complex
numbers. The technique of analytic continuation is also used, in the form of formal
patching, in inverse Galois theory [71]. But note that our use of analytic continuation is rather different from the one used in inverse Galois theory. Indeed, in ourcontext, we are patching data defined on “tubular neighborhoods”
Spf B pband

Spf A[[T ]]Spec B/pB

andSpec B/(T )of two closed subsets

18

INTRODUCTIONof the scheme Spec B, where the data are required to coincide on the “tubularneighborhood”
Spf Apb[[T ]]of the closed subset
Spec B/(p, T ) = (Spec B/pB) ∩ (Spec B/(T ))of Spec B. This is in contrast with the use of patching in Galois theory [71] where
one patches data defined on two open sets covering a formal scheme.
Of course, the trivial adelic connection δ0 = (δ0p ), δ0p x = 0, is global along 1
so it induces ring endomorphisms φ0p : A[[T ]] → A[[T ]],φ0p (T ) = (1 + T )(p) − 1.
We may morally view δ0 as an analogue of a flat connection in real geometry (whereA ⊂ C ∞ (Rm , R)). Alternatively we may view δ0 as an arithmetic analogue of thederivations δi = ∂/∂z i on A[x, det(x)−1 ] which kill x, where A ⊂ C ∞ (Cm , C).
Following this second analogy we may consider an arbitrary adelic connection δ =(δp ) global along 1, with attached Frobenius lifts (φp ), and introduce the following:
Definition 0.33. The (1, 1)-curvature of δ is the matrix of “divided commutators” (ϕpp0 )1(0.74)
ϕpp0 := 0 [φp , φ0p0 ] : A[[T ]] → A[[T ]], p 6= p0 ,pp(0.75)

ϕpp :=

1[φp , φ0p ] : A[[T ]] → A[[T ]].
p

Let us discuss next the curvature of real Chern connections. (The case ofcomplex Chern connections is easier and will be skipped here.) We will prove:
theorem 0.34. Let q ∈ GLn (A) with q t = ±q. If all the entries of q are rootsof unity or 0 then the real Chern connection δ attached to q is global along 1; inparticular δ has a well defined curvature and (1, 1)-curvature.
So we may address the question of computing the curvature and (1, 1)-curvatureof real Chern connections for various q’s whose entries are 0 or roots of unity. A
special case of such q’s is given by the following:
Definition 0.35. A matrix q ∈ GLn (A) is split if it is one of the following:



 

1 0 0
0
1r
0 1r
(0.76),,  0 0 1r  ,
−1r 0
1r 0
0 1r 0where 1r is the r × r identity matrix and n = 2r, 2r, 2r + 1 respectively.
We will prove:
theorem 0.36. Let q be split and let (ϕpp0 ) and (ϕpp̄0 ) be the curvature and the(1, 1)-curvature of the real Chern connection on G attached to q. Then the following
hold:1) Assume n ≥ 4. Then for all p 6= p0 we have ϕpp0 6= 0.
2) Assume n = 2r ≥ 2. Then for all p, p0 we have ϕpp0 (T ) ≡ 0 mod (T )3 .
3) Assume n = 2 and q t = −q. Then for all p, p0 we have ϕpp0 = 0.

0.1. OUTLINE OF THE THEORY

194) Assume n ≥ 2. Then for all p, p0 we have ϕpp0 6= 0.
5) Assume n = 1. Then for all p, p0 we have ϕpp0 = ϕpp0 = 0.
Assertion 1 morally says that Spec Z is “curved,” while assertion 2 morally saysthat Spec Z is only “mildly curved.” Assertions 1 and 2 will imply, in particular,
assertion 1 in the following:
theorem 0.37. Assume q split and n ≥ 4 is even. Then, for the real Chern
connection attached to q, the following hold:c is non-zero and pronilpotent.
1) hol2) holQ is not spanned over Q by the components of the curvature.
Assertion 1 is in stark contrast with the fact, to be explained in the body ofthe book, that holonomy Lie algebras arising from Galois theory are never nilpotent
unless they vanish. Assertion 2 should be viewed as a statement suggesting thatthe flavor of our arithmetic situation is rather different from that of classical locallysymmetric spaces; indeed, for the latter, the Lie algebra of holonomy is spanned by
the components of the curvature.
Note that the above theorem says nothing about the vanishing of the curvature
ϕpp0 in case n = 2, 3 and q t = q; our method of proof does not seem to apply tothese cases.
By the way if q ∈ GL2r is symmetric and split then the real Chern connection
δp : B pb → B pb attached to q does not send the ideal of SLn into itself; this is
in contrast with the situation encountered in classical differential geometry. To
remedy this situation we will construct connections on GLn that do send the idealof SLn into itself; there are many such connections and the “simplest” one will becalled the special linear connection.
Other curvatures can be introduced and vanishing/non-vanishing results forthem will be proved; they will be referred to as 3-curvature, first Chern form, first
Chern (1, 1)-form, mean curvature, scalar curvature, etc. We will also introduce
real Chern connections attached to hermitian metrics and results will be proved fortheir curvature. A similar theory can be developed for complex Chern connections.
Similar results will also be proved for the curvature of Lax connections. As
we shall see the open sets G∗ and G∗∗ where isospectral and isocharacteristic Laxconnections are defined (cf. 0.69 and 0.70) do not contain the identity of thegroup G = GLn hence curvature cannot be defined by analytic continuation along
the identity; however these open sets will contain certain torsion points of the
diagonal maximal torus of G and we will use analytic continuation along suchtorsion points to define curvature and (1, 1)-curvature. We will then prove thenon-vanishing of the (1, 1)-curvature of isocharacteristic Lax connection for n = 2.
The (canonical) isospectral Lax connection has, immediately from its definition, avanishing curvature.
Note that the concept of curvature discussed above was based on what wecalled analytic continuation between primes; this was the key to making Frobenius
lifts corresponding to different primes act on a same ring and note that it only
works as stated for adelic connections that are global along 1. This restricts theapplicability of our method to “metrics” q with components roots of unity or 0. One
can generalize our method to include q’s with more general entries by replacing
the condition of being global along 1 with the condition of “being global along

20

INTRODUCTION

certain tori;” this will be discussed in the body of the book. However, there isa different approach towards making Frobenius lifts comparable; this approachis based on algebraizing Frobenius lifts via correspondences and works for adelicconnections that are not necessarily global along 1 (or along a torus). The priceto pay for allowing this generality is that endomorphisms (of A[[T ]]) are replacedby correspondences (on GLn ). Let us explain this alternative road to curvature inwhat follows.
We will prove the following:
theorem 0.38. Let δ = (δp ) be the real Chern connection on G = GLn attached
to a matrix q ∈ GLn (A) with q t = ±q. Then there exists a family of diagrams ofA-schemesYp
πp
ϕp(0.77).
&
G
G
where πp is affine and étale, inducing, via p-adic completion, a family of diagrams
Yppb(0.78)

πppb

b
ϕp
p

.

&
φp

Gpb

Gpb

−→

where πppb are isomorphisms.
The triples(0.79)

Γp = (Yp , πp , ϕp )can be viewed as correspondences on G which provide “algebraizations” of our
Frobenius lifts φp .
The family (Γp ) is not uniquely attached to (δp ) but does have some “uniquenessfeatures” to be explained in the body of the book.
On the other hand if E is the field of rational functions on the scheme G and
Fp is the finite E-algebra
Fp := O(Yp ⊗ E)
then each correspondence Γp induces, via tensorization by E, a diagram
Spec Fp

πp(0.80).

ϕp&Spec ESpec EThe family of diagrams 0.80 will be referred to as a correspondence structure for(δp ). Note that each Γp induces an additive map Γ∗p : E → E,
(0.81)

Γ∗p (z) := Trπp (ϕ∗p (z)), z ∈ E,

where
Trπp : Fp → E
is the trace of the finite algebra map
πp∗ : E → Fp

0.1. OUTLINE OF THE THEORY21

andϕ∗p : E → Fp
is induced by ϕp .
Definition 0.39. The curvature of (Γp ) is the matrix (ϕ∗pp0 ) with entries the
additive homomorphisms
1(0.82)
ϕ∗pp0 := 0 [Γ∗p , Γ∗p0 ] : E → E, p, p0 ∈ V.
pp
Note that, in this way, we have defined a concept of curvature for real Chern
connections attached to arbitrary q’s (that do not necessarily have entries zeroes
or roots of unity). There is a (1, 1)-version of the above as follows. Indeed thetrivial adelic connection δ0 = (δ0p ) has a canonical correspondence structure (Γ0p )induced byΓ0p = (G, π0p , ϕ0p ),where π0p is the identity, and ϕ0p (x) = x(p) .
Definition 0.40. The (1, 1)-curvature of (Γp ) is the family (ϕ∗pp0 ) where ϕ∗pp0
is the additive endomorphism
1(0.83)
ϕ∗pp0 := 0 [Γ∗0p0 , Γ∗p ] : E → E for p 6= p0 ,pp(0.84)

ϕ∗pp :=1 ∗[Γ , Γ∗ ] : E → E.
p 0p p

Then we will prove the following:
theorem 0.41. Let q ∈ GL2 (A) be split. Then, for the real Chern connection
attached to q, the following hold:1) Assume q t = −q. Then for all p, p0 we have ϕ∗pp0 = 0 and ϕ∗pp0 6= 0.
2) Assume q t = q. Then for all p, p0 we have ϕ∗pp0 6= 0.
Once again our results say nothing about curvature in case n = 2 and q t = q;
our method of proof does not seem to apply to this case.
Later in the body of the book the curvature and (1, 1)-curvature of (Γp ) introduced above will be called upper ∗-curvatures and we will also introduce theconcepts of lower ∗-curvature and lower ∗-(1, 1)-curvature coming from actions on“cycles.”
Finally note that one can define curvature for the Levi-Cività connection.
Definition 0.42. Let (δ1p , ..., δnp ) be the Levi-Cività connection attached toa symmetric q ∈ GLn (A). The curvature is defined as the family (ϕijp ), indexed byi, j = 1, ..., n given by the divided commutators
1
p
b
p
b
(0.85)
ϕij
p := [φip , φjp ] : O(G ) → O(G ).
pThis is a “vertical” curvature (indexed by the index set of the columns and rowsof x) rather than a “horizontal” curvature, in the style of the previously introduced
curvatures (which are indexed by primes). We will prove non-vanishing results forthese curvatures. For instance we have:

22

INTRODUCTION

theorem 0.43. Let i, j, k, l be fixed indices between 1 and n. Then, for theLevi-Cività connection, the following hold:kl1) Assume δp qjk + δp qil 6≡ δp qik + δp qjl mod p. Then ϕij
p , ϕp 6≡ 0 mod p.

2) Assume n = 2r and q is split. Then ϕij
p 6≡ 0 mod p for i 6= j.
We will actually prove a more precise result than assertion 1 of the above
Theorem. Indeed set
Φij := Φij (x) := ϕij (x),22

Ψij := Ψij (x) := x(p )t q (p ) Φij (x),and let Rijkl be the (k, l)-entry of the matrix Ψij , so
(0.86)

Ψij = (Rijkl ).

Then the Rijkl in 0.86 can be viewed as an arithmetic analogue of the covariant
Riemann tensor in classical differential geometry. We will prove that (Rijkl ) in 0.86
satisfies congruences mod p:(0.87) Rijkl ≡ −Rjikl , Rijkl ≡ −Rijlk , Rijkl +Riklj +Rijlk ≡ 0, Rijkl ≡ Rklij ,which are, of course, an arithmetic analogue of 0.37. In addition we will prove thecongruence1(δqjk + δqil − δqik − δqjl )p mod (p, x − 1).
2which can be viewed as an analogue of the condition of constant sectional curvature
0.38 for a metric that “degenerates into a square of a 1-form at the boundary;” theanalogy will be explained in the body of the book.
(0.88)

Rijkl ≡

Going back to arithmetic differential geometry, in case the entries of q areroots of unity or 0, we will prove that, for each i, the adelic connection (δip )appearing in the Levi-Cività connection is global along 1 so we will be able to0
define a mixed curvature (ϕij
pp0 ) indexed by i, j = 1, ..., n and p, p ∈ V given by thedivided commutators
1(0.89)
ϕij[φip , φjp0 ] : A[[T ]] → A[[T ]], p 6= p0 ,pp0 :=
pp0(0.90)

ϕij
pp :=

1
[φip , φjp ] : A[[T ]] → A[[T ]].
p

For a fixed p and the Fedosov connection (δ1p , δ2p ) relative to any anti-symmetricq ∈ GL2 (A) the formula 0.85 defines, again, a curvature; we will prove that thiscurvature does not vanish in general even if q is split.
0.2. Comparison with other theories
0.2.1. Analogies. From a number theoretic viewpoint there are at least two
perspectives on the integers:
A) Integers can be viewed as numbers.
B) Integers can be viewed as analogous to functions.
The two perspectives sometimes coexist in various specific contexts in spite of
the obvious tensions between them.

0.2. COMPARISON WITH OTHER THEORIES

23

This being said, Perspective A is mainly the basis for subjects such as: thegeometry of numbers [49]; diophantine approximation [50]; p-adic analysis [90];Fourier analysis on adeles [94]; p-adic mathematical physics [120]; etc.
On the other hand perspective B interprets Z as a ring of “functions” on a hypothetical “space;” cf. [102]. Here are some possibilities that have been suggested:
B1) Integers are analogous to functions on a Riemann surface.
B2) Integers are analogous to functions on a 3-dimensional manifold.
B3) Integers are analogous to functions on an infinite dimensional manifold.
Accordingly the hypothetical “space” referred to above should be an analogue ofa Riemann surface, a 3-manifold, or an infinite dimensional manifold, respectively.
Analogy B1 is classical, it has a complex geometric flavor, and is the basisof work stretching from Dedekind to Arakelov and beyond. According to thisviewpoint the primes p in Z are analogous to complex functions on a Riemannsurface that vanish at one point only and hence they define points on the surface.
A similar picture holds for integers in number fields. Note that Riemann surfaces
are objects of complex dimension 1 (or real dimension 2). Genera of number fields
are classically defined and finite, as in the case of Riemann surfaces. There is arelated viewpoint according to which Z is analogous to a ring of functions on a
Riemann surface of infinite genus; cf., e.g., [55].
Analogy B2, in its topological form, originates in suggestions of Mazur, Manin,
Kapranov, and others. According to this viewpoint the primes p in Z correspondto embedded circles (knots) in a 3-dimensional manifold; so primes can be viewed
as analogues of complex functions on the manifold that vanish along these knots.
The Legendre symbol is then an analogue of linking numbers. Cf. Morishita’smonograph [104] and the references therein. A different form of analogy B2, witha dynamical flavor, was proposed by Deninger [59].
By contrast, the present book and previous work by the author adopt theanalogy B3 and have a differential geometric flavor. The primes, in this setting, areanalogous to global coordinate functions on an infinite dimensional manifold. Other
viewpoints that fit into analogy B3 have also been proposed; cf. the discussion below
on the field with one element.
There is a possibility that our theory has connections with viewpoints B1 andB2, as shown, for instance, by the presence of the Legendre symbol in our real Chern
connections. Indeed the underlying Galois theory of reciprocity is an analogue ofthe monodromy in either the 2 or 3-dimensional picture; in the same way ourarithmetic curvature theory could be an analogue of the identity component of anatural “holonomy” in one of these pictures.
There is also a possibility that our arithmetic differential geometry could play
the role of framework for an arithmetic physics in the sense of Manin [100] which
would then have perspective B in its background (as opposed to [120] which has
perspective A in its background).
0.2.2. Field with one element. According to the viewpoint B describedabove integers can be viewed as functions on a hypothetical “space.” As noted,
at least in its variant B1, this viewpoint is entirely classical. More recently thisviewpoint was adopted in the search for a geometry over the field with one element,
F1 ; cf. the pioneering suggestions of Tits [118]. See also the articles in [117], andthe bibliography therein, for a survey of some approaches to the subject.

24

INTRODUCTION

A remarkable idea appearing in the context of F1 is the consideration by
Kurokawa et. al., [93] and Haran [70] of the absolute derivation operators,a∂p : Z → Z, ∂p a := vp (a) ,pwhere vp (a) is the p-adic valuation of a. This is in line with paradigm B3. However
the flavor of the operators ∂p is rather different from that of the Fermat quotient
operators δp and it seems unlikely that Haran’s theory and ours are directly related. Even more remote from our theory are the approaches to F1 in papers of
Kapranov-Smirnov [86], Soulé [115], Deitmar [56, 117], Manin-Marcolli [117],
Connes-Consani [55], etc., which do not directly provide an analogue of derivation
for the integers.
Borger’s philosophy of F1 , cf. [11], is, in some sense, perpendicular to theabove mentioned approaches to F1 . On the other hand, in the “case of one prime”
Borger’s approach is consistent with ours: roughly speaking, in the case of one
prime, Borger’s theory [11, 13] can be viewed as an algebraization of our analytic
theory in [23, 35]. In the case of more (all) primes Borger’s F1 theory can also
be viewed as a viewpoint consistent with B3 above: indeed Borger’s remarkablesuggestion is to take λ-structures (in the sense of Grothendieck) as descent datafrom Z to F1 . Recall that a λ-structure on a scheme X flat over Z is the same as acommuting family (φp ) of Frobenius lifts φp : X → X. So our theory would fit into
“λ-geometry” as long as:1) the Frobenius lifts are defined on the schemes X themselves (rather than on
the various p-adic completions X pb) and2) the Frobenius lifts commute.
However conditions 1 and 2 are almost never satisfied in our theory: the failureof condition 2 is precisely the origin of our curvature, while finding substitutes forcondition 1 requires taking various convoluted paths (such as analytic continuation
between primes or algebraization by correspondences). So in practice our approach
places us, most of the times, outside the paradigm of λ-geometry.
0.2.3. Ihara’s differential. Next we would like to point out what we thinkis an important difference between our viewpoint here and the viewpoint proposedby Ihara in [78]. 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 [78]Ihara proposed to see the mapYa − apmod p(0.91)
d:Z→
Fp , a 7→pp
as an analogue of differentiation for integers and he proposed a series of conjecturesconcerning the “zeroes” of the differential of an integer. These conjectures arestill completely open; they are in the spirit of paradigm B1 listed above, in thesense that counting zeroes of 1-forms is a Riemann surface concept. But what
we see as 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 quotientsthemselves. This allows the possibility of considering compositions between our0.2. COMPARISON WITH OTHER THEORIES

25

δp ’s which leads to the possibility of considering arithmetic analogues of differential
equations, curvature, etc.
0.2.4. Fontaine-Colmez calculus. The Fontaine-Colmez theory of p-adic
periods [64] also speaks of a differential calculus with numbers. Their calculus isperpendicular to ours in the following precise sense. For a fixed prime p our calculus
based on the Fermat quotient operator δp should be viewed as a a differential
calculus in the “unramified direction,” i.e., the “direction” given by the extension[
Q⊂
Q(ζN ),
p-N

whereas the Fontaine-Colmez calculus should be viewed as a differential calculus
in the “totally ramified direction,” i.e., roughly, in the “direction” given by theextension[
Q⊂
Q(ζpn ).
n

The Fontaine-Colmez theory is based on the usual Kähler differentials of totally
ramified extensions and hence, unlike ours, it is about usual derivations. It is notunlikely, however, that, for a fixed p, a theory unifying the unramified and the totally ramified cases, involving two “perpendicular” directions, could be developed
leading to arithmetic partial differential equations in two variables. Signs of possibility of such a theory can be found in our papers [32, 33]. This two variablesetting would fit into paradigm B1 listed above but would be rather different fromthe classical forms (Dedekind style) of paradigm B1 discussed above.
0.2.5. Grothendieck’s p-curvature. It is worth pointing out that the studyof our curvature here resembles the study of the p-curvature appearing in thearithmetic theory of differential equations that has been developed around theGrothendieck conjecture (cf., e.g., [87]). Both curvatures measure the lack of commutation of certain operators and both theories rely on technical matrix computations. However we should also point out that the nature of the above mentioned
operators in the two theories is quite different. Indeed our curvature here involves
the “p-differentiation” of numbers with respect to primes p (in other words it isabout d/dp and fits into paradigm B) whereas the theory in [87], and related papers, is about usual differentiation d/dt of power series in a variable t and essentiallyfits into paradigm A. In spite of these differences the two types of curvatures could
interact; a model for such an interaction between d/dp and d/dt is in the papers[32, 33]. A similar remark can be made about the difference between our approachand that in [17] which, again, is about usual Kähler differentials, hence about usual
derivations.
0.2.6. Discrete geometry as Euclidean geometry. Finally we would liketo point out that the theory in this book is a priori unrelated to topics such as thegeometry of numbers [49] on the one hand and discrete differential geometry [10]on the other. Indeed in both these geometries what is being studied are discreteconfigurations of points in the Euclidean space Rm ; in the geometry of numbers theconfigurations of points typically represent rings of algebraic numbers (making the
subject fit into paradigm A) while in discrete differential geometry the configurations of points approximate smooth submanifolds of the Euclidean space (making
the subject a priori unrelated to number theory). This framework is, therefore,

26

INTRODUCTION

that of the classical geometry of Euclidean space, based on R-coordinates, and notthat of an analogue of this geometry, based on “prime coordinates.” It may very
well happen, however, that (one or both of) the above topics are a natural home forsome (yet to be discovered) Archimedian counterpart of our (finite) adelic theory.

CHAPTER 1

Open problems
At this point arithmetic differential geometry is far from being a fully developed
theory. What are then the next expected steps? And where is the theory headed?
A number of puzzles, internal to the theory, as well as suggestive of connections
with the rest of number theory, may be showing the right direction. We collect
here a sample of such puzzles/problems, both technical and conceptual, that seemto indicate a general common tendency.

1.1. Unifying holQ and ΓQ
One of the main objects of the theory presented in this book is the holonomy
algebra holQ of an adelic connection; cf. Definition ??. (This is defined only for
adelic connections that are global along appropriate subschemes; and as we saw thisis the case with real Chern and Levi-Cività connections attached to large classes of
metrics, in particular to δ-constant metrics.) We would like to view the Lie algebra
holQ as an infinitesimal analogue of the absolute Galois group ΓQ = Gal(Qa /Q).
On the other hand if Hol is the holonomy group of a connection on principal G-bundle over a manifold M in classical differential geometry and if Hol0 is
its connected component then the quotient Hol/Hol0 is usually referred to as themonodromy group of the connection; it is isomorphic to the image ΓM of the monodromy representation(1.1)

π1 (M ) → G

defined by our connection. (We have ignored here the base points.) The exact
sequence(1.2)

1 → Hol0 → Hol → ΓM → 1

gives rise to a natural homomorphism(1.3)

ΓM → Out(Hol0 ).

Now it is classical to see the absolute Galois group ΓQ as an arithmetic analogue of a fundamental group π1 (M ). Galois representations are then analogous tomonodromy representations 1.1 and their images are analogous to the groups ΓM .
Since our Lie algebras holQ attached to real Chern or Levi-Cività adelic connections
are an arithmetic analogue of the Lie algebra of Hol0 and since we attached Liealgebras to Galois connections it is reasonable to pose the following:
Problem 1. “Unify” our holonomy on GLn with Galois theory by constructing
canonical extensions of Lie algebras(1.4)

0 → hol1 → hol2 → hol3 → 0
27

28

1. OPEN PROBLEMS

where hol3 is attached to a Galois connection and hol1 is attached to a real Chern
or Levi-Cività (adelic) connection on GLn . Such an extension 1.4 should function
as an arithmetic analogue of the extension 1.2 and the induced representation(1.5)

hol3 → Out(hol1 )should be an analogue of the representation 1.3. This construction could involve
interesting Galois representations; indeed an extension such as 1.4 could be, in its
turn, analogous to the basic exact sequence(1.6)

1 → π1 (X a ) → π1 (X) → ΓQ → 1

attached to a geometrically connected scheme X over Q (where X a = X ⊗ Qa andwe suppressed, again, reference to base points); cf. [57, 77]. The representation
1.5 could then be analogous to the representation(1.7)

ΓQ → Out(π1 (X a ))arising from 1.6. The representations 1.7 play a prominent role in work of Grothendieck, Ihara, Deligne, and many others.
Here are some related problems:
Problem 2. Find links between our curvature of adelic connections and reciprocity (as it appears in arithmetic topology [104]); the appearance of the Legendre
symbol in our Christoffel symbols may be an indication that such links may exist.
Problem 3. Develop a Galois correspondence for Ehresmann connections; extend the Galois theory from Ehresmann connections to other types of connections,especially real Chern and special linear connections.
Problem 4. Find links between our adelic (flat) connections and Galois representations similar to the classical link between vector bundles with flat connections
and monodromy.
Problem 5. Find an arithmetic analogue of Atiyah’s theory [3] according towhich:
1) the algebraic connections on an algebraic vector bundle over a variety are in
bijection with the splittings of the Atiyah extension;2) the obstruction to the splitting of the Atiyah extension generates the characteristic ring of the vector bundle.
Problem 6. The curvature of adelic Chern connections led, in our theory, toobjects that we called Chern (1, 1)-forms; it is tempting to use these forms, andhigher versions of them, to define cohomology classes in the style of Chern-Weiltheory; the target cohomology groups could be Galois (étale) cohomology groups
of appropriate arithmetic objects.
1.2. Unifying “∂/∂p” and “∂/∂ζp∞ ”
As discussed in the Introduction the theory of p-adic periods of Fontaine andColmez involves, in particular, a “totally ramified arithmetic calculus” that is perpendicular to our (unramified) arithmetic calculus. In its simplest form this totally
ramified calculus morally comes from the fact that the modules of Kähler differentials ΩZ[ζpn ] are torsion but non-trivial: dζpn 6= 0. It is conceivable that thissituation could lead to a “derivation like” operator, “∂/∂ζp∞ ” on an appropriate
space on which our p-derivation δp also acts. Then one is led to:1.3. UNIFYING Sh AND GLn

29

Problem 7. Unify our unramified arithmetic calculus (based on p-derivations
δp , thought of as “∂/∂p”) with the totally ramified arithmetic calculus of FontaineColmez [64] (possibly involving the conjectural “derivation like” operators “∂/∂p∞ ”).
One would presumably get, for each p, a partial differential situation with two “directions” and, in particular, one could attempt to study the resulting “curvature.”
A way to proceed could be to start from the paper [32] where the base ring was
the ring of power series R[[q]] in the variable q, equipped with the “standard” pderivation δp (extending that of R = W (Fap ) and killing q) and with the usual
derivation δq = qd/dq. Then one could try to specialize the theory in [32] via
q 7→ ζpn with n → ∞. Such a specialization does not a priori make sense, ofcourse, and no recipe to give it a meaning seems available at this point. However
an indication that this procedure might be meaningful can be found in the formulae
giving the “explicit reciprocity laws” of local class field theory due to Shafarevich,Brückner, Vostokov, etc. Cf. [63].
1.3. Unifying Sh and GLn
Problem 8. Relate the arithmetic differential calculus on Shimura varieties Sh(appearing in our book [35]) with the arithmetic differential calculus on the classicalgroups (appearing in our present book). In particular examine the case of
Sh = SL2 (Z)\SL2 (R)/SO2 (R).
An indication that such a relation may exist is the δ-algebraic theory of δ-Hodge
structures in [22] giving δ-algebraic correspondences between classical groups andmoduli spaces of abelian varieties; one would have to develop an arithmetic analogueof [22].
Here are some related questions:
Problem 9. Find a link between the concept of curvature in the present bookand the concept of curvature based on the arithmetic Laplacian in [34]. Moregenerally find a common ground between our theory here and [34].
Problem 10. The theory in this book is mainly about adelic connections on
GLn possibly compatible with involutions of GLn . It is natural to explore what
happens if one replaces GLn by a general reductive group.
Problem 11. Explore the arithmetic differential geometry of homogeneous
spaces such as “spheres,” S n := SOn+1 /SOn .
Problem 12. Develop an arithmetic analogue of the Cassidy-Kolchin theory ofdifferential algebraic groups. Compute the rings of invariant δ-functions for actionsof such groups on the varieties naturally appearing in Riemannian geometry. For
instance compute the δ-functions on SOn \GLn /Γ where Γ is a δ-subgroup of GLn .
The δ-invariants of the curvature of the Levi-Cività connection may be playing arole in this problem.
Problem 13. Find interactions between the classical theory of quadratic forms
over rings of integers in number fields [107] and our theory here (around symmetricmatrices that play the role of metrics). Both theories are about the geometry of
GLn /SOn .

30

1. OPEN PROBLEMS

1.4. Further concepts and computations
Problem 14. Decide if the curvature of the real Chern connection attached toa split symmetric q ∈ GLn vanishes for n = 2, 3. Compute (or prove non-vanishingof) curvature for the correspondence structures attached to real Chern connections
on GLn in case q is symmetric and n = 2. Same problem for curvature and (1, 1)curvature in case n ≥ 3.
Problem 15. Prove vanishing/non-vanishing for the mixed curvature of the LeviCività connection attached to split q’s. Perform analytic continuation of Levi-Cività
connection along partition tori and analyze the corresponding mixed curvature andgauge action.
Problem 16. Investigate the existence of correspondence structure for LeviCività connection; same question for the canonical isospectral and isocharacteristic
Lax connections. In case such correspondence structures exist compute (or provevanishing/non-vanishing of) their curvatures.
Problem 17. Study the impact of conformal changes of metrics on curvature.
Problem 18. Find the explicit structure of holonomy algebras hol, holΩ attached
to real Chern connections.
Problem 19. Investigate the analytic properties of the Dirichlet series arisingas our mean and scalar (1, 1)-curvatures.
Problem 20. Further develop the formalism of curvature in the context of thering of correspondences on a field.
Problem 21. Develop an arithmetic analogue of Ricci curvature, scalar curvature, and constant sectional curvature for the Levi-Cività connection. If this is tobe done in the case of several (all) primes one will need a better understanding of
the problem of soldering i.e., the problem of how to view GL∞ as an analogue of theframe bundle of Spec Z. On the other hand, in the case of one prime p, one should
better understand the analogy between our formula for curvature mod p and theformula for the classical Riemann tensor of a metric with constant scalar curvature;
cf. Remark ??. On a related note it would be important to develop an arithmetic
analogue of covariant differentiation. This might require passing, in a systematicway, from adelic connections on GLn to adelic connections on GLN for any given
rational representation (group scheme homomorphism) ρ : GLn → GLN . Such a
construction is, of course, available in classical differential geometry for Ehresmannconnections. It would be desirable to have a similar construction in arithmetic
differential geometry for Chern or Levi-Cività connections, at least in the case ofrepresentations ρ obtained from the tautlogical representation id : GLn → GLn by
iterating the operations ⊕, ⊗, Hom .
Problem 22. Develop an arithmetic analogue of connections that are Hamiltonian with respect to the Lie-Poisson bracket. Find an arithmetic analogue ofthe theory of algebraically completely integrable systems [2], along the lines of
the arithmetic analogue of the Euler equations for the rigid body discussed in thebook; cf. [48]. Find a strengthening of the concept of Euler connection by requiring
equality instead of congruence in the linearization condition.

1.5. WHAT LIES AT INFINITY?

31

Problem 23. Find an arithmetic analogue of Fedosov connections; furthermore
find an arithmetic counterpart of deformation quantization. Indeed, the “strong”
arithmetic analogue of Fedosov connections introduced in the text was proved not
to exist in dimension ≥ 4; what is needed is the “correct weakening” of that concept.
Problem 24. Many of the curvature computations in our theory have the flavorof computations in algebraic dynamics (study of semigroups of endomorphisms ofalgebraic varieties); an instance of this is our use of ideas from [65] in the study ofadelic Ehresmann connections. The connection with algebraic dynamics deserves
further study.
Problem 25. Find an arithmetic analogue of geodesics. A hint could comefrom [32] where “time” was a geometric variable and “space” was represented byarithmetic variables (primes).
Problem 26. Perform curvature computations for complex Chern connections;as a rule they are easier than the corresponding computations for real Chern connections which were presented in the book. In particular complex Chern connections have obvious correspondence structure which is represented by field endomorphisms; and if attached to diagonal matrices they are always global along the
diagonal maximal torus so Jordan multiplication ceases to play the role it playedin the real case.
1.5. What lies at infinity?
The theory of the present book involves non-Archimedian (finite) adeles only.
A natural question is whether there is an Archimedian (infinite) counterpart to ourtheory. So one can ask, for instance:
Problem 27. Could Borger’s positivity approach to study “Witt vectors at
infinity” [15] interact with our theory? This sounds reasonable in view of thecompatibility between our theory and Borger’s in the case of a single prime.
Problem 28. Could concepts of discrete differential geometry [10] play the roleof Archimedian counterpart to our theory?
Problem 29. Is there an Archimedian counterpart to our theory that has theflavor of the classical geometry of numbers [49] or more generally of Arakelov geometry?
Problem 30. Could an arithmetic physics (in the sense of [100]) be developed,with our arithmetic differential geometry as background, that would have “realphysics” as its Archimedian counterpart? This would offer a picture rather different from that in [120], as explained in the Introduction. Indeed the underlying
analogy in [120] is between p-adic numbers and real numbers whereas the underlying analogy in our arithmetic differential geometry is between p-adic numbers andfunctions.

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