ARITHMETIC DIFFERENTIAL GEOMETRY
ALEXANDRU BUIUM

1. Introduction
The aim of these notes is to explain past work and proposed research aimed atdeveloping an arithmetic analogue of classical differential geometry. In the theory we are going to describe the ring of integers Z will play the role of a ring offunctions on an infinite dimensional manifold. The role of coordinate functions on
this manifold will be played by the prime numbers p ∈ Z. The role of the partial
derivative of an integer n ∈ Z with respect to a prime p will be played (in the spiritp∈ Z. The role of metrics/2-formsof [16, 28]) by the Fermat quotient δp n := n−npwill be played by symmetric/antisymmetric matrices with entries in Z. The roleof connection/curvature attached to metrics or 2-forms will be played by certain
adelic/global objects attached to integral matrices. The resulting geometry will
be referred to as arithmetic differential geometry. One of the somewhat surprisingconclusions of our theory is that (the space whose ring of functions is) Z appears
as “intrinsically curved.”
A starting point for this circle of ideas can be found in our paper [16] where
we showed how one can replace derivation operators by Fermat quotient operatorsin the construction of the classical jet spaces of Lie and Cartan; the new objects
thus obtained were referred to as arithmetic jet spaces. With these objects at hand
it was natural to try to develop an arithmetic analogue of differential calculus, inparticular of differential equations and differential geometry. A theory of arithmetic
differential equations was developed in a series of papers [16]-[35], [4] and waspartly summarized in our monograph [28] (cf. also the survey papers [36, 73]); thistheory led to a series of applications to invariant theory [20, 4, 21, 28], congruences
between modular forms [20, 30, 31], and Diophantine geometry of Abelian andShimura varieties [17, 29]. On the other hand an arithmetic differential geometry
was developed in a series of papers [35]-[40], [5] and in the book [42]. The present
notes follow closely the Introduction to [42].
We should note that our book [28] on arithmetic differential equations and thebook [42] on arithmetic differential geometry, although based on the same ideology introduced in [16], are concerned with rather different objects. In particular
the two books are independent of each other and the overlap between them isminimal. Indeed the book [28] was mainly focussed on the arithmetic differential
equations naturally occurring in the context of Abelian and Shimura varieties. By
contrast, the book [42] naturally concentrates on the arithmetic differential equations related to the classical groups GLn , SLn , SOn , Spn , and their corresponding
symmetric spaces. Of course, the world of Abelian and Shimura varieties on the one
hand and the world of classical groups on the other, although not directly related
within abstract algebraic geometry, are closely related through analytic concepts
such as uniformization and representation theory. The prototypical example of this1

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ALEXANDRU BUIUM

relation is the identification M1 (C) ' SL2 (Z)\SL2 (R)/SO2 (R) where M1 is themoduli space of Abelian varieties, C/lattice, of dimension one; the curve M1 is,of course, one the simplest Shimura varieties. It is conceivable that the analytic
relation between the two worlds referred to above be realized via certain arithmetic
differential equations/correspondences; in particular it is conceivable that the book[42] and our previous book [28] be related in ways that go beyond what can be seenat this point. A suggestion for such a possible relation comes from the theory ofδ-Hodge structures developed in [15] and from a possible arithmetic analogue of thelatter that could emerge from [4, 28].
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 this theory
turn out to be related to classical number theoretic concepts (e.g., Christoffel symbols are related to Legendre symbols); existence and uniqueness results for the main
objects (such as the arithmetic analogues of Ehresmann, Chern, Levi-Civita, andLax connections) are being proved; the non-trivial problem of defining curvature issolved 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 investigationand further discovery.
As a final general comment let us consider the question: what is the position of
our theory among more established mathematical theories? The answer we would
like to suggest is that our curvature of Z, suitably encoded into a Lie Q-algebra holQthat we refer to as the holonomy algebra of Q, could be viewed as an infinitesimalcounterpart of the absolute Galois group ΓQ := Gal(Q/Q) of Q: our Lie algebra holQ
should be to the absolute Galois group ΓQ what the identity component Hol0 of theholonomy group Hol is to the monodromy group Hol/Hol0 in classical differential
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” ofholQ and ΓQ may be expected in the same way in which Hol0 and Hol/Hol0 are“unified” by Hol.
Here is the plan of these notes. Section 2 will be devoted to an outline of
our arithmetic differential geometry; this will be done by presenting the theory inparallel with classical differential geometry. In order to stress the analogies wewill revisit the main concepts of classical differential geometry from a somewhat
non-standard angle. Section 3 will be devoted to comparing our theory with other“geometric theories of the discrete.”
2. Outline of the theory
2.1. Classical versus arithmetic setting. In classical differential geometry one
starts with an m-dimensional smooth manifold M and its ring of smooth real valued
functions C ∞ (M, R). For our purposes here it is enough to think of M as being the
Euclidean space M = Rm . In these notes the arithmetic analogue of the manifold
Rm will be the scheme Spec Z. Next, in order to stress our analogy between
functions and numbers it is convenient to fix a subring(1)

A ⊂ C ∞ (Rm , R)

ARITHMETIC DIFFERENTIAL GEOMETRY

3that is stable under partial differentiation and to frame the classical differential
geometric definitions in terms of this ring. The arithmetic analogue of A in (1) willthen be the ring of integers Z or, more generally, rings of fractions A of rings ofintegers in an abelian extension of Q. A prototypical example of such a ring is the
ring
(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.
Most of the times the ring A in (1) can be thought of as equal to C ∞ (Rm , R). But
it will be sometimes useful to think of the ring A in (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(3)

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

on Rm will be a (possibly infinite) set of primes,
V = {p1 , p2 , p3 , ...} ⊂ Z(4)not dividing M N . We denote by m = |V| ≤ ∞ the cardinality of V.
Next one considers the partial derivative operators on A ⊂ C ∞ (Rm , R),∂f, i ∈ {1, ..., m}.
(5)
δi := δiA : A → A, δi f :=∂ξi
Following [16] we propose to take, as an analogue of the operators (5), the operators
δp on A = Z[1/M, ζN ] defined by(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. [16, 28].

Definition 1. Assume B is a ring and assume, for simplicity, that p is a non-zerodivisor in B; then a p-derivation on B is a set theoretic map(7)δp := δpB : B → B

with the property that the map(8)

φp := φB
p :B →B

defined by(9)φp (b) := bp + pδp b

is a ring homomorphism.
We will always denote by φp the ring homomorphism (8) attached to a pderivation δp as in (7) via the formula (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.

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ALEXANDRU BUIUM

2.2. Classical connections and curvature revisited. As our next step, in classical differential geometry, we consider the frame bundle P → M of a rank n vector
bundle over the manifold M = Rm . Then P is a principal homogeneous space forthe group GLn ; and if the vector bundle is trivial (which we assume from now on)then P is identified with M × GLn . (Note that the rank n of the vector bundle
and the dimension m of M in this picture are unrelated.) We want to review theclassical 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 ann × n matrix x = (xij ) of indeterminates and consider the ring of polynomials inthese n2 indeterminates over some A ⊂ C ∞ (Rm , R) as in (1), with the determinant
inverted,(10)

B = A[x, det(x)−1 ].

Note that B is naturally a subring of the ring C ∞ (M ×GLn , R). Denote by End(B)the Lie ring of Z-module endomorphisms of B.
Definition 2. A connection on P = M × GLn is a tuple δ = (δi ) of derivations(11)

δi := δiB : B → B, i ∈ {1, ..., m},extending the derivations (5). The curvature of δ is the matrix (ϕij ) of commutators
ϕij ∈ End(B),(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(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(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 (11) extending the derivations (5); one can then define the curvatureby the same formula (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 essentially a“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,

ARITHMETIC DIFFERENTIAL GEOMETRY

5(7) Cartan connections,(8) Riccati connections,(9) Weierstrass connections,(10) Painlevé connections.
In some cases our terminology above is not entirely standard. For instance what
we call Fedosov connection is usually called symplectic connection [51]; we chose a
different name in order to avoid confusion with the symplectic paradigm underlying
the Hamiltonian case; our choice of name is based on the appearance of these connections in [51]. Also there are many connections in classical differential geometry
which are known under the name of Cartan connection; the Cartan connections that
we will be considering are usually introduced as Cartan distributions [1], p. 133,and “live” on jet bundles; they are unrelated, for instance, to the Maurer-Cartan
connections.
In what follows we discuss the connections (1) through (7) in some detail. For
these connections we assume A as in (1); for the connections (1) through (5) we
assume, in addition, that B is as in (10); for connections (6) and (7) B will bedefined when we get to discuss these connections. Connections (8), (9), (10) will
not be discussed here but will appear in [42]; 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 equations attached to (10)are closely related to the Hamiltonian connections (6). Both (9) and (10) lead toelliptic curves.
Definition 3. A connection (δi ) on B in (10) is an Ehresmann connection if itsatisfies one of the following two equivalent conditions:
1a) There exist n × n matrices Ai with coefficients in A such that(15)

δi x = Ai x1b) The following diagrams are commutative:
(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(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
logarithmic derivative map, lδ : GLn (A) → gln (A)m , with coordinates(18)lδi (u) = δi u · u−1 ,where gln is the Lie algebra of GLn . The fibers of the map lδ are solution sets ofsystems of linear equations(19)

δi u = Ai · u.

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ALEXANDRU BUIUM

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. Let us consider again the rings A as in (1)and B as in (10), and let q ∈ GLn (A) be an n × n invertible matrix with coefficientsin A which is either symmetric or antisymmetric,
q t = ±q;(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,

(21)

on M = Rm while an antisymmetric q is viewed as defining a 2-form on (21);here metrics/2-forms on vector bundles mean smoothly varying non-degenerate
symmetric/antisymmetric bilinear forms on the fibers of the bundle. Set G = GLn
and consider the maps of schemes over A,
Hq : G → G,

Bq : G × G → G

Hq (x) = xt qx,

Bq (x, y) = xt qy.

(22)
defined by(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 trivial:
Theorem 4. There is a unique connection δ = (δi ) on G such that the following
diagrams are commutative:
(24)

δ

B
Hq ↑

i
←−

B

0i
←−

δ

B
↑ Hq
B

B
δ0i ⊗ 1 + 1 ⊗ δi ↑B ⊗A Bδi ⊗1+1⊗δ0i

←−
Bq

←−

B ⊗A B↑ Bq
B

For an argument and an explicit formula for (δi ) (that are useful to comparewith the arithmetic case) see (29) below.
Definition 5. We say that a connection δ is Hq -horizontal (respectively Bq -symmetric) with respect to δ0 if the left (respectively right) diagram in (24) is commutative. The unique connection which is Hq -horizontal and Bq -symmetric withrespect to δ0 (cf. Theorem 4) is called the Chern connection attached to q.
Chern connections turn out to be automatically Ehresmann connections; theanalogue of this in arithmetic will cease to be true.
The definition just given may look non-standard. It turns out that the Chern
connection we just defined is a “real” analogue of the Chern connection on a hermitian vector bundle in differential geometry [54, 66] (in which δ0 is an analogue
of a complex structure). A special case of this real analogue is the connection
introduced in [50] which we shall refer to as the Duistermaat connection.
To see the analogy of our Chern connection with the Chern connection in [54, 66]
and to see an argument plus an explicit formula for δ in Theorem 4 we introduce
more notation as follows.

ARITHMETIC DIFFERENTIAL GEOMETRY

7

For any connection (δi ) on B in (10) set
Ai= Ai (x):= δi x · x−1 ∈ gln (B),Γkij

= −Aikj=(25)(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 (20), set(26)

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:(27)

δi qjk = Γijk ± Γikj .

Classically if (27) 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:(28)

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 (27) and (28) the upper (respectively lower) sign correspond to the upper(respectively lower) sign in (20).
Using (27) and (28) it is trivial to see that the Chern connection attached to q
exists and is unique, being given by1(29)
Γijk = δi qjk .
2This proves Theorem 4 and gives the explicit formula for δ which is analogous tothe classical formulas for the Chern and Duistermaat connections. 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 Chern connection has the following compatibility with the special
linear group SLn : if δi (det(q)) = 0 then the Chern connection δi : B → B attachedto q sends the ideal of SLn into itself and hence induces a “connection on SLn .”(This compatibility with SLn will fail to hold in the arithmetic case.)
We next discuss Levi-Cività connections. Assume A as in (1), B as in (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 matrix xij ; such a bijection plays the role of what is classicallycalled a soldering.)
Definition 6. The connection (δi ) is symmetric or torsion free if(30)

Γijk = Γjik .

Note the difference between the condition (30) defining symmetry and the condition (28) 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:

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ALEXANDRU BUIUM

Theorem 7. Let q ∈ GLn (A), q t = q. Then there is a unique connection δ thatis Hq -horizontal (i.e., satisfies equation (27) with the + sign) and is torsion free(i.e., it satisfies equation (30)); it is given by(31)

Γkij =

1(δk qij + δi qjk − δj qki )2Definition 8. The connection in Theorem 7 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 to hold in thearithmetic case.) The Levi-Cività connection is, in a precise sense to be discussed
later, “dual” to the Chern connection and is generally different from the Chern
connection; it coincides with the Chern connection if and only if(32)

δi qjk = δj qik

in which case q is called Hessian (the “real” analogue of Kähler).
For (Fij ) the curvature of the Levi-Cività connection attached to a metric q =(qij ) we set:(33)kFij = (Fijkl ), Rlij:= −Fijkl ,mRijkl = qim Rjkl.

One refers to Rijkl as the Riemann curvature tensor; the latter has following symmetries:(34) Rijkl = −Rjikl ,Rijkl = −Rijlk ,Rijkl + Riklj + Rijlk = 0, Rijkl = Rklij .

We next discuss Fedosov connections. To explain these we start, again, with A
as in (1) and B as in (10).
Definition 9. 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 (27)with the − sign) and is torsion free (i.e. it satisfies (30)).
One trivially checks that a Fedosov connection relative to q exists if and only if
q is symplectic in the sense that it satisfies(35)

δi qjk + δj qki + δk qij = 0.

Fedosov connections 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 Ehresmann connection is given by(36)

Γijk =

1(δi qjk + δj qik ) .
3

We next discuss Lax connections. Let A be as in (1) and B as in (10).
Definition 10. A connection (δi ) on B is called a Lax connection if it satisfies(37)

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

ARITHMETIC DIFFERENTIAL GEOMETRY

9

Note 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↑

(38)

δ

0i
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),(39)

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

The commutativity of (38) expresses the fact that the Lax connections describe“isospectral flows” on GLn .
By the way the “real” theory summarized above has a “complex” analogue (andhence a “(1, 1)”-analogue) for which we refer to [42]. Suffices to say here that, forthe complex theory, one may start with M = Cm and with a subringA ⊂ C ∞ (M, C)(40)of the ring of smooth complex valued functions on M which is stable under thederivations
∂
∂
(41)
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(42)

ϕij := [δi , δj ] : B → B.

The theory proceeds from here.
So far we discussed connections on the ring B in (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 (10).
To discuss Hamiltonian connections consider the ring(43)

B = A[x],with A as in (1) and x = {x1 , ..., xd } a d-tuple of variables and consider a nondegenerate “vertical” 2-form(44)ω := ω ij · dxi ∧ dxj ,ω ij ∈ B, with ω symplectic (i.e. closed) “in the vertical directions”∂ i := ∂/∂xi
in the sense that(45)

∂ k ω ij + ∂ j ω ki + ∂ i ω jk = 0.

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ALEXANDRU BUIUM

Definition 11. A Hamiltonian connection with respect to ω is a connection on a
B as in (43) for which the corresponding “Lie derivatives” annihilate ω:(46)

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

Note the contrast with the case of Fedosov connections which are attached to amatrix q that is symplectic “in the horizontal directions”
δj = ∂/∂ξj ;cf. (35). Hamiltonian connections with respect to symplectic forms naturally appear, by the way, in the background of some of the basic differential equations ofmathematical physics, in particular in the background of the Painlevé VI equations.
This kind of Hamiltonian connections turn out to have an arithmetic analogue [35]which will be discussed in [42].
On the other hand one can consider Hamiltonian connections on a B as in (43)with respect to a Poisson structure. 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 toLax connections. A classical example of equations arising from a Lie-Poisson structure are the Euler equations for the rigid body; the Euler equations are related toLax equations in at least two different ways. Euler equations have an arithmetic
analogue [41], although this arithmetic analogue is not be a priori related to ourarithmetic analogues of Lax equations.
Finally we mention Cartan connections. In the same way we considered principal
bundles and attached to them algebras B as in (10) one can consider “infinite jetbundles” (which we do not define here) and attach to them polynomial algebras,
(47)

(α)

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

where A is as in (1) and xj are indeterminates; these algebras come equippedwith a natural flat connection
X X (α+e ) ∂
∂
,
(48)
δi :=
+
xj i
(α)
∂ξi
∂x
α
j
j

where ei is the canonical basis of Zm .
Definition 12. The flat connection (δi ) on the ring B in (47) defined by (48) iscalled the Cartan connection.
Other names for (δi ) above are: the total derivative or the Cartan distribution.
The Cartan connections have an arithmetic analogue which was thoroughly studied
in [28, 27] and plays a central role in the whole theory.
2.3. Arithmetic connections and curvature. We would like to introduce now
our arithmetic analogues of connection and curvature and to state some of our main
results about them. We start with the analogue of the real case. The first step isto consider the ring
B = A[x, det(x)−1 ]ARITHMETIC DIFFERENTIAL GEOMETRY

11

defined as in (10), with x = (xij ) an n × n matrix of indeterminates, but where A
is given now by
A = Z[1/M, ζN ]as in (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 (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 will encounter in practice will almost never lead to p-derivations B → B! What we shallbe 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.

(49)

←

Then we make the following:
Definition 13. An adelic connection on G = GLn is a family (δp ) of p-derivations(50)

δp := δpB : B pb → B pb, p ∈ V,

extending the p-derivations in (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 inclassical differential geometry: Ehresmann, Chern, Levi-Cività, Fedosov, and Lax.
(The stories of Hamiltonian connections and Cartan connections will be discussed
in [42] and will be skipped here.)
In what follows we need an analogue of the trivial connection in (14). It will begiven by the adelic connection (δ0p ) defined by(51)

δ0p x = 0.

The associated Frobenius lifts will be denoted by (φ0p ); they satisfy(52)

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 by noting
that, for n ≥ 2, there are no adelic connections δ = (δp ) whose attached Frobenius
lifts (φp ) make the following diagrams commute:µ(53)

Gpb × Gpb −→φp × φ0p ↓µGpb × Gpb −→

Gpb
↓ φp
Gpb

12

ALEXANDRU BUIUM

Since (53) is an analogue of (16) one can view this as saying that there are no adelicconnections that are analogues of right invariant connections. This is an elementary
observation; one can in fact prove a less elementary result:
Theorem 14. [42] For n ≥ 2 and p 6 |n there are no adelic connections (δp ) and(δ1p ) whose attached Frobenius lifts (φp ) and (φ1p ) make the following diagrams
commute:µ

Gpb × Gpb −→φp × φ1p ↓µ
Gpb × Gpb −→

(54)

Gpb
↓ φp
Gpb

There is a useful property, weaker than the commutativity of (53), namely an
invariance property with respect to the action on G by right translation of the
group
(55)

N := (normalizer of the diagonal maximal torus T of G).

Indeed 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

Gpb × N pb −→φp × φ0p ↓µGpb × N pb −→

(56)

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,” (15). What one can do [38] is toreplace the Lie algebra gln by an arithmetic analogue of it, gln,δp , and then introduce
an arithmetic analogue of the logarithmic derivative. This new framework naturally
leads to the following:
Definition 15. An adelic connection (δp ) is an Ehresmann connection if
δp x = αp · x(p) ,(57)

where αp are matrices with coefficients in A.
By the way, clearly, Ehresmann connections are right invariant with respect to
N . One can attach Galois groups to such Ehresmann connections [39] and develop
the basics 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.
This expectation is not always realized but one can prove that something close to it
is realized for (αp ) “sufficiently general;” cf. [39]. 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,

Bq : G × G → G;

Hq (x) = xt qx,

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 one can prove the following:ARITHMETIC DIFFERENTIAL GEOMETRY

13

Theorem 16. [38] 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
commutative:
(58)

φp

Gpb
↓ Hq

Gpb
φp × φ0p ↓

φ0p

Gpb

Gpb × Gpb

Gpb −→Hq ↓Gpb −→

φ0p ×φp−→
Bq−→

Gpb × Gpb
↓ Bq
Gpb

Definition 17. We say that an adelic connection δ = (δp ) is Hq -horizontal (respectively Bq -symmetric) with respect to δ0 = (δ0p ) if the left (respectively right)
diagram in (58) is commutative. The unique connection which is Hq -horizontal andBq -symmetric with respect to δ0 (cf. Theorem 16) is called the Chern connection
attached to q.
Unlike in the case of classical differential geometry our adelic Chern connections
are not be special cases of Ehresmann connections (although they are right invariant
with respect to N ).
Note the following relation between the “Christoffel symbols” defining our Chern
connection and the Legendre symbol. We explain this in a special case. Letq ∈ GL1 (A) = A× ,A = Z[1/M ],and let δ = (δp ) be the Chern connection associated to q. Then it turns out that
φp : Gpb → Gpb is defined by φp : Zp [x, x−1 ]pb → Zp [x, x−1 ]pb, q(p−1)/2
xp ,(59)
φp (x) = qp where pq is the Legendre symbol of q ∈ A× ⊂ Z(p) .
We can also introduce adelic connections that are analogues of Levi-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.
Theorem 18. [42] For any symmetric q ∈ GLn (A) with q t = q there is a unique
n-tuple (δ1p , ..., δnp ) of adelic connections on G = GLn with attached Frobenius lifts(φ1p , ..., φnp ), such that the following diagrams are commutative for i = 1, ..., n,
(60)

φip

Gpb
↓ Hq

φ0p

Gpb

Gpb −→Hq ↓Gpb −→and such that, for all i, j = 1, ..., n, we have:(61)

δip xkj = δjp xki .

Definition 19. The tuple (δ1p , ..., δnp ) in Theorem 18 is called the Levi-Cività
connection attached to q,
The condition (60) says of course that for each i, (δip ) is Hq -horizontal withrespect to (δ0p ) and hence is analogous to the condition of parallelism (27) (withthe + sign). The condition (61) is analogous to the condition of torsion freeness (30).
This justifies our Definition 19 of the Levi-Cività connection. But note that, unlikein the case of classical differential geometry, our adelic Levi-Cività connections are
not special cases of Ehresmann connections.

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ALEXANDRU BUIUM

The adelic Levi-Cività connection and the adelic Chern connection attached toq are 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
P Levi-Cività connections of central metrics, by which
we understand metrics g = gij dξi dξj satisfying(62)

δk gij = δl gij .

Continuing to assume V = {p} one can also attempt to develop an arithmetic analogue of Fedosov connections as follows. Consider an antisymmetricq ∈ 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 attached Frobenius
lifts (φ1p , ..., φnp ) make the diagrams (60) commutative and, in addition, the equalities (61) hold. One can prove that for n = 2 and any antisymmetric q Fedosov
connections relative to q exist. However, in contrast with the Levi-Cività story, one
can 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 are not defined on the whole of G but rather
on certain Zariski open sets G∗ and G∗∗ of G respectively. Let us give some detailsof this in what follows. Let T ∗ ⊂ T be the open set of regular matrices in the
diagonal maximal torus T ; regular means here “with distinct diagonal entries.”
One can prove:
Theorem 20. [38] There exists a unique adelic connection δ = (δp ) with Frobenius
lifts (φp ) such that each φp makes the following diagram commute:(63)(T ∗ )pb × Gpb
C↓(G∗ )pb

φ0p ×φ0p−→(T ∗ )pb × Gpb↓Cφp(G∗ )pb−→

where T ∗ := G∗ ∩ T and C(t, x) := x−1 tx.
Definition 21. The adelic connection δ in Theorem 20 is called the canonicalisospectral Lax connection.
On the other hand isocharacteristic Lax connections make commutative the following diagrams that are analogous to (38):(64)(G∗∗ )pb −→P↓

φp(G∗∗ )pb↓P

φ0p(An )pb,(An )pb −→ARITHMETIC DIFFERENTIAL GEOMETRY

15

where An = Spec A[z]. Among isocharacteristic Lax connections there is a canonical one.
Note that the canonical Frobenius lifts φp in the two diagrams (63) and (64) do
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 firstthe case of Ehresmann connections, (57). 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(65)
ϕpp0 := 0 [φp , φp0 ] : A[x] → A[x], p, p0 ∈ V.
pp
The family (ϕpp0 ) can be referred to as the curvature of the adelic connection (δp ).
The situation for general adelic connections (including the cases of Chern and Laxconnections) is quite different. Indeed, in defining curvature we face the following
dilemma: our p-derivations δp in (50) do not act on the same ring, so there is no apriori way of considering their commutators and, hence, it does not seem possible
to define, in this way, the notion of curvature. It turns out, however, that some of
our adelic connections satisfy a remarkable property which we call globality along
the identity (more generally along various subvarieties); this property allows us todefine curvature via commutators.
Definition 22. 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 the family of“divided” commutators (ϕpp0 ),(66)

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 , ..., [φs−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 ).

(67)

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

b0⊂ Ap [[T ]] ⊃ B p↓ φ p0↓ φ p0b0
pb0⊂ A [[T ]] ⊃ B p

16

ALEXANDRU BUIUM

The idea of comparing p-adic phenomena for different p’s by “moving along theidentity section” was introduced in [27] where it was referred to as analytic continuation between primes. Analytic continuation is taken here in the sense of Zariski[?], 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 [56]. Note that in our context we are patchingdata defined on “tubular neighborhoods”
Spf B pband

Spf A[[T ]]Spec B/pB

andSpec B/(T )of two closed subsetsof the scheme Spec B; the data are required to coincide on the “tubular neighborhood”
Spf Apb[[T ]]of the intersection
Spec B/(p, T ) = (Spec B/pB) ∩ (Spec B/(T )).
This is in contrast with the use of patching in Galois theory [56] 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 soit 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 ), with attached Frobenius lifts (φp ), and introduce the following:
Definition 23. The (1, 1)-curvature of δ is the matrix of “divided commutators”(ϕpp0 )
(68)

(69)

ϕpp0 :=

1[φp , φ0p0 ] : A[[T ]] → A[[T ]],pp0

ϕpp :=

p 6= p0 ,1[φp , φ0p ] : A[[T ]] → A[[T ]].
p

Going back to our discussion of curvature for Chern connections one can prove:
Theorem 24. [5] Let q ∈ GLn (A) with q t = ±q. If all the entries of q are roots ofunity or 0 then the Chern connection δ attached to q is global along 1; in particularδ has a well defined curvature and (1, 1)-curvature.
So we may address the question of computing the curvature and (1, 1)-curvatureof 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:ARITHMETIC DIFFERENTIAL GEOMETRY

17

Definition 25. A matrix q ∈ GLn (A) is split if it is one of the following:



 

1 0 0
0
1r
0 1r
,
,  0 0 1r  ,
(70)
−1r 0
1r 0
0 1r 0where 1r is the r × r identity matrix and n = 2r, 2r, 2r + 1 respectively.
One can prove:
Theorem 26. [5] Let q be split and let (ϕpp0 ) and (ϕpp̄0 ) be the curvature and the(1, 1)-curvature of the 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.
4) 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 morallysays that Spec Z is only “mildly curved.” Assertions 1 and 2 imply, in particular,
assertion 1 in the following:
Theorem 27. [42] Assume q split and n ≥ 4 is even. Then 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 that holonomy Lie algebras arisingfrom Galois theory are never nilpotent unless they vanish. Assertion 2 should beviewed as a statement suggesting that the flavor of our arithmetic situation is rather
different from that of classical locally symmetric spaces; indeed, for the latter, theLie 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 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 one constructs connections on GLn that do send the ideal of
SLn into itself; there are many such connections and the “simplest” one will becalled the special linear connection. Cf. [42].
Other curvatures can be introduced and vanishing/non-vanishing results for themcan be proved; they can be referred to as 3-curvature, first Chern form, first Chern(1, 1)-form, mean curvature, scalar curvature, etc. One can also introduce Chern
connections attached to hermitian matrices and results can be proved for theircurvature. Cf. [42].
Similar results can be proved for the curvature of Lax connections. What happens is that the open sets G∗ and G∗∗ where isospectral and isocharacteristic Laxconnections are defined (cf. (63) and (64)) do not contain the identity of thegroup G = GLn hence curvature cannot be defined by analytic continuation along

18

ALEXANDRU BUIUM

the identity; however these open sets contain certain torsion points of the diagonal
maximal torus of G and one can use analytic continuation along such torsion points
to define curvature and (1, 1)-curvature. One can then prove the non-vanishing ofthe (1, 1)-curvature of isocharacteristic Lax connection for n = 2. The (canonical) isospectral Lax connection has, immediately from its definition, a vanishingcurvature.
Note that the concept of curvature discussed above was based on what we called
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 the applicability 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 thecondition of being global along 1 with the condition of “being global along certain
tori.” However, there is a different approach towards making Frobenius lifts comparable; this approach is based on algebraizing Frobenius lifts via correspondences
and works for adelic connections that are not necessarily global along 1 (or alonga torus). The price to pay for allowing this generality is that endomorphisms (ofA[[T ]]) are replaced by correspondences (on GLn ). Let us explain this alternativeroad to curvature in what follows.
Definition 28. Let φp : X pb → X pb be a Frobenius lift on X pb where X is ascheme of finite type over A. An algebraizing correspondence for φp is a triple
Γp := (Yp , πp , ϕp ) where πp : Yp → G and ϕp : Yp → G are morphisms of A-schemesof finite type such that:1) πp is affine and étale;2) the p-adic completion of πp , πppb : Yppb → Gpb, is an isomorphism;3) we have an equality of maps, ϕppb = φp ◦ πppb : Yppb → Gpb.
For the Chern connection one can prove the following:
Theorem 29. [40] Let δ = (δp ) be the Chern connection on G = GLn attached toa matrix q ∈ GLn (A) with q t = ±q and let (φp ) be the attached Frobenius lifts on
Gpb. Then for each p there exists an algebraizing correspondence Γp = (Yp , πp , ϕp )for φp .
Similarly for the Levi-Civitá connection we have the following (in which, tosimplify, we assume A = Z[1/M ]):Theorem 30. [43] Let (δ1p , ..., δnp ) be the Levi-Cività connection attached to q ∈GLn (A), q t = q, and let (φ1p , ..., φnp ) be the attached Frobenius lifts on Gpb. Then
there exist algebraizing correspondences Γpi = (Yp , πp , ϕpi ) for φpi .
Definition 31. The family (Γp ) (respectively (Γpi )) is called a correspondence
structure for (δp ) (respectively (δpi )).
This structure is not unique but does have some “uniqueness features.”On the other hand correspondences as above act on the field E of rational functions of G = GLn . Let us explain this in the Chern case; the Levi-Civitá case issimilar. The action is by the formula Γ∗p : E → E,(71)Γ∗p (z) := Trπp (ϕ∗p (z)), z ∈ E,

ARITHMETIC DIFFERENTIAL GEOMETRY

19

where
Trπp : Fp → E
is the trace of the extensionπp∗ : E → Fp := Yp ⊗G E
andϕ∗p : E → Fp
is induced by ϕp .
Definition 32. The curvature of (Γp ) is the matrix (ϕ∗pp0 ) with entries the additive
homomorphisms
1(72)
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 Chern
connections attached to arbitrary q’s (that do not necessarily have entries zeroes orroots of unity). There is a (1, 1)-version of the above as follows. Indeed the trivial
adelic connection δ0 = (δ0p ) has a canonical correspondence structure (Γ0p ) givenbyΓ0p = (G, π0p , ϕ0p ),where π0p is the identity, and ϕ0p (x) = x(p) .
Definition 33. The (1, 1)-curvature of (Γp ) is the family (ϕ∗pp0 ) where ϕ∗pp0 is the
additive endomorphism
1(73)
ϕ∗pp0 := 0 [Γ∗0p0 , Γ∗p ] : E → E for p 6= p0 ,pp
(74)

ϕ∗pp :=1 ∗[Γ , Γ∗ ] : E → E.
p 0p p

Then one can prove the following:
Theorem 34. [40] Let q ∈ GL2 (A) be split. Then 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.
Finally note that one can define curvature for the Levi-Cività connection in caseV = {p}.
Definition 35. Let (δ1p , ..., δnp ) be the Levi-Cività connection attached to a 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
(75)
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). This curvature satisfies congruences modp that are analogous to the symmetries of the classical Riemann tensor. One canprove non-vanishing results for these curvatures. For instance we have:

20

ALEXANDRU BUIUM

Theorem 36. [42] Let i, j, k, l be fixed indices between 1 and n. Then 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(76)

Ψij = (Rijkl ).

Then the Rijkl in 76 can be viewed as an arithmetic analogue of the covariant
Riemann tensor in classical differential geometry. We will prove that (Rijkl ) in 76
satisfies congruences mod p:(77) Rijkl ≡ −Rjikl ,Rijkl ≡ −Rijlk ,Rijkl + Riklj + Rijlk ≡ 0, Rijkl ≡ Rklij ,which are, of course, an arithmetic analogue of the classical symmetries of the
Riemann curvature tensor. In addition we will prove the congruence1(δqjk + δqil − δqik − δqjl )p mod (p, x − 1).
2This can be viewed as an analogue of the expression of curvature at a point, innormal coordinates.
In case the entries of q are roots of unity or 0 one can prove that for each i, (δip )is global along 1 so we will be able to define a mixed curvature (ϕij
pp0 ) indexed byi, j = 1, ..., n and p, p0 ∈ V given by the divided commutators
1
(79)
ϕij
[φip , φjp0 ] : A[[T ]] → A[[T ]], p 6= p0 ,pp0 :=
pp0
(78)

(80)

Rijkl ≡

ϕij
pp :=

1
[φip , φjp ] : A[[T ]] → A[[T ]].
p

For a fixed p and the Fedosov connection (δ1p , δ2p ) relative to any antisymmetricq ∈ GL2 (A) the formula (75) defines, again, a curvature; one can prove that thiscurvature does not vanish in general even if q is split.
Finally, using the algebraization by correspondences one can define a mixed curvature for the Levi-Civitá connection for V consisting of more than one prime andfor q with entries not necessarily roots of unity or 0 and one can prove various
non-vanishing results for this curvature; cf. [43].
3. Comparison with other theories
3.1. Three perspectives. A number of analogies between primes and geometricobjects have been proposed. Here are three of them:A) Primes are analogous to points on a Riemann surface.
B) Primes are analogous to knots in a 3-dimensional manifold.
C) Primes are analogous to directions in an infinite dimensional manifold.
The viewpoint A is classical, it has a complex analytic flavor, and goes backto Dedekind, Hilbert, etc. The framework of Grothendieck, Arakelov, etc., also

ARITHMETIC DIFFERENTIAL GEOMETRY

21

fits into viewpoint A. According to this viewpoint the ring of integers Z, or moregenerally rings of integers in number fields, can be viewed as analogues of ringsof functions on Riemann surfaces or affine algebraic curves; these are objects ofcomplex dimension 1 (or real dimension 2). Genera of number fields are classicallydefined and finite, as in the case of Riemann surfaces. There is a related viewpointaccording to which Z is the analogue of an algebraic curve of infinite genus; cf. e.g.,[49].
The viewpoint B has a topological flavor and originates in suggestions of Mazur,
Manin, Kapranov, and others. According to this viewpoint Spec Z should be viewed
as an analogue of a 3-dimensional manifold, while the embeddings Spec Fp →Spec Z should be viewed as analogues of embeddings of circles. The Legendre
symbol is then an analogue of linking numbers. This analogy goes rather deep [76].
Our approach here and previous papers by the author adopt the viewpoint C
and have a differential geometric flavor.
There is a possibility that our theory has connections with viewpoint B, as shown,for instance, by the presence of the Legendre symbol in our Chern connections.
Indeed the underlying Galois theory of reciprocity is an analogue of the monodromy
in the 3-dimensional picture; in the same way our arithmetic curvature theory
could be an analogue of the identity component of a natural “holonomy” in the
3-dimensional picture.
3.2. Field with one element. There are other approaches that adopt the viewpoint C. For instance Haran’s theory of the field with one element, F1 , cf. [55], (andprevious F1 flavored work of Kurokawa and others [70]) considers the operators∂: Z → Z,∂p∂aa:= vp (a) ,∂ppwhere vp (a) is the p-adic valuation of a. These operators have a flavor that is rather
different from that of Fermat quotients, though, and it seems unlikely that Haran’stheory and ours are directly related. Even more remote from our theory are the F1theories of Soulé [80] and Connes-Consani [49] which do not directly provide a way
to differentiate integers.
Borger’s philosophy of F1 , cf. [8], is, in some sense, perpendicular to the above
mentioned approaches to F1 and, in the “case of one prime” is consistent with ourapproach: roughly speaking, in the case of one prime, Borger’s theory [8, 10] canbe viewed as an algebraization of our analytic theory in [16, 28]. In the case of
more (all) primes Borger’s F1 theory can also be viewed as a viewpoint consistent
with C above: indeed Borger’s beautiful suggestion is to take λ-structures (in thesense of Grothendieck) as descent data from Z to F1 . Recall that a λ-structure ona scheme X flat over Z is the same as a commuting 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

22

ALEXANDRU BUIUM

between primes or algebraization by correspondences). So in practice our approach
places us, most of the times, outside the paradigm of λ-geometry.
3.3. Ihara’s differential. Next we would like to point out what we think is an
important difference between our viewpoint here and the viewpoint proposed by
Ihara in [61]. 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 [61]Ihara proposed to see the mapYa − ap
mod p
(81)
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 the approach A 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 ourδp ’s which leads to the possibility of considering arithmetic analogues of differential
equations, curvature, etc.
3.4. Fontaine-Colmez calculus. The Fontaine-Colmez theory of p-adic periods[52] also speaks of a differential calculus with numbers. Their calculus is perpendicular to ours in the following precise sense. For a fixed prime p our calculus basedon 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 ),
p6|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 thetotally ramified cases, involving two “perpendicular” directions, could be developed leading to arithmetic partial differential equations in two variables. Signs ofpossibility of such a theory can be found in our papers [25, 26].
3.5. Grothendieck’s p-curvature. It is worth pointing out that the study of ourcurvature here resembles the study of the p-curvature appearing in the arithmetic
theory of differential equations that has been developed around the Grothendieckconjecture (cf., e.g., [65]). Both curvatures measure the lack of commutation of certain operators and both of theories rely on technical matrix computations. However
we should also point out that the nature of the above mentioned operators in the twotheories is quite different. Indeed our curvature here involves the “p-differentiation”

ARITHMETIC DIFFERENTIAL GEOMETRY

23

of numbers with respect to primes p (in other words it is about d/dp) whereas thetheory in [65] (and related papers) is about usual differentiation d/dt with respect
to a variable t of power series in t with arithmetically interesting coefficients. In
spite of these differences the two types of curvatures could interact; a model forsuch an interaction between d/dp and d/dt is in the papers [25, 26]. A similar
remark can be made about the difference between our approach and that in [13]which, again, is about usual Kähler differentials and hence about usual derivations.
3.6. Discrete geometry as Euclidean geometry. Finally we would like to pointout that the theory presented here is a priori unrelated to topics such as the geometryof numbers [44] on the one hand and discrete differential geometry [7] on the other.
Indeed in both these geometries what is being studied are discrete configurations ofpoints in the Euclidean space Rm ; in the geometry of numbers the configurationsof points typically represent rings of algebraic numbers while in discrete differential geometry the configurations of points approximate smooth submanifolds of the
Euclidean space. This framework is, therefore, that of the classical geometry of
Euclidean space, based on R-coordinates, and not that of an analogue of this geometry, based on “prime coordinates.” It may very well happen, however, that (oneor both of) the above topics are a natural home for some (yet to be discovered)Archimedian counterpart of our (finite) adelic theory.

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