DIFFERENTIAL CALCULUS WITH INTEGERS
ALEXANDRU BUIUM

Abstract. Ordinary differential equations have an arithmetic analogue inwhich functions are replaced by numbers and the derivation operator is replaced by a Fermat quotient operator. In this survey we explain the main
motivations, constructions, results, applications, and open problems of thetheory.

The main purpose of these notes is to show how one can develop an arithmetic
analogue of differential calculus in which differentiable functions x(t) are replaced
by integer numbers n and the derivation operator x 7→ dx
dt is replaced by the Ferp
mat quotient operator n 7→ n−n
,
where
p
is
a
prime
integer. The Lie-Cartanpgeometric theory of differential equations (in which solutions are smooth maps) isthen replaced by a theory of “arithmetic differential equations” (in which solutions
are integral points of algebraic varieties). In particular the differential invariantsof groups in the Lie-Cartan theory are replaced by “arithmetic differential invariants” of correspondences between algebraic varieties. A number of applications todiophantine geometry over number fields and to classical modular forms will be
explained.
This program was initiated in [11] and pursued, in particular, in [12]-[35]. For
an exposition of some of these ideas we refer to the monograph [16]; cf. also thesurvey paper [58]. We shall restrict ourselves here to the ordinary differential case.
For the partial differential case we refer to [20, 21, 22, 7]. Throughout these notes
we assume familiarity with the basic concepts of algebraic geometry and differential
geometry; some of the standard material is being reviewed, however, for the sake ofintroducing notation, and “setting the stage”. The notes are organized as follows.
The first section presents some classical background, the main concepts of thetheory, a discussion of the main motivations, and a comparison with other theories.
The second section presents a sample of the main results. The third section presents
a list of open problems.
Acknowledgement. The author is indebted to HIM for support during part ofthe semester on Algebra and Geometry in Spring 2013. These notes are partiallybased on lectures given at the IHES in Fall 2011 and MPI in Summer 2012 when
the author was partially supported by IHES and MPI respectively. Partial support
was also received from the NSF through grant DMS 0852591.
1. Main concepts
1.1. Classical analogies. The analogies between functions and numbers haveplayed a key role in the development of modern number theory. Here are some
1

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ALEXANDRU BUIUM

classical analogies. All facts in this subsection are well known and entirely classical;
we review them only in order to introduce notation and put things in perspective.
1.1.1. Polynomial functions. The ring C[t] of polynomial functions with complex
coefficients is analogous to the ring Z of integers. The field of rational functions
C(t) is then analogous to the field of rational numbers Q. In C[t] any non-constantpolynomial is a product of linear factors. In Z any integer different from 0, ±1 is
up to a sign a product of prime numbers. To summarize
C ⊂ C[t] ⊂ C(t)are analogous to{0, ±1} ⊂ Z ⊂ Q
1.1.2. Regular functions. More generally rings O(T ) of regular functions on complex
algebraic affine non-singular curves T are analogous to rings of integers OF in
number fields F . Hence curves T themselves are analogous to schemes Spec OF .
Compactifications
T ⊂ T = T ∪ {∞1 , ..., ∞n } ' (compact Riemann surface of genus g)
are analogous to “compactifications”
Spec OF ⊂ Spec OF = (Spec OF ) ∪Hom(F, C)conjugation

1.1.3. Formal functions. The inclusions
C ⊂ C[[t]] ⊂ C((t))(where C[[t]] is the ring of power series and C((t)) is the ring of Laurent series) areanalogous to the inclusion{0} ∪ µp−1 = {c ∈ Zp ; cp = c} ⊂ Zp ⊂ Qp
(where Zp is ring of p-adic integers and Qp = Zp [1/p]). Recall that
Zp = lim Z/pn Z = {←∞X

ci pi ; ci ∈ {0} ∪ µp−1 }n=0

So {0} ∪ µp−1 plays the role of “constants” in Zp . Sometimes we need more “constants” and we are led to consider, instead, the inclusions:[ur ⊂ Zur [1/p]dd{0} ∪µpν −1 ⊂ Z
p
p
ν

where
∞
X
[
ν
ur = Z [ζ; ζ p −1 = 1, ν ≥ 1]ˆ = {dZci pi ; ci ∈ {0} ∪µpν −1 }.
p
p
i=0

ν

Here the upper hat on a ring A means its p-adic completion:
b := lim A/pn A.
A←S
So in the latter case the monoid {0} ∪ ν µpν −1 should be viewed as the set ofur ; this is consistent with the “philosophy of the field with oned“constants” of Zpelement” to which we are going to allude later. Let us say that a ring is a local
p-ring if it is a discrete valuation ring with maximal ideal generated by a prime
ur are local p-rings. Also for any local p-ring R we denote bydp ∈ Z. Then Zp and Zp

DIFFERENTIAL CALCULUS WITH INTEGERS

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k = R/pR the residue field and by K = R[1/p] the fraction field of R. Sometimes wewill view local p-rings as analogues of rings C{x} of germs of analytic functions on
Riemann surfaces and even as analogues of rings of global analytic (respectively C ∞functions) on a Riemann surface T (respectively on a 1-dimensional real manifold
T , i.e. on a circle S 1 or R).
1.1.4. Topology. Fundamental groups of complex curves (more precisely Deck transformation groups of normal covers T 0 → T of Riemann surfaces) have, as analogues,
Galois groups G(F 0 /F ) of normal extensions number fields F ⊂ F 0 . The genus of a
Riemann surface has an analogue for number fields defined in terms of ramification.
All of this is very classical. There are other, less classical, topological analogies likethe one between primes in Z and nodes in 3-dimensional real manifolds [61].
1.1.5. Divisors. The group of divisors
Div(T ) = {X

nP P ; nP ∈ Z}

P ∈T

on a non-singular complex algebraic curve T is analogous to the group of divisors
XνP P ; νP ∈ Z if P is finite, νP ∈ R if P is infinite}
Div(Spec OF ) = {One can attach divisors to rational functions f on T (Div(f ) is the sum of poles
minus the sum of zeroes); similarly one can attach divisors to elements f ∈ F . In
both cases one is lead to a “control” of the spaces of f s that have a “controlled”
divisor (the Riemann-Roch theorem). One also defines in both settings divisor classgroups. In the geometric setting the divisor class group of T is an extension of Z
by the Jacobian
Jac(T ) = Cg /(period lattice of T )where g is the genus of T . In the number theoretic setting divisor class groups
can be interpreted as “Arakelov class groups”; one recaptures, in particular, theusual class groups Cl(F ). Exploring usual class groups “in the limit”, when one
adjoins roots of unity, leads to Iwasawa theory. We will encounter Jacobians later in
relation, for instance, to the Manin-Mumford conjecture. This conjecture (provedby Raynaud) says that if one views T as embedded into Jac(T ) (via the “AbelJacobi map”) the the intersection of T with the torsion group of Jac(T ) is a finite
set. This particular conjecture does not seem to have an analogue for numbers.
1.1.6. Families. Maps
M →Tof complex algebraic varieties, complex analytic, or real smooth manifolds, where
dim T = 1, are analogous to arithmetic schemes i.e. schemes of finite type
X → Spec R
where R is either the ring of integers OF in a number field F or a complete local
p-ring respectively. Note however that in this analogy one “goes arithmetic only
half way”: indeed for X → Spec R the basis is arithmetic yet the fibers are still
geometric. One can attempt to “go arithmetic all the way” and find an analogue of
M → T for which both the base and the fiber are “arithmetic”; in particular one
would like to have an analogue of T ×T which is “arithmetic” in two directions. Thisis one of the main motivations in the search for F1 , the “field with one element”.

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1.1.7. Sections. The set of sections
Γ(M/T ) = {s : T → M ; π ◦ σ = 1}
of a map π : M → T is analogous to the set
X(R) = {s : Spec R → X; π ◦ s = 1}of R-points of X where π : X → Spec R is the structure morphism. This analogy
suggests that finiteness conjectures for sets of the form X(R), which one makes inDiophantine geometry, should have as analogues finiteness conjectures for sets ofsections Γ(M/T ). A typical example of this phenomenon is the Mordell conjecture(Faltings’ theorem) saying that if X is an algebraic curve of “genus” ≥ 2, definedby polynomials with coefficients in a number field F , then the set X(F ) is finite.
Before the proof of this conjecture Manin [55] proved a parallel finiteness result forΓ(M/T ) where M → T is a “non-isotrivial” morphism from an algebraic surface
to a curve, whose fibers have genus ≥ 2. Manin’s proof involved the considerationof differential equations with respect to vector fields on T . Faltings’ proof went
along completely different lines. This raised the question whether one can developa theory of differential equations in which one can differentiate numbers.
All these examples of analogies are classical; cf. work of Dedekind, Hilbert,
Hensel, Artin, Weil, Lang, Tate, Iwasawa, Grothendieck, and many others.
1.2. Analogies proposed in [11]-[35]. One thing that seems to be missing fromthe classical picture is a counterpart, in number theory, of the differential calculus (in particular of differential equations) for functions. Morally the question iswhether one can meaningfully consider (and successfully use) “arithmetic differential equations” satisfied by numbers. In our research on the subject [11] - [35] we
proposed such a theory based on the following sequence of analogies:d: C[t] → C[t] is analogous to the1.2.1. Derivatives. The derivative operator δt = dt
a−ap
Fermat quotient operator δ = δp : Z → Z, δp a = p . More generally the derivativedoperator δt = dt
: C ∞ (R) → C ∞ (R) (with t the coordinate on R) is analogous topthe operator δ = δp : R → R on a complete local p-ring R, δp α = φ(α)−α(wherep
φ : R → R is a fixed homomorphism lifting the p-power Frobenius map on R/pR).
The map δ = δp above is, of course, not a derivation but, rather, it satisfies thefollowing conditions:

δ(1)
= 0
δ(a + b) = δ(a) + δ(b) + Cp (a, b)
δ(ab)= ap δ(b) + bp δ(a) + pδ(a)δ(b),where Cp (x, y) ∈ Z[x, y] is the polynomial Cp (x, y) = p−1 (xp + y p − (x + y)p ). Any
set theoretic map δ : A → A from a ring A to itself satisfying the above axioms
will be referred to a p-derivation; such operators were introduced independently in[49, 11] and they implicitly arise in the theory of Witt rings. For any such δ, the map
φ : A → A, φ(a) = ap + pδa is a ring homomorphism lifting the p-power Frobenius
on A/pA; and vice versa, given a p-torsion free ring A and a ring homomorphismp
φ : A → A lifting Frobenius the map δ : A → A, δa = φ(a)−ais a p-derivation.
p

DIFFERENTIAL CALCULUS WITH INTEGERS

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1.2.2. Differential equations. Differential equations F (t, x, δt x, ..., δtn x) = 0 (withF smooth) satisfied by smooth functions x ∈ C ∞ (R) are replaced by arithmetic
differential equations F (α, δp α, ..., δpn α) = 0 with F ∈ R[x0 , x1 , ..., xn ]ˆ satisfied
by numbers α ∈ R. As we shall see it is crucial to allow F to be in the p-adic
completion of the polynomial ring rather than in the polynomial ring itself; indeed
if one restricts to polynomial F ’s the main interesting examples of the theory areleft out.
1.2.3. Jet spaces. More generally, the Lie-Cartan geometric theory of differential
equations has an arithmetic analogue which we explain now. Let M → T be asubmersion of smooth manifolds with dim T = 1, and let
J n (M/T ) = {Jtn (s); s ∈ Γ(M/T ), t ∈ T }be the space of n-jets Jtn (s) of smooth sections s of M → T at points t ∈ T . Cf.
[1, 48, 62] for references to differential geometry. If M = Rd × R, T = R, M → T
is the second projection, and x = (x1 , ..., xd ), t are global coordinates on Rd and R
respectively then J n (M/T ) = R(n+1)d × R with jet coordinates x, x0 , ..., x(n) , t. So
for general M → T the map J n (M/T ) → M is a fiber bundle with fiber Rnd whered + 1 = dim(M ). One has the total derivative operator
δt : C ∞ (J n (M/T )) → C ∞ (J n+1 (M/T ))which in coordinates is given by
δt =

n X
d
X
∂
(j+1) ∂.
+xi(j)∂t j=0 i=1∂xi

In the arithmetic theory the analogues of the manifolds J n (M/T ) are certain formal
schemes (called arithmetic jet spaces or p-jet spaces) J n (X) = J n (X/R) defined asfollows. Assume X is affine, X = Spec R[x](f ) with x, f tuples; the constructionthat follows is easily globalized to the non-affine case. Let x0 , ..., x(n) , ... be newtuples of variables, consider the polynomial ring R{x} := R[x, x0 , x00 , ...], let φ :R{x} → R{x} be the unique ring homomorphism extending φ on R and sending
φ(x) = xp + px0 , φ(x0 ) = (x0 )p + px00 ,..., and letδ = δp : R{x} → R{x}be the p-derivation
δF =φ(F ) − F p.
p

Then one defines
J n (X) = Spf

R[x, x0 , ..., x(n) ]ˆ
.
(f, δf, ..., δ n f )1.2.4. Differential equations on manifolds. Usual differential equations are definedgeometrically as elements of the ring C ∞ (J n (M/T )); alternatively such elements
are referred to as (time dependent) Lagrangians on M . Their analogue in the arithmetic theory, which we call arithmetic differential equations [16], are the elementsof the ring On (X) := O(J n (X)). For group schemes the following concept [11]plays an important role: if G is a group scheme of finite type over R then we mayb a ) ⊂ On (G).
consider the R-module Xn (G) = Homgr (J n (G), G
Going back to arbitrary schemes of finite type X/R note that δp : R{x} → R{x}induces maps δ = δp : On (X) → On+1 (X) which can be viewed as arithmetic

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ALEXANDRU BUIUM

analogues of the total derivative operator δt : C ∞ (J n (M/T )) → C ∞ (J n+1 (M/T )).
The latter is a “generator” of the Cartan distribution defined by dxi − x0i dt, dx0i −x00i dt, etc. Note however that the forms in differential geometry defining the Cartan
distribution do not have a direct arithmetic analogue; for one thing there is no form“dp” analogous to dt. On the other hand in the arithmetic case we have induced
ring homomorphisms φ : On (X) → On+1 (X) which have no analogue in differential
geometry.
1.2.5. Differential functions. Any differential equation F ∈ C ∞ (J n (M/T )) definesa natural differential function F∗ : Γ(M/T ) → C ∞ (T ); in coordinates sections s ∈Γ(M/T ) correspond to functions x = x(t) and then F∗ sends x(t) into F∗ (x(t)) =F (x(t), δt x(t), ..., δtn x(t)). Analogously any arithmetic differential equation f ∈O(J n (X)) defines a map of sets f∗ : X(R) → R, referred to as a δ-function, whichin affine coordinates sends α ∈ X(R) ⊂ RN into f∗ (α) := F (α, δp α, ..., δpn α) ∈ R
if F ∈ R[x, x0 , ..., x(n) ]ˆ represents f . If X = G is in addition a group scheme andψ ∈ Xn (G) then ψ∗ : G(R) → Ga (R) = R is a group homomorphism called a
δ-character of G. For X/R smooth and R/pR algebraically closed f is uniquelydetermined by f∗ so one can identify f with f∗ and ψ with ψ∗ .
1.2.6. Prolongations of vector fields. For any vertical vector field
ξ :=

d
X
i=1

ai (t, x)∂∂xi

on M/T one can consider the canonical prolongations
ξ (n) :=n X
dX∂
(δtj ai (t, x)) (j)∂xi
j=0 i=1

on J n (M/T ). The map
ξ (n) : C ∞ (J n (M/T )) → C ∞ (J n (M/T ))is the unique R-derivation whose restriction to C ∞ (M ) is ξ and which commuteswith the total derivative operator δt . The above construction has an arithmetic
analogue that plays a key technical role in the development of the theory. Indeed
for any affine smooth X/R and any R-derivation ξ : O(X) → O(X) the canonicalprolongation
ξ (n) : On (X)[1/p] → On (X)[1/p]is defined as the unique K = R[1/p]-derivation whose restriction to O(X) is ξ andwhich commutes with φ. This construction then obviously globalizes.
1.2.7. Infinitesimal symmetries of differential equations. Some Galois theoretic concepts based on prolongations of vector fields have arithmetic analogues. Indeedrecall that if L ⊂ C ∞ (J n (M/T )) is a linear subspace of differential equations
then a vertical vector field ξ on M/T is called an infinitesimal symmetry of L if
ξ (n) L ⊂ L; it is called a variational infinitesimal symmetry if ξ (n) L = 0. Similarly given an R-submodule L ⊂ On (X) an infinitesimal symmetry of L is an Rderivation ξ : O(X) → O(X) such that ξ (n) L ⊂ L[1/p]. One says ξ is a variational
infinitesimal symmetry of L if ξ (n) L = 0.

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1.2.8. Total differential forms on manifolds. Recall that a total differential form([62], p. 351) on M/T is an expression that in coordinates looks like a sum of
expressions
F (t, x, x0 , ..., x(n) )dxj1 ∧ ... ∧ dxji ,It is important to introduce an arithmetic analogue of this which we now explain.
We consider the case of top forms (i = d) which leads to what we will call δline bundles. Denote by On the sheaf U 7→ On (U ) on X for the Zariski topology.
Define a δ-line bundle of order n on X to be a locally free On -module of rank
1. Integral powers of bundles need to be generalized as follows. Set
Z[φ]
PW =
s
(ring of polynomials
with
Z-coefficients
inthesymbol
φ).
For
w
=
a
φ
write
s
P
P
deg(w) =as . Also let W+ be the set of all w =as φs ∈ W P
with as ≥ 0 forn
all s. If L is a line bundle on X given by a cocycle (gij ) and w = s=0 as φs ∈ W ,w
w
w 6= 0,Q ans 6= 0,asthen define a δ-line bundle L of order n by the cocycle (gij ),w
gij = s φ (gij ) . With all these definitions in place we may define the following
rings which, by the way, are the main objects of the theory:
MRδ (X, L) =H 0 (X, Lw ).
06=w∈W+

Note that the above is a graded ring without unity. The homogeneous elements ofRδ (X, L) can be viewed as arithmetic analogues of Lagrangian densities and canalso be referred to as arithmetic differential equations [16].
1.2.9. Differential forms on jet spaces and calculus of variations. The spacesΩi↑ (J n (M/T ))of vertical smooth i-forms on J n (M/T ) (generated by i-wedge products of forms
dx, dx0 , ..., dx(n) ) play an important role in the calculus of variations [62]. These
spaces fit into a deRham complex where the differential is the vertical exteriordifferentialnd↑ : Ωi↑ (J n (M/T )) → Ωi+1
↑ (J (M/T ))with respect to the variables x, x0 , ..., x(n) . On the other hand we have uniqueoperators
δt : Ωi↑ (J n (M/T )) → Ωi↑ (J n+1 (M/T ))that commute with d↑ , induce a derivation on the exterior algebra, and for i = 0 coincide with the total derivative operators. Then one can define spaces of functional
forms ([62], p. 357)Ωi↑ (J n (M/T ))Ωi∗ (J n (M/T )) =Im(δt )and the (vertical part of the) variational complex ([62], p. 361) with differentialsnd : Ωi∗ (J n (M/T )) → Ωi+1
∗ (J (M/T )).

R
The class of ω ∈ Ωi↑ (J n (M/T )) in Ωi∗ (J n (M/T )) is denoted by ωdt. For i = 0,RR
ω = F , have the formula d( F dt) = EL(F )dt in Ωi∗ (J 2n (M/T )) where EL(F ) =Pd
i=1 Fi dxi is the Euler-Lagrange total differential form,
!
n
X
∂F
j j
Fi =
(−1) δt
.
(j)
∂xi
j=0

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ALEXANDRU BUIUM

Noether’s theorem then says that for any vertical vector field ξ on M/T we havethe formulahξ, EL(F )i − ξ (n) (F ) = δt G
for some G ∈ C ∞ (J 2n−1 (M/T )); G is unique up to a constant. If ξ is a variational
infinitesimal symmetry of F then G is referred to as the conservation law attachedto this symmetry; in this case if x(t) is a solution of all Fi = 0 then G evaluated at
x(t) will be a constant.
Analogously, for X/R smooth and affine, one can consider the modules ΩiOn (X)/Rdefined as the exterior powers of the inverse limit of the Käher differentialsΩOn (X)⊗(R/pn R)/(R/pn R) ,and the exterior differential
d : ΩiOn (X)/R → Ωi+1
On (X)/R .
Also one may consider the operatorsφ∗ : ΩiOn (X)/R → ΩiOn+1 (X)/R ;again φ∗ and d commute. Also note that any element in the image of φ∗ is uniquely
divisible by pi ; for any ω ∈ ΩiOn (X)/R and r ≥ 0 we then set ωr = p−ir φ∗r ω. The
operation p−ir φ∗r is a characteristic zero version of the inverse Cartier operator.
For any element µ ∈ R (which we refer to as eigenvalue) one can define groupsΩi∗ (J n (X)) =ΩiOn (X)/RIm(φ∗ − µ)that fit into a variational complex with differentialsnd : Ωi∗ (J n (X)) → Ωi+1
∗ (J (X)).

R
The class of ω ∈ ΩiOn (X)/R in Ωi∗ (J n (X)) is denoted by ωdp. For i = 0, µ = 1,RR Pd
ω = F ∈ On (X) we have d( F dp) = { i=1 Fi ωni }dp in Ω1∗ (J 2n (X)) whereFi ∈ O2n (X), ω i is a basis of ΩO(X)/R , and ωni = p−n φ∗n ω i . Also an analoguePdof the Noether theorem holds in this context with (F ) := i=1 Fi ωni playing therole of the Euler-Lagrange form; indeed for any vector field ξ on X there exists
G ∈ O2n−1 (X) such thathξ (n) , (F )i − ξ (n) (F ) = Gφ − G.
If ξ is a variational infinitesimal symmetry of F then G can be referred to as aconservation law; in this case if P ∈ X(R) is a point which is a solution to allFi∗ (P ) = 0 then G∗ (P )φ = G∗ (P ).
1.2.10. Flows and Hamiltonian formalism [31]. We place ourselves in either thesmooth or the complex analytic setting. Assume one is given a section σ : M →J 1 (M/T ) of the projection J 1 (M/T ) → M . To give such a σ is equivalent to giving
a vector field δM on M lifting δt ; such a vector field can be referred to as a δt -flow
on M/T . In coordinates, if σ is defined by (t, x) 7→ (t, x, s(t, x)), thenX∂δM =si (t, x).
∂xi
The composition
δσ∗tLδM : Ωi↑ (M/T ) −→Ωi↑ (J 1 (M/T )) −→ Ωi↑ (M/T )DIFFERENTIAL CALCULUS WITH INTEGERS

9

induces a derivation on the exterior algebra of vertical forms on M/T , commuteswith exterior vertical differentiation d↑ , and coincides with δM for i = 0; one canrefer to LδM as the “Lie derivative” attached to the vector field δM (it is a relative
version of the usual Lie derivative).
We consider now the “time dependent” Hamiltonian formalism. The classicaltreatment of this formalism (e.g. [56]) usually involves the differential of time, dt.
Since “dp” does not exist in the arithmetic setting what we will do will be to firstfind a presentation of the classical time dependent Hamiltonian formalism without
reference to dt and then we shall transpose the latter to the arithmetic case. Assume
for simplicity that M has dimension 3 so the fibers of M → T are 2-dimensional.
A vertical symplectic form on M is a nowhere vanishing vertical 2-form on M . A
vertical contact form on M is a vertical 1-form ν on M such that dν is symplectic.
We shall drop the word “vertical” in what follows. Recall that a δt -flow on M isa vector field δM on M that lifts the vector field δt on T ; and recall that to givea δ-flow on M is the same as to give a section of the projection J 1 (M/T ) → M .
Given a δ-flow δM as above we may consider the relative Lie derivative LδM on thevertical forms Ωi (M/T ). A δt -flow δM will be called Hamiltonian with respect toa symplectic form η on M if LδM η = µ · η for some function µ on T . Note that if
δt γ = −µ · γ for some function γ on T and if η1 = γη then LδM η1 = 0. If ν is acontact form, η = d↑ ν, and δM is a δt -flow which is Hamiltonian with respect toη (hence LδM η = µη) then  := LδM ν − µν is closed; the latter can be referred toas an Euler-Lagrange form attached to the symplectic form η and the δt -flow δM .
If  happens to be exact, i.e. LδM ν − µν = dL for some function L on M , then
L can be referred to as a Lagrangian attached to η and δM . (This Euler-Lagrange
formalism is related to, but does not coincide with, the Euler-Lagrange formalism
in 1.2.9.) In a series of interesting applications (especially in the complex analytic
setting of the above, e.g. to Painlevé equations [56]) M is taken to be J 1 (N/T )where N → T is a family of curves; in this case a 1-form on M is called canonical if
it is a function on M times the pull-back of a 1-form on N . Also one is interested,in this situation, in δt -flows defined by sections of J 1 (M/T ) → M = J 1 (N/T ) thatarise from sections of J 2 (N/T ) → J 1 (N/T ) (where J 2 (N/T ) is naturally embedded
into J 1 (M/T )); such δt -flows are referred to as canonical and are defined by order
2 differential equations (functions on J 2 (N )).
A justification for the above terminology comes, for instance, from the following
local algebraic considerations. Let F be a field equipped with a derivation δ : F →F (e.g. F = C((t)), δ = δt ) and let A be a ring isomorphic to a power series ringin 2 variables with coefficients in F . Choose formal coordinates x, y ∈ A (i.e. A =F [[x, y]]), and set ΩA = A · dx + A · dy, Ω2A = A · dx ∧ dy. A form η = udx ∧ dy ∈ Ω2A
is called invertible if u(0, 0) 6= 0. (These definitions are independent of the choiceof the formal coordinates x, y.) The following is a translation into the language ofdifferential algebra of the classical Hamiltonian/Lagrangian formalism in 2 analytic
coordinates. Assume one is given a derivation δ : A → A such that δF ⊂ F andsending the maximal ideal of A into itself. Then the following are easy to check:
1) Assume we are given an invertible η ∈ Ω2A . Then there exist formal coordinatesx, y ∈ A such that η = dy ∧ dx. Furthermore if δη = 0 then and there exists anelement H ∈ A such thatδy = −

∂H
,
∂x

δx =

∂H
.
∂y

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ALEXANDRU BUIUM

2) Assume there exist formal coordinates x, y ∈ A, and a series H ∈ A such that∂H∂H
δy = − ∂H∂x and δx = ∂y . Let η = dy ∧ dx, ν = ydx,  = δν, and L = y ∂y − H =yδx − H. Thendν = η, δη = 0, dL = .
3) Assume the notation in 2) and assume the series ∂∂yH
2 is invertible in F [[x, y]].
∂Then there exists a unique F -derivation denoted by ∂δx on F [[x, y]] sending x 7→ 0
and δx 7→ 1. Moreover we have


∂L
∂L
∂H
∂L
∂L
= y,
=−
, δ
=
.
∂δx
∂x
∂x
∂δx
∂x
2

We want to introduce an arithmetic analogue of the above formalism; cf. [31]. To
simplify our discussion let X be a smooth surface over R (or the p-adic completionof such a surface). Recall that by an i-form on X we understand a global section ofΩiX (projective limit of usual Kähler differentials of X mod pn ). A symplectic formon X is an invertible 2-form on X. A contact form on X is a 1-form on X such thatdν is symplectic. By a δ-flow on X we will understand a section X → J 1 (X) of the
projection J 1 (X) → X. Let us further specialize the situation to the case when
X = J 1 (Y ) where Y is a curve (affine, to simplify). A 1-form ν on X = J 1 (Y ) is
called canonical if ν = gβ with g ∈ O(X) and β a 1-form on Y . If ν is a canonical
1-form on X = J 1 (Y ) and is closed, i.e. dν = 0, then ν is (the pull-back of) a 1-form
on Y . Note that J 2 (Y ) naturally embeds into J 1 (J 1 (Y )). A δ-flow on X = J 1 (Y )will be called canonical if the section J 1 (Y ) = X → J 1 (X) = J 1 (J 1 (Y )) factors
through a section J 1 (Y ) → J 2 (Y ). This concept is, however, too restrictive forapplications. We will need to relax the concept of canonical δ-flow as follows. To
simplify the discussion we restrict to the case of order 2 differential equations. Letf ∈ O(J 2 (Y )) and let Z = Z 2 (f ) ⊂ J 2 (Y ) be the closed formal subscheme definedby f . For i ≥ 1 we defineΩ0iZ =ΩiJ n (Y )hf ΩiJ n (Y ) , df ∧ Ωi−1
J n (Y ) i.

0
So Ω01Z = ΩZ . Also we have natural wedge products
0j
0i+j∧ : Ω0iZ × Ω Z → ΩZ ,and exterior differentiation maps
0i+1
d : Ω0i
Z → ΩZ .

We say that f defines a generalized canonical δ-flow on J 1 (Y ) if the natural map
π ∗ ΩJ 1 (Y ) → Ω0Z
is injective and has a cokernel annihilated by a power of p. Note now that thenatural maps φ∗ : ΩiJ 1 (Y ) → ΩiJ 2 (Y ) induce natural maps denoted byφ∗Z : ΩiJ 1 (Y ) → Ω0i
Z.
We say that the generalized canonical δ-flow is Hamiltonian with respect to a sympletic form η on J 1 (Y ) if φ∗Z η = µ · η in Ω02
Z 2 (f ) for some µ ∈ pR. If in addition1
η = dν for some canonical 1-form ν on J (Y ) we call  := p1 (φ∗Z ν − µν) ∈ Ω0Z 2 (f ) an
Euler-Lagrange form. (This arithmetic Euler-Lagrange formalism is related to, butdoes not coincide with, the arithmetic Euler-Lagrange formalism in 1.2.9.) In theDIFFERENTIAL CALCULUS WITH INTEGERS

11

notation above p is closed in the sense that d(p) = 0 in Ω02
Z 2 (f ) . The above is theframework for the main results in [31] about the arithmetic analogue of PainlevéVI.
1.2.11. “Category of differential equations”. A categorical framework for differential equations on manifolds can be introduced (cf. [1], for instance). In one variantof this the objects are locally isomorphic to projective systems of submanifolds of
J n (M/T ) compatible with the total derivative operator and morphisms are smooth
maps between these, again compatible with the total derivative operator. This
categorical framework has an arithmetic analogue as follows. Note first that φacts naturally on Rδ (X, L) but δ does not. Nevertheless for any homogeneouss ∈ Rδ (X, L) of degree v the ring Rδ (X, L)hsi of all fractions f /sw with f homogeneous of degree wv has a natural p-derivation δ on it. This inductive system
Xδ (L) = (Rδ (X, L)hsi , δ)s of rings equipped with p-derivations δ is an object ofa natural category underlying a geometry more general than algebraic geometry
which we refer to as δ-geometry [16]. This geometry is an arithmetic analogue ofthe categorical setting in [1] and also an arithmetic analogue of the Ritt-Kolchin δalgebraic geometry [52, 9]. If one restricts to étale maps of smooth schemes we havea functor X 7→ Xδ = Xδ (K ν ) from “algebraic geometry” to our “δ-geometry” bytaking L = K ν to be a fixed power of the canonical bundle; the natural choice laterin the theory turns out to be the anticanonical bundle L = K −1 . In δ-geometryXδ should be viewed as an infinite dimensional object.
1.2.12. Differential invariants. Recall the concept of differential invariant whichplays a key role in the “Galois theoretic” work of Lie and Cartan. Assume thata group G acts on M and T such that M → T is G-equivariant. (In this discussion assume d := dim T is arbitrary.) Then G acts on J n (M/T ) and the ringC ∞ (J n (M/T ))G of G-invariant elements in the ring C ∞ (J n (M/T )) is called thering of differential invariants. There are two extreme cases of special interest: thecase when G acts trivially on T and the case when G acts transitively on T . A special case of the first extreme case mentioned above is that in which M = F ×T , with
G a Lie group acting trivially on T and transitively on F ; this leads to the context ofCartan’s moving frame method and of Cartan connections. For dim T = 1, T is the“time” manifold, F is the “physical space”, and sections in Γ(M/T ) correspond toparticle trajectories. A special case of the second extreme case mentioned above isthe situation encountered in the study of “geometric structures” and in the formulation of field theories, in which G is the group Diff(T ) of diffeomorphisms of T , T isviewed as the “physical space” or “physical space-time”, and M is a natural bundle
over T , i.e. a quotient M = Γ\Repn (T ) of the bundle Repn (T ) → T of n-jets offrames (Rd , 0) → T by a Lie subgroup Γ of the group Autn (Rd , 0) of n-jets of diffeomorphisms of (Rd , 0); cf. [1], pp. 150-153 and 183. (E.g. the Riemannian metrics
on T identify with the sections in Γ(M1,O(d) /T ) where O(d) < GL(d) = Aut1 (Rd , 0)is the orthogonal group.) In the case when the action of G on T is trivial the ring ofdifferential invariants above turns out to have an interesting arithmetic analogue,
namely the ring of δ-invariants of a correspondence; cf. the discussion below. The
special case of Cartan connections has also an arithmetic analogue: the arithmetic
logarithmic derivative attached to a δ-flow. The latter has a flavor different fromthat of δ-invariants of correspondences and will be discussed later. There are also

12

ALEXANDRU BUIUM

interesting candidates for arithmetic analogues of the situation when G = Diff(T );cf. our discussion of Lie groupoids below.
We next explain the ring of δ-invariants of a correspondence. Pursuing an analogy with usual geometric invariant theory assume we are given (X, L), a correspondence on X (i.e. a morphism σ = (σ1 , σ2 ) : X̃ → X × X), and a linearization of L(i.e. an isomorphism β : σ1∗ L ' σ2∗ L). Then we may define the ring of δ-invariantsof σ byRδ (X, L)σ = {f ∈ Rδ (X, L); βσ1∗ f = σ2∗ f }.
(This ring, again, has no unity!) The homogeneous elements s of Rδ (X, L)σ can beviewed as arithmetic analogues of total differential forms invariant under appropriate symmetries. The inductive system of rings(Rδ (X, L)σhsi , δ)equipped with p-derivations δ can be viewed as an incarnation of the quotient
space “Xδ /σδ ” in δ-geometry (and, under quite general hypotheses, is indeed thecategorical quotient of Xδ by σδ ); note that, in most interesting examples (like theones in Theorem 2.6 below), the quotient X/σ does not exist in usual algebraic
geometry (or rather the categorical quotient in algebraic geometry reduces to apoint).
It is worth revisiting the sequence of ideas around differential invariants. Classical Galois theory deals with algebraic equations (satisfied by numbers). Lie andCartan extended Galois’ ideas, especially through the concept of differential invariant, to the study of differential equations (satisfied by functions); roughly speaking
they replaced numbers by functions. In the theory presented here functions are
replaced back by numbers. But we did not come back to where things started
because we have added new structure, the operator δ = δp . In particular the δinvariants mentioned above, although arithmetic in nature, and although attached
to algebraic equations (defining X, X̃, σ), are nevertheless not “Galois theoretic”
in any classical sense.
1.2.13. Lie groupoids. Let us go back to the (interrelated) problems of finding arithmetic analogues of T × T and of Diff(T ). The arithmetic analogue of S := T × T is
usually referred to as the hypothetical tensor product “Z⊗F1 Z” over the “field with
one element”; cf. [57] for some history of this. The arithmetic analogue of Diff(T )could be thought of as the Galois group “Gal(Q/F1 )”. According to a suggestionof Borger [6] one should take “Z ⊗F1 Z” to be, by definition, the big Witt ring of Z.
Since we are here in a local arithmetic situation it is convenient, for our purposes,
to take, as an arithmetic analogue of S = T × T , the schemes Σm = Spec Wm (R),where Wm is the p-typical functor of Witt vectors of length m + 1 (we use Borger’s
indexing) and R is a complete local p-ring. On the other hand an infinitesimalanalogue of Diff(T ) is the Lie groupoid of T defined as the projective system ofgroupoids Gn (T ) := J n (S/T )∗ (where the upper ∗ means taking “invertible” elements and S = T × T → T is the second projection.) So an arithmetic analogue ofthe system J n (S/T )∗ (which at the same time would be an infinitesimal analogue
of “Gal(Q/F1 )”) would be the system J n (Σm ) equipped with the natural maps
induced by the comonad map. This system is a rather non-trivial object [29]. It
is worth noting that a good arithmetic analogue of J n (S/T )∗ could also be the“usual” jet spaces (in the sense of [8]) of Wm (R)/R (which in this case can be constructed directly from the module of Käher differentials ΩWm (R)/R ). By the way

DIFFERENTIAL CALCULUS WITH INTEGERS

13ΩWm (R)/R is also the starting point for the construction of the deRham-Wit complex [44]. However Ω involves usual derivations (rather than Fermat quotients) sotaking Ω as a path to an arithmetic analogue of J n (S/T )∗ seems, again, like “goingarithmetic halfway”. On the contrary taking the system J n (Σm ) as the analogueof J n (S/T )∗ seems to achieve, in some sense, the task of “going arithmetic all theway”.
We end by remarking that if we denote π1 and π2 the source and target projections from Gn (T ) into T then there are natural “actions” ρ of the groupoids Gn (T )on all natural bundles Mn,Γ := Γ\Repn (T ) → T fitting into diagrams:
ρ

Mn,Γ ×T,π1 Gn (T ) −→p2 ↓π2Gn (T ) −→

Mn,Γ
↓
T

where p2 is the second projection. The above induce “actions”
ρ

J n (Mn,Γ /T ) ×T,π1 Gn (T ) −→p2 ↓π2Gn (T ) −→J n (Mn,Γ /T )↓T

where p2 is again the second projection. One can consider rings of differential
invariants
C ∞ (J n (Mn,Γ /T ))ρ := {F ∈ C ∞ (J n (Mn,Γ /T )); F ◦ ρ = F ◦ p1 }where p1 : J n (Mn,Γ /T ) ×T,π1 Gn (T ) → J n (Mn,Γ /T ) is the fist projection. There
should be arithmetic analogues of the above “actions” and rings of differential invariants. One can argue that the analogue of Repn (T ) is, again, the system J n (Σm ).
Then for Γ = 1 the analogue of J n (Mn,Γ /T ) might be J n (J n (Σm )); the analogue,for Γ = 1, of the “action” ρ above could then be the map J n (J n (Σm0 ,m00 )) →J n (J n (Σm0 +m00 )) where Σm0 ,m00 = Spec Wm0 (Wm00 (R)). A challenge would thenbe to find arithmetic analogues of non-trivial Γs.
1.2.14. Differential Galois theory of linear equations. This subject is best explainedin a complex (rather than real) situation. Classically (following Picard-Vessiot andKolchin [51]) one starts with a differential field F of meromorphic functions on a
domain D in the complex plane C and one fixes an n × n matrix A ∈ gln (F) in the
Lie algebra of the general linear group GLn (F). The problem then is to develop a“differential Galois theory” for equations of the formδz U = A · U
where U ∈ GLn (G), G a field of meromorphic functions on a subdomain of D, andz a coordinate on D. The start of the theory is as follows. One fixes a solution
U and introduces the differential Galois group GU/F of U/F as the group of all
F−automorphisms of the field F(U ) that commute with d/dz. One can ask foran arithmetic analogue of this theory. There is a well developed difference algebra
analogue of linear differential equations [65]; but the arithmetic differential theory is
still in its infancy [33, 34, 35]. What is being proposed in loc. cit. in the arithmetic
theory is to fix a matrix α ∈ gln (R) and define δ-linear equations as equations of
the formδu = α · u(p)14

ALEXANDRU BUIUM

where u = (uij ) ∈ GLn (R), δu := (δuij ), and u(p) := (upij ). Note that the above
equation is equivalent to φ(u) =  · u(p) where  = 1 + pα which is not a difference
equation for φ in the sense of [65]; indeed difference equations for φ have the form
φ(u) =  · u. To define the δ-Galois group of such an equation start with a δ-subringO ⊂ R and let u ∈ GLn (R) be a solution of our equation. Let O[u] ⊂ R the ring
generated by the entries of u; clearly O[u] is a δ-subring of R. We define the δGalois group Gu/O of u/O as the subgroup of all c ∈ GLn (O) for which there exists
an O-algebra automorphism σ of O[u] such that σ ◦ δ = δ ◦ σ on O[u] and such that
σ(u) = uc. The theory starts from here.
1.2.15. Maurer-Cartan connections. The Maurer-Cartan connection attached to a
Lie group G is a canonical map T (G) → L(G) from the tangent bundle T (G)to the Lie algebra L(G); for G = GLn it is given by the form dg · g −1 and its
algebraic incarnation is Kolchin’s logarithmic derivative map [51] GLn (G) → gln (G),u 7→ δz u · u−1 . In our discussion of linear equations above the arithmetic analogueof the Kolchin logarithmic derivative map is the map GLn (R) → gln (R),u 7→ δu · (u(p) )−1 .
[n → GL[n whoseThis map is naturally attached to the lift of Frobenius φGLn ,0 : GL−1(p)[effect on the ring O(GLn ) = R[x, det(x) ]ˆ is φGLn ,0 (x) = x . The latter lift of
Frobenius behaves well (in a precise sense to be explained later) with respect to the
maximal torus T ⊂ GLn of diagonal matrices and with respect to the Weyl group
W ⊂ GLn of permutation matrices but it behaves “badly” with respect to othersubgroups like the classical groups SLn , SOn , Spn . (This phenomenon does notoccur in the geometry of Lie groups where the Maurer-Cartan form behaves wellwith respect to any Lie subgroup of GLn , in particular with respect to the classicalgroups.) In order to remedy the situation one is lead to generalize the above[n → GL[nconstructions by replacing φGLn ,0 with other lifts of Frobenius φGLn : GLthat are adapted to each of these classical groups. Here is a description of theresulting framework.
First we define an arithmetic analogue of the Lie algebra gln of GLn as theset gln of n × n matrices over R equipped with the non-commutative group law+δ : gln × gln → gln given bya +δ b := a + b + pab,where the addition and multiplication in the right hand side are those of gln , viewed
as an associative algebra. There is a natural “δ-adjoint” action ?δ of GLn on gln
given bya ?δ b := φ(a) · b · φ(a)−1 .
[n ) lifting Frobenius,
Assume now one is given a ring endomorphism φGLn of O(GLi.e. a ring endomorphism whose reduction mod p is the p-power Frobenius on[n → GL[n the inducedO(GLn )ˆ/(p) = k[x, det(x)−1 ]; we still denote by φGLn : GL[n ormorphism of p-formal schemes and we refer to it as a lift of Frobenius on GL[
simply as a δ-flow on GLn . Consider the matrices Φ(x) = (φGLn (xij )) and x(p) =(xpij ) with entries in O(GLn )ˆ; then Φ(x) = x(p) +p∆(x) where ∆(x) is some matrix
with entries in O(GLn )ˆ. Furthermore given a lift of Frobenius φGLn as above oneφ(f )−f p
defines, as usual, a p-derivation δGLn on O(GLn )ˆ by setting δGLn (f ) = GLn p.

DIFFERENTIAL CALCULUS WITH INTEGERS

15

Assume furthermore that we are given a smooth closed subgroup scheme G ⊂GLn . We say that G is φGLn -parallel if φGLn sends the ideal of G into itself; in thisb equivalently on O(G)ˆ.
case we have a lift of Frobenius endomorphism φG on G,
Assume the ideal of G in O(GLn ) is generated by polynomials fi (x). Then recall
that the Lie algebra L(G) of G identifies, as an additive group, to the subgroup of
the usual additive group (gln , +) consisting of all matrices a satisfying“−1 ”fi (1 + a) = 0,(φ)

where 2 = 0. Let fi be the polynomials obtained from fi by applying φ to thecoefficients. Then we define the δ-Lie algebra Lδ (G) as the subgroup of (gln , +δ )consisting of all the matrices a ∈ gln satisfying(φ)

p−1 fi (1 + pa) = 0.
The analogue of Kolchin’s logarithmic derivative (or of the Maurer-Cartan connection) will then be the map, referred to as the arithmetic logarithmic derivative,lδ : GLn → gln , defined by
1φ(a)Φ(a)−1 − 1 = (δa − ∆(a))(a(p) + p∆(a))−1 .
lδa :=p
For G a φGLn -parallel subgroup lδ above induces a map lδ : G → Lδ (G). Now
any α ∈ Lδ (G) defines an equation of the form lδu = α, with unknown u ∈ G;
such an equation will be referred to as a δ-linear equation (with respect to our
δ-flow). Later in the paper we will explain our results about existence of δ-flows on
GLn compatible with the classical groups. These δ-flows will produce, as explained
above, corresponding δ-linear equations.
1.2.16. Symmetric spaces. In the classical Cartan theory of symmetric spaces one
starts with a (real) Lie group G equipped with a Lie group automorphism x 7→ xτ
of order 1 or 2 (which we refer to as an involution). One defines the fixed group of
the involution(1.1)

G+ := {a ∈ G; aτ = a}and homogeneous spaces G/S where (G+ )◦ ⊂ S ⊂ G+ are Lie groups intermediate
between the identity component (G+ )◦ and G+ . Then τ acts on the Lie algebra
L(G) and one denotes by L(G)+ and L(G)− the + and − eigenspaces of this action.
We have a decomposition L(G) = L(G)+ ⊕ L(G)− which can be referred to as a
Cartan decomposition corresponding to τ . One can also consider the closed subset
of G defined by(1.2)

G− := {a ∈ G; aτ = a−1 }.

This is not a subgroup of G. Set a−τ = (aτ )−1 . The map H : G → G− , H(a) :=a−τ a identifies G/G+ with a subset of G− and there is an obvious compatibility
between the tangent map to H and the Cartan decomposition.
All of the above has an arithmetic analogue as follows. We start with a linear
smooth group scheme G over R and an involution τ on G. We define G+ and G− asin 1.1 and 1.2 respectively. One also has an action of τ on the δ-Lie algebra Lδ (G).
One considers the subgroup Lδ (G)+ ⊂ Lδ (G) of all b ∈ Lδ (G) such that bτ = b.
One also considers the closed subscheme Lδ (G)− ⊂ Lδ (G) whose points b satisfybτ +δ b = 0; Lδ (G)− is not a subgroup. For G = GLn one easily proves that themap +δ : Lδ (G)+ × Lδ (G)− → Lδ (G) is a bijection on points; this can be viewed

16

ALEXANDRU BUIUM

as an analogue of the Cartan decomposition. Later we will state results having theabove as background.
The classical theory of symmetric spaces also considers bilinear (positive definite)forms B on L(G) such that G+ acts on L(G) via the adjoint action by orthogonaltransformations with respect to B; it would be interesting to find an arithmetic
analogue of this condition.
1.3. Main task of the theory. At this point we may formulate the main technicalur . First given a specific scheme X (ordtasks of the theory. Let, from now on, R = Z
pgroup scheme G) the task is to compute the rings On (X) (respectively the modules
Xn (G)). More generally given a specific pair (X, L) we want to compute the ringRδ (X, L). Finally given (X, L), a correspondence σ on X, and a linearization of L,
we want to compute the ring Rδ (X, L)σ . The main applications of the theory (cf.
the subsection below on motivations) arise as a result of the presence of interesting/unexpected elements and relations in the rings On (X), Rδ (X, L), Rδ (X, L)σ .
1.4. Motivations of the theory. There are a number of motivations for developing such a theory.
1.4.1. Diophantine geometry. Usual differential equations satisfied by functions canbe used to prove diophantine results over function fields (e.g. the function field analogues of the Mordell conjecture [55] and of the Lang conjecture [8]). In the samevein one can hope to use “arithmetic differential equations” satisfied by numbers
to prove diophantine results over number fields. This idea actually works in certain situations, as we will explain below. Cf. [12, 19]. The general strategy is asfollows. Assume one wants to prove that a set S of points on an algebraic variety
is finite. What one tries to do is find a system of arithmetic differential equations
Fi (α, δp α, ..., δpn α) = 0 satisfied by all α ∈ S; then one tries, using algebraic operations and the “differentiation” δp , to eliminate δp α, ..., δpn α from this system toobtain another system Gj (α) = 0 satisfied by all α ∈ S, where Gj do not involvethe “derivatives” anymore. Finally one proves, using usual algebraic geometry thatthe latter system has only finitely many solutions.
1.4.2. “Impossible” quotient spaces. If X is an algebraic variety and σ : X̃ → X ×X
is a correspondence on X then the categorical quotient X/σ usually reduces to apoint in the category of algebraic varieties. In some sense this is an illustration of
the limitations of classical algebraic geometry and suggests the challenge of creating
more general geometries in which X/σ becomes interesting. (The non-commutative
geometry of A. Connes [38] serves in particular this purpose.) As explained above
we proposed, in our work, to replace the algebraic equations of usual algebraic geometry by “arithmetic differential equations”; the resulting new geometry is calledδp -geometry (or simply δ-geometry). Then it turns out that important class ofquotients X/σ that reduce to a point in usual algebraic geometry become interesting in δ-geometry (due to the existence of interesting δ-invariants). A generalprinciple seems to emerge according to which this class coincides with the class of“analytically uniformizable” correspondences. Cf. [16].
1.4.3. “Impossible” liftings to characteristic zero. A series of phenomena belonging
to algebraic geometry in characteristic p, which do not lift to characteristic 0 in
algebraic geometry, can be lifted nevertheless to characteristic 0 in δ-geometry.
This seems to be a quite general principle with various incarnations throughout theDIFFERENTIAL CALCULUS WITH INTEGERS

17

theory (cf. [15, 16, 25]) and illustrates, again, how a limitation of classical algebraic
geometry can be overcome by passing to δ-geometry.
1.5. Comparison with other theories. It is worth noting that the paradigm of
our work is quite different from the following other paradigms:
1) Dwork’s theory of p-adic differential equations [41] (which is a theory about
δt acting on functions in Qp [[t]] and not about δp acting on numbers; also Dwork’s
theory is a theory of linear differential equations whereas the theory here is about
analogues of non-linear differential equations),
2) Vojta’s jet spaces [66] (which, again, although designed for arithmetic pur1 n
δt ” acting
poses, are nevertheless constructed from Hasse-Schmidt derivations “ n!
on functions and not from operators acting on numbers),3) Ihara’s differentiation of integers [47] (which, although based on Fermat quotients, goes from characteristic zero to characteristic p and hence, unlike our δp ,cannot be iterated),4) the point of view of Kurokawa et. al. [53] (which uses an operator on numbers
n−1
β for α = pn β ∈ Z, p 6 |β),very different from δp namely ∂α∂p := np
5) the theory of Drinfeld modules [39](which is entirely in characteristic p),6) the difference geometry in the work of Cohn, Hrushovski-Chatzidakis [37],and others (in which the jet spaces are n-fold products of the original varieties as
opposed to the jet spaces in our work which are, as we shall see, bundles over theoriginal varieties with fibers affine spaces),7) Raynaud’s deformation to Witt vectors W2 (k) over a field k of characteristic
p [64] (which again leads to operators from characteristic zero to characteristic p
which cannot be iterated; loosely speaking W2 (k) in Raynaud’s approach is replacedin our theory by W2 (W (k))).
8) the work of Soulé, Deitmar, Connes, Berkovich, and many others on the “geometry over the field F1 with one element”. In their work passing from the geometry
over Z to the geometry over F1 amounts to removing part of the structure defining
commutative rings, e.g. removing addition and hence considering multiplicative
monoids instead of rings. On the contrary our theory can be seen as a tentative
approach to F1 (cf. the Introduction to [16]) that passes from Z to F1 by adding
structure to the commutative rings, specifically adding the operator(s) δp . This
point of view was independently proposed (in a much more systematic form) byBorger [6]. Borger’s philosophy is global in the sense that it involves all the primes(instead of just one prime as in our work) and it also proposes to see “positivity” as
the corresponding story at the “infinite” prime; making our theory fit into Borger’s
larger picture is an intriguing challenge.
9) the work of Joyal [49] and Borger [4, 5] on the Witt functor; the Witt functor
is a right adjoint to the forgetful functor from “δ-rings” to rings as opposed to thearithmetic jet functor which is a left adjoint to the same forgetful functor. As it isusually the case the left and right stories turn out to be rather different.
10) the theory of the Greenberg transform, cf. Lang’s thesis and [42] (whichattaches to a scheme X/R varieties Gn (X) over k; one can show [12] that Gn (X) 'J n (X) ⊗R k so the arithmetic jet spaces are certain remarkable liftings to characteristic zero of the Greenberg transforms. The operators δ on On (X) do not survive
after reduction mod p as operators on the Greenberg transforms.)11) the work on the deRham-Witt complex, cf., e.g. [44] (which has as its starting
point the study of Kähler differential of Witt vectors; on the contrary, what our18

ALEXANDRU BUIUM

research suggests, cf. [29], is to push arithmetization one step further by analyzing
instead the arithmetic jet spaces of Witt vectors.)
12) the theory φ-modules (which is a theory about linear equations as opposedto the theory here which is non-linear).
2. Main results
We present in what follows a sample of our results. We always set A = A⊗Z Fp =A/pA, X = X ⊗Z Fp for any ring A and any scheme X respectively. Recall thatb the p-adic completion of A; for X Noetherian we denote by Xb the
we denote by A
ur , k := R/pR. We begindp-adic completion of X. Also, in what follows, R := Zpwith completely general facts:
2.1. Affine fibration structure of p-jet spaces.
Theorem 2.1. [11]1) If X/R is a smooth scheme of relative dimension d then X has an affinednd in the category of p-adic formal schemes.
ci × A
covering Xi such that J n (Xi ) ' X
2) If G/R is a smooth group scheme of relative dimension d, with formal groupb is isomorphic
law F ∈ R[[T1 , T2 ]]d (T1 , T2 d-tuples), then the kernel of J n (G) → Gdnd
to A with composition law obtained from the formal series
(n)

(n)

δF, ..., δ n F ∈ R[[T1 , T2 , ..., T1 , T2 ]]dby setting T1 = T2 = 0.
Note that after setting T1 = T2 = 0 the series δ n F become restricted i.e. elements(n)(n)
of (R[T10 , T20 , ..., T1 , T2 ]ˆ)d so they define morphisms in the category of p-adic
formal schemes. Assertion 1) in the theorem makes p-jet spaces resemble the usual
jet spaces of the Lie-Cartan theory. Note however that, even if G is commutative,b is not, in general the additive group raised to somethe kernel of J n (G) → G
power. Here is the idea of the proof of 1) for n = 1. We may assume X =Spec A, A = R[x]/(f ), and there is an étale map R[T ] ⊂ A with T a d-tupleof indeterminates. Consider the unique ring homomorphism R[T ] → W1 (A[T 0 ])sending T 7→ (T, T 0 ) where W1 is the functor of Witt vectors of length 2 and T 0 is
a d-tuple of indeterminates. Using the fact that R[T ] ⊂ A is formally étale and thefact that the first projection W1 (A[T 0 ]/(pn )) → A[T 0 ]/(pn ) has a nilpotent kernel,
one constructs a compatible sequence of homomorphisms A → W1 (A[T 0 ]/(pn )),T 7→ (T, T 0 ). Hence one gets a homomorphism A → W1 (A[T 0 ]ˆ), a 7→ (a, δa).
Then one defines a homomorphism R[x, x0 ]ˆ/(f, δf ) → A[T 0 ]ˆ by sending x 7→
a := class(x) ∈ A, x0 7→ δa. Conversely one defines a homomorphism A[T 0 ]ˆ →R[x, x0 ]ˆ/(f, δf ) by sending T 0 7→ δT . The two homomorphisms are inverse to eachother which ends the proof of 1) for n = 1.
2.2. δ-functions and δ-characters on curves. The behavior of the rings On (X)for smooth projective curves X depends on the genus of X as follows:
Theorem 2.2. [11, 12, 32] Let X be a smooth projective curve over R of genus g.
1) If g = 0 then On (X) = R for all n ≥ 0.
2) Let g = 1. If X is not a canonical lift then O1 (X) = R (hence X1 (X) = 0)
and X2 (X) is a free module of rank 1; in particular O2 (X) 6= R. If, on the otherc1 ) and X1 (X) is free of rank one.
hand, X is a canonical lift then O1 (X) = O(ADIFFERENTIAL CALCULUS WITH INTEGERS

193) If g ≥ 2 then J n (X) is affine for n ≥ 1; in particular O1 (X) separates thepoints of X(R).
The proof of 1) is a direct computation. The idea of proof of the statements aboutb Then one first proves (using
Xn in 2) is as follows. Let N n = Ker(J n (X) → X).
n b
Theorem 2.1) that Hom(N , Ga ) has rank at least n over R and one computes theranks of Xn (X) by looking at the exact squenceb a ) → Hom(N 2 , G
b a ) → H 1 (X, O).
Hom(J 2 (X), G
Here Hom = Homgr . The proof of 3) is based on representing J 1 (X) as P(E)\D
where E is a rank 2 vector bundle on X, and D is an ample divisor.
If g = 1 and X is not a canonical lift then a basis ψ for X2 (X) can be viewed
as an analogue of the “Manin map” of an elliptic fibration [55]. Also note that
assertion 3) in Theorem 2.2 implies the effective version of the Manin-Mumford
conjecture [12]. Indeed Manin and Mumford conjectured that if X is a complex
curve of genus ≥ 2 embedded into its Jacobian A then X ∩ Ators is a finite set. Thiswas proved by Raynaud [64]. Mazur later asked [59] if ](X ∩ Ators ) ≤ C(g) whereC(g) is a constant that depends only on the genus g of X. Using 3) in Theorem
2.2 one can prove:
Theorem 2.3. [12] For a smooth projective complex curve X in its Jacobian A wehave ](X ∩ Ators ) ≤ C(g, p) where C(g, p) is a constant that depends only on thegenus g and (in case X is defined over Q) on the smallest prime p of good reductionof X.
Roughly speaking the idea of proof is as follows. First one can replace thecomplex numbers by R and Ators by its prime to p torsion subgroup Γ < A(R).
Then one embeds X(R) ∩ Γ (via the “jet map”) into the the set of k-points ofJ 1 (X) ∩ pJ 1 (A). But the latter is a finite set because J 1 (X) is affine, pJ 1 (A) is
projective, and both are closed in J 1 (A). Moreover the cardinality of this finite set
can be bounded using Bézout’s theorem.
One can ask for global vector fields on a smooth projective curve X/R that are
infinitesimal symmetries for given δ-functions on X. The only non-trivial care isthat of genus 1 (elliptic curves); indeed for genus 0 there are no non-constant δfunctions (cf. Theorem 2.2) while for genus ≥ 2 there are no non-zero vector fields.
Here is the result:
Theorem 2.4. [11, 16] Let X be an elliptic curve over R with ordinary reduction.
1) If X has Serre-Tate parameter q(X) 6≡ 1 mod p2 then there exists a non-zeroglobal vector field on X which is a variational infinitesimal symmetry for all themodules Xn (X), n ≥ 1.
2) If X is a canonical lift (equivalently has Serre-Tate parameter q(X) = 1) thenthere is no non-zero global vector field on X which is a variational infinitesimal
symmetry of X1 (X).
Finally here is a computation of differentials of δ-characters.
Theorem 2.5. [13, 16] Assume X is an elliptic curve over R which is not a canonical lift and comes from an elliptic curve XZp over Zp . Let ap ∈ Z be the trace of
Frobenius of the reduction mod p of XZp and let ω be a basis for the global 1-forms
on XZp . Then there exists an R-basis ψ of X2 (X) such thatp · dψ = (φ∗2 − ap φ∗ + p)ω.

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ALEXANDRU BUIUM

In particular for the eigenvalue µ = 1 we haveZZψdp = (1 − ap + p) · ωdp.
p·d
A similar result holds in case X is a canonical lift.
2.3. δ-invariants of correspondences. Here is now a (rather roughly stated)
result about δ-invariants of correspondences on curves; for precise statements werefer to [16].
Theorem 2.6. [16] The ring Rδ (X, K −1 )σ is “δ-birationally equivalent” to thering Rδ (P1 , O(1)) if the correspondence σ on X “comes from” one of the following
cases:
1) (spherical case) The standard action of SL2 (Zp ) on P1 .
2) (flat case) A dynamical system P1 → P1 which is post-critically finite with(orbifold) Euler characteristic zero.
3) (hyperbolic case) The action of a Hecke correspondence on a modular (orShimura) curve.
Here by saying that σ “comes from” a group action on X (respectively froman endomorphism of X) we mean that (“up to some specific finite subschemes”
for which we refer to [16]) X̃ is the disjoint union of the graphs of finitely many
automorphisms generating the action (respectively X̃ is the graph of the endomorphism). Also δ-birational equivalence means isomorphism (compatible with theactions of δ) between the p-adic completions of the rings of homogeneous fractions of degree zero with denominators not divisible by p. The proofs behind thespherical case involve direct computations. The proofs behind the flat case use thearithmetic Manin map, i.e. the space X2 (X) in Theorem 2.2, plus an induction inwhich canonical prolongations of vector fields play a crucial role. The proofs behindthe hyperbolic case of Theorem 2.6 are based on a theory of δ-modular forms which
we quickly survey next.
2.4. δ-modular forms. Cf. [15, 2, 16, 26, 17, 25]. Let X1 (N ) be the complete
modular curve over R of level N > 4 and let L1 (N ) → X1 (N ) be the line bundle
with the property that the sections of its various powers are the classical modular
forms on Γ1 (N ) of various weights. Let X be X1 (N ) minus the cusps and thesupersingular locus (zero locus of the Eisenstein form Ep−1 of weight p − 1). Let
L → X be the restriction of the above line bundle and let V be L with the zerosection removed. So L2 ' K. The elements of M n = On (V ) are called δ-modular
functions. Set M ∞ = ∪M n . The elements of On (X) are called δ-modular forms
of weight 0. For w ∈ W , w 6= 0, the elements of H 0 (X, Lw ) ⊂ M ∞ are calledδ-modular forms of weight w. We let σ = (σ1 , σ2 ) : X̃ → X × X be the union of all
the (prime to p) Hecke correspondences. Any δ-modular function f ∈ M n has a “δFourier expansion” in R((q))[q 0 , ..., q (n) ]ˆ; setting q 0 = q 00 = ... = 0 in this δ-Fourier
expansion one gets a series in R((q))ˆ called the Fourier expansion of f . Finally let
a4 and a6 be variables; then consideration of the elliptic curve y 2 = x3 + a4 x + a6yields an R-algebra map
R[a4 , a6 , ∆−1 ] → M 0

DIFFERENTIAL CALCULUS WITH INTEGERS

21

where ∆ = ∆(a4 , a6 ) is the discriminant polynomial. By universality we haveinduced homomorphisms
(n)

(n)

R[a4 , a6 , ..., a4 , a6 , ∆−1 ]ˆ → M n
that are compatible with δ.
P
Example 2.7. [17, 19] For f =an q n a classical newform over Z of weight 2, we
get a δ-modular form of weight 0 and order 2
2

J (Φ)
ψ
ca
f ] : J 2 (X) ⊂ J 2 (X1 (N )) → J 2 (Ef ) → G

where Φ : X1 (N ) → Ef is the Eichler-Shimura map to the corresponding elliptic
curve Ef (assumed for simplicity to be non-CM), and ψ the “unique” δ-characterof order 2. Then the δ-Fourier expansion of f ] is congruent mod p toX
 q 0 X  q 0 pX an
2
q n − ap
am q mp p +am q mp;nq
qp(n,p)=1

hence the Fourier expansion of f ] is congruent mod p toan n(n,p)=1 n q .

P

Example 2.8. [15] There exists a unique δ-modular form f 1 ∈ H 0 (X, L−φ−1 ) with
δ-Fourier expansion
 0 n
n−1
X
q
q0
n−1 p
(−1)
=
+ p (...) .
n
qp
qp
n≥1

Hence the Fourier expansion of f 1 is 0. By the way the above δ-Fourier expansion
has the same shape as a certain function playing a role in “explicit” local class field
theory. Here is the rough idea for the construction of f 1 . One considers the universal
Sbi → J 1 (Ui ) ofelliptic curve E = Ui → Spec M ∞ , one considers sections si : U
the projection, one considers the differencesb a,bi ∩ Ubj → N 1 ' G
si − sj : Ub O) = H 1 (E, O). Thenand one considers the cohomology class [si − sj ] ∈ H 1 (E,1∞f ∈ M is defined as the cup product of this class with the canonical generatorof the 1-forms; in fact f 1 is the image of some element
f 1 (a4 , a6 , a04 , a06 ) ∈ R[a4 , a6 , a04 , a06 , ∆−1 ]ˆ.
By the way it is a result of Hurlburt [45] thatf 1 (a4 , a6 , a04 , a06 ) ≡ Ep−1

2ap4 a06 − 3ap6 a04+ f0 (a4 , a6 )
∆p

mod p

where f0 ∈ R[a4 , a6 , ∆−1 ] and we recall that Ep−1 is the Eisenstein form of weightp − 1. (The polynomial f0 is related to the Kronecker modular polynomial mod p2 .)This plus the δ-Fourier expansion of f 1 should be viewed as an arithmetic analogueof the fact (due to Ramanujan) thatdq
2a4 da6 − 3a6 da4
7→∆q
via the map between the Kähler differentials induced by the Fourier expansion map
C[a4 , a6 , ∆−1 ] → C((q)).

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ALEXANDRU BUIUM

Example 2.9. [2] There exists unique δ-modular forms f ∂ ∈ H 0 (X, Lφ−1 ) andf∂ ∈ H 0 (X, L1−φ ) with δ-Fourier expansions 1 (and hence Fourier expansions 1).
They satisfy f ∂ · f∂ = 1. The form f ∂ is constructed by applying the “top part” ofthe canonical prolongation of the “Serre operator” to the form f 1 . More preciselyf ∂ is a constant times the image of∂∂(72φ(a6 ) 0 − 16φ(a4 )2 0 − p · φ(P ))(f 1 (a4 , a6 , a04 , a06 ))
∂a4
∂a6
−1
where P ∈ R[a4 , a6 , ∆−1 , Ep−1
]ˆ is the Ramanujan form.

Theorem 2.10. [2, 15, 16] The ring Rδ (X, K −1 )σ is “δ-generated” by f 1 and f ∂ .
Note that f 1 and f ∂ do not actually belong to Rδ (X, K −1 ) so the above theorem
needs some further explanation which we skip here; essentially, what happens is thatf 1 and f ∂ belong to a ring slightly bigger than Rδ (X, K −1 ) and they “δ-generate”
that ring. We also note the following structure theorem for the kernel and imageof the δ-Fourier expansion map, in which the forms f 1 and f ∂ play a key role:
Theorem 2.11. [25]1) The kernel of the Fourier expansion map M ∞ → R((q))ˆ is the p-adic closureof the smallest δ-stable ideal containing f 1 and f ∂ − 1.
2) The p-adic closure of the image of the Fourier expansion map M ∞ → R((q))ˆ
equals Katz’ ring W of generalized p-adic modular forms.
The proof of the Theorem above is rather indirect and heavily Galois-theoretic.
Statement 2) in Theorem 2.11 says that all Katz’ divided congruences between
classical modular forms can be obtained by taking combinations of “higher pderivatives” of classical modular forms. Statement 1) above is a lift to characteristic zero of the Serre and Swinnerton-Dyer theorem about the kernel of theclassical Fourier expansion map for classical modular forms mod p. Theorem 2.10
can also be viewed as a lift to characteristic zero of results of Ihara [46] about the
Hasse invariant in characteristic p. These mod p results do not lift to characteristic
zero in usual algebraic geometry but do lift, as we see, to characteristic zero inδ-geometry.
We mention the following remarkable infinitesimal symmetry property; recall
the classical Serre operator ∂ : O(V ) → O(V ). Also consider the Euler derivation operator D : O(V ) → O(V ) given by multiplication by the degree on each
graded component of O(V ). Finally let P be the Ramanujan form (in the degree 2
component of O(V )) and let θ : O(V ) → O(V ) be the derivation θ = ∂ + P D.
Theorem 2.12. [16] The operator θ is an infinitesimal symmetry of the R-module
generated by f 1 , f ∂ , and f∂ in M 1 = O1 (V ). Also θ is a variational infinitesimal
symmetry of the R-module generated by f ∂ and f∂ in M 1 = O1 (V ).
Here is a calculation of differentials of the forms f 1 , f ∂ , f∂ .
Theorem 2.13. [16] Let ω, α be the basis of ΩO(V )/R dual to θ, D. Thend(f ∂ )=f ∂ · (φ∗ α − α)d(f∂ )=−f∂ · (φ∗ α − α)d(f 1 )=

−f 1 · (φ∗ α + α) − f∂ · ω + f ∂ · p−1 φ∗ ω.

DIFFERENTIAL CALCULUS WITH INTEGERS

23

In particular, for the eigenvalue µ = 1 we haveZZd(f ∂ )d(f∂ ){ ∂ }dp = {}dp = 0.
f
f∂
By the way, the forms f ] and f 1 introduced above can be used to prove someinteresting purely diophantine results. For instance we have the following:
Theorem 2.14. [19] Assume that Φ : X = X1 (N ) → A is a modular parametrization of an elliptic curve. Let p be a sufficiently large “good” prime and let Q ∈ X(R)be an ordinary point. Let S be the set of all rational primes that are inert in theimaginary quadratic field attached to Q. Let C be the S-isogeny class of Q in X(R)(consisting of points corresponding to isogenies of degrees only divisible by primesin S). Then there exists a constant c such that for any subgroup Γ ≤ A(R) with
r := rank(Γ) < ∞ the set Φ(C) ∩ Γ is finite of cardinality at most cpr .
Other results of the same type (e.g. involving Heegner points) were proved in[19]. In particular an analogue of the above Theorem is true with C replaced by
the locus CL of canonical lifts. To have a rough idea about the arguments involved
assume we want to prove that Φ(CL) ∩ Γ is finite (and to bound the cardinalityof this set) in case Γ is the torsion group of A(R). One considers the order 2 δmodular form f ] : X1 (N )(R) → R and one constructs, using f 1 , a δ-function oforder 1, f [ : X(R) → R, on an open set X ⊂ X1 (N ) which vanishes exactly on
CL ∩ X(R). Then any point P in the intersection X(R) ∩ Φ(CL) ∩ Γ satisfies thesystem of “differential equations of order ≤ 2 in 1 unknown”(
f ] (P ) = 0
f [ (P ) = 0
One can show that this system is “sufficiently non-degenerate” to allow the elimination of the “derivatives” of the unknown; one is left with a differential equation
f 0 (P ) = 0 “of order 0” which has, then, only finitely many solutions (by Krasner’s
theorem). By that theorem one can also bound the number of solutions.
We end the discussion here by noting that the main players in the theory above
enjoy a certain remarkable property which we call δ-overconvergence. Morally thisis a an overconvergence property (in the classical sense of Dwork, Monsky, Washnitzer) “in the direction of the variables x0 , x00 , ..., x(n) ” (but not necessarily in the
direction of x). We prove:
Theorem 2.15. [27] The δ-functions f ] , f 1 , f ∂ are δ-overconvergent.
2.5. δ-Hecke operators. Next we discuss the Hecke action on δ-modular forms.
For (n, p) = 1 the Hecke operators Tm (n) act naturally on δ-series (i.e. series inR((q))[q 0 , ..., q (r) ]ˆ) by the usual formula that inserts roots of unity of order prime
to p which are all in R. However no naive definition of Tm (p) seems to work.
Instead we consider the situation mod p and make the following definition. Let
x be a p-tuple x1 , ..., xp of indeterminates and let s be the p-tuple s1 , ..., sp offundamental symmetric polynomials in x. An element f ∈ k[[q]][q 0 , ..., q (r) ] is calledδ-p-symmetric mod p if the sum
(r)

0
(r)
f (x1 , ..., x1 ) + ... + f (xp , ..., x(r)]p ) ∈ k[[x]][x , ..., xis the image of an element
(r)

0
(r)
f(p) = f(p) (s1 , ..., sp , ..., s1 , ..., s(r)].
p ) ∈ k[[s]][s , ..., s24

ALEXANDRU BUIUM

For f that is δ-p-symmetric mod p define“pTm (p)”f = f(p) (0, ..., 0, q, ..., 0, ..., 0, q (r) ) + pm f (q p , ..., δ r (q p )) ∈ k[[q]][q 0 , ..., q (r) ].
Eigenvectors of “pTm (p)” will be automatically understood to be δ-p-symmetric.
Also let us say that a series in k[[q]][q 0 , ..., q (r) ] is primitive if the series in k[[q]]obtained by setting q 0 = ... = q (r) = 0 is killed by the classical U -operator. Then
one can give a complete description (in terms of classical Hecke eigenforms modp) of δ-“eigenseries” mod p of order 1 which are δ-Fourier expansions of δ-modular
forms of arbitrary order and weight:
Theorem 2.16. [26] There is a 1 − 1 correspondence between:1) Series in k[[q]] which are eigenvectors of all Tm+2 (n), Tm+2 (p), (n, p) = 1, andwhich are Fourier expansions of classical modular forms over k of weight ≡ m + 2
mod p − 1.
2) Primitive series in k[[q]][q 0 ] which are eigenvectors of all nTm (n), “pTm (p)”,(n, p) = 1, and which are δ-Fourier expansions of δ-modular forms of some order≥ 0 with weight w, deg(w) = m.
Note that the δ-Fourier expansion of the form f ] discussed in Example 2.7 is an
example of series in 2) of Theorem 2.16. (Note that f ] has order 2 although its
δ-Fourier expansion reduced mod p has order 1!) More generally the series in 1)
and 2) of Theorem 2.16 are related in an explicit way, similar to the way f and f ] of
Example 2.7 are related. The proof of Theorem 2.16 involves a careful study of theaction of δ-Hecke operators on δ-series plus the use of the canonical prolongationsof the Serre operator acting on δ-modular forms.
2.6. δ-functions on finite flat schemes. The p-jet spaces of finite flat schemes
over R seem to play a key role in many aspects of the theory. These p-jet spaces areneither finite nor flat in general and overall they seem quite pathological. There
are two remarkable classes of examples, however, where some order seems to berestored in the limit; these classes are finite flat p-group schemes that fit into pdivisible groups and finite length p-typical Witt rings. Recall that for any ring A
we write A = A/pA. Then for connected p-divisible groups we have:
Theorem 2.17. [28] Let F be a formal group law over R in one variable x, assume
F has finite height, and let F[pν ] be the kernel of the multiplication by pν viewed asa finite flat group scheme over R. Thenlim
On (F[pν ]) '→nk[x, x0 , x00 , ...]
(xpν , (x0 )pν , (x00 )pν , ...)

sending x, δx, δ 2 x, ... into the classes of x, x0 , x00 , ....
A similar result is obtained in [28] for the divisible group E[pν ] of an ordinary
elliptic curve; some of the components of J n (E[pν ]) will be empty and exactly which
ones are so is dictated by the valuation of q −1 where q is the Serre-Tate parameter.
The components that are non-empty (in particular the identity component) behavein the same way as the formal groups examined in Theorem 2.17 above.
In the same spirit one can compute p-jet spaces of Witt rings. Let us consider
the ring Wm (R) of p-typical Witt vectors of length m + 1, m ≥ 1, and denote byΣm = Spec Wm (R) its spectrum. Set vi = (0, ..., 0, 1, 0, ..., 0) ∈ Wm (R), (1 precededby i zeroes, i = 1, ..., m), set π = 1 − δv1 ∈ O1 (Σm ), and let Ωm = {1, ..., m}. The

DIFFERENTIAL CALCULUS WITH INTEGERS

25

following is a description of the identity component of the limit of p-jet spaces modp:
Theorem 2.18. [29] For n ≥ 2 the image of π p in On (Σm ) is idempotent and wehave an isomorphism(r)

On (Σm )π 'lim→nk[xi ; i ∈ Ωm ; r ≥ 0](r)

(xi xj , (xi )p ; i, j ∈ Ωm , r ≥ 1)(r)sending each δ r vi into the class of the variable xi .
A similar description is obtained in [29] for the p-jet maps induced by the Wittcomonad maps. We recall that the data consisting of On (Σm ) and the maps induced by the comonad maps should be viewed as an arithmetic analogue of the Liegroupoid of the line.
2.7. δ-Galois groups of δ-linear equations. Recall that for any solution u ∈GLn (R) of a δ-linear equation
δu = α · u(p)(where α ∈ gln (R)) and for any δ-subring O ⊂ R we defined the δ-Galois group
Gu/O ⊂ GLn (O). We want to explain a result proved in [34]. Consider the maximal
torus T ⊂ GLn (R) of diagonal matrices, the Weyl group W ⊂ GLn (R) of permutation matrices, the normalizer N = W T = T W of T in GLn (R), and the subgroupN δ of N consisting of all elements of N whose entries are in the monoid of constantsRδ . We also use below the notation K a for the algebraic closure of the fraction
field K of R; the Zariski closed sets Z of GLn (K a ) are then viewed as varieties
over K a . The next result illustrates some “generic” features of our δ-Galois groups;
assertion 1) of the next theorem shows that the δ-Galois group is generically “nottoo large”. Assertion 2) show that the δ-Galois group are generically “as large as
possible”. As we shall see presently, the meaning of the word generic is different
in each of the 2 situations: in situation 1) generic means outside a Zariski closed
set; in situation 2) generic means outside a set of the first category (in the sense of
Baire category).
Theorem 2.19.
1) There exists a Zariski closed subset Ω ⊂ GLn (K a ) not containing 1 such thatfor any u ∈ GLn (R)\Ω the following holds. Let α = δu · (u(p) )−1 and let O be a
δ-subring of R containing α. Then Gu/O contains a normal subgroup of finite index
which is diagonalizable over K a .
2) There exists a subset Ω of the first category in the metric space
X = {u ∈ GLn (R); u ≡ 1 mod p}
such that for any u ∈ X\Ω the following holds. Let α = δu · (u(p) )−1 . Then
there exists a δ-subring O of R containing Rδ such that α ∈ gln (O) and such that
Gu/O = N δ .
The groups W and N δ should be morally viewed as “incarnations” of the groups“GLn (F1 )” and “GLn (Fa1 )” where “F1 ” and “Fa1 ” are the “field with element” and“its algebraic closure” respectively [6]. This suggests that the δ-Galois theory weare proposing here should be viewed as a Galois theory over “F1 ”.

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ALEXANDRU BUIUM

2.8. δ-flows attached to outer involutions of GLn . The main results in [34][n that are attached to the outerconcern the existence of certain δ-flows on GL
involutions defining the (forms of the) classical groups GLn , SLn , SOn , Spn . Theseδ-flows can be viewed as analogues of the Chern connections attached to hermitian
vector bundles on complex manifolds.
Let G = GLn and let H be a smooth closed subgroup scheme of G. We say thata δ-flow φG is left (respectively right) compatible with H if H is φG -parallel andthe left (respectively right) diagram below is commutative:
b ×G
b → G
b
H
φH × φG ↓
↓ φG
b ×G
b → G
b
H

b×H
b
G
φG × φH ↓
b×H
b
G

→

b
G
↓ φG
b
→ G

b →H
b is induced by φG and the horizontal maps are given by multiwhere φH : H
plication.
Next recall that by an involution on G we understand an automorphism τ :G → G over R, x 7→ xτ , of order 1 or 2. For each such τ one may consider the
subgroup G+ = {a ∈ G; aτ = a}. The identity component (G+ )◦ is referred to as
the subgroup defined by τ . Also set H : G → G, H(x) = x−τ x and B : G × G → G,
B(x, y) = x−τ y. More generally, for any g ∈ G, we may consider the maps Hg :G → G, Hg (x) = gx−τ x, and Bg : G × G → G, Bg (x, y) = gx−τ y.
b A lift of Frobenius φG on G
b is
Let us fix now a lift of Frobenius φG,0 on G.
said to be Hg -parallel (respectively Bg -symmetric) with respect to φG,0 if the left
(respectively right) diagram below is commutative:

(2.1)

b
G
Hg ↓
b
G

φG

−→
φG,0

−→

b
G
↓ Hg
b
G

b
G
φG × φG,0 ↓b×G
b
G

φG,0 ×φG

−→
Bg

−→

b×G
b
G
↓ Bg
b
G

If this is the case for g = 1 and φG,0 (1) = 1 then the group S := (G+ )◦ defined byb Alsoτ is φG -parallel; in particular there is an induced lift of Frobenius φS on S.
note that if we set φG,0 (x) = x(p) , viewing H as a matrix H(x) with entries inR[x, det(x)−1 ]ˆ, we have that δG H = 0, which can be interpreted as saying that H
is a prime integral for our δ-flow φG .
The basic split classical groups GLn , SOn , Spn are defined by involutions on
G = GLn as follows. We start with GLn itself which is defined by xτ = x. We
call τ the canonical involution defining GLn . We also recall that if T ⊂ G is the
maximal torus of diagonal matrices, N is its normalizer in G, and W ⊂ G is thegroup of permutation matrices then N = T W = W T and W is isomorphic to the
Weyl group N/T . Throughout our discussion we let φG,0 (x) be the lift of Frobenius[n defined by φG,0 (x) := x(p) ; one can prove that this φG,0 (x) is the uniqueon GLb that is left and right compatible with N and extends to alift of Frobenius on G
cn (where we view GL
cn ). On the other[n as an open set of gl
lift of Frobenius on gl
hand the groups Sp2r , SO2r , SO2r+1 are defined by the involutions on G = GLn
given by xτ = q −1 (xt )−1 q where q is equal to



 

1 0 0
0
1r
0 1r
,
,  0 0 1r  ,
−1r 0
1r 0
0 1r 0DIFFERENTIAL CALCULUS WITH INTEGERS27

n = 2r, 2r, 2r +1 respectively, xt is the transpose, and 1r is the r ×r identity matrix.
We call this τ the canonical involution defining Sp2r , SO2r , SO2r+1 respectively. All
these groups are smooth over R. If S is any of these groups and we set TS = T ∩ S
and NS = N ∩ S then these groups are smooth, TS is a maximal torus in S and
NS is the normalizer of TS in S. Call a root of one of these groups abnormal if it
is a shortest root of a group SOn with n odd.
Here is our result for arbitrary outer involutions:
Theorem 2.20. Let q ∈ GLn be any matrix with q t = ±q and let τ be the involution
on GLn defined by xτ = q −1 (xt )−1 q. Let Hq (x) = qx−τ x = xt qx and Bq (x, y) =dn that isqx−τ y = xt qy. Then there is a unique lift of Frobenius φGLn on GL
Hq -parallel and Bq -symmetric with respect to φGLn ,0 .
The above theorem can be viewed as an analogue of results on the existence anduniqueness result for Chern connections on hermitian vector bundles on complex
manifolds. In our context vector bundles are replaced by principal bundles. The
role of the hermitian metric is played by the matrix q (or, rather, by the involutionτ ). The condition that the left diagram in 2.1 is commutative is the analogue ofthe condition that the Chern connection is compatible with the hermitian metric.
The condition that the right diagram in 2.1 is commutative is an analogue of thecondition that the Chern connection is compatible with the complex structure.
These analogies are not direct and not obvious; for a detailed explanation of theseanalogies see [34]. On the other hand let φGLn be the lift of Frobenius in Theorem
2.20 and consider the matrix φGLn (x) = Φ(x) with entries in R[x det(x)−1 ]ˆ. Thenour proof of Theorem 2.20 provides the following formula for the value of Φ(x) at
the identity matrix 1:

∞ 
X
−1/2
pi ((q (p) )−1 δq)i ,(2.2)
Φ(1) = (1 + p(q (p) )−1 δq))−1/2 := 1 +
i
i=1


−1/2
where
are the binomial coefficients. For n = 1 and q ∈ Z this simplifies
ito the following formula (cf. [15], Introduction): q(p−1)/2(2.3)
Φ(1) = q·,p where pq denotes the Legendre symbol. (Indeed q (p−1)/2 · Φ(1) has square 1 andis ≡ q (p−1)/2 mod p.) This makes 2.2, “up to a polynomial function of the entries
of in q”, a matrix analogue of the Legendre symbol.
For the involutions defining the split classical groups we have the following result:
Theorem 2.21. Let S be any of the groups GLn , SOn , Spn and let τ be the canonical involution on G = GLn defining S. Then the following hold.
1) (Compatibility with involutions) There exists a unique lift of Frobenius φGb that is H-parallel and B-symmetric with respect to φG,0 . In particular ifon Glδ, lδ0 : GLn → gln are the arithmetic logarithmic derivative associated to φG , φG,0
then for all a ∈ S we have that the Cartan decompositionlδ0 (a) = (lδ0 (a))+ +δ (lδ0 (a))−
of lδ0 (a) satisfies (lδ0 (a))+ = lδ(a).

28

ALEXANDRU BUIUM

2) (Compatibility with the normalizer of maximal torus.) φG is right compatible
with N and left and right compatible with NS . In particular for all a ∈ NS andb ∈ GLn (alternatively for all a ∈ GLn and b ∈ N ) we havelδ(ab) = a ?δ lδ(b) +δ lδ(a).
3) (Compatibility with root groups.) If χ is a root of S which is not abnormal
then the corresponding root group Uχ ' Ga is φGLn -parallel.
Note that a similar result can be proved for SLn ; the involution τ lives, in thiscase, on a cover of GLn rather than on GLn itself. Note also that the exception inassertion 3 of the Theorem (occurring in case χ is a shortest root of SOn with nodd) is a curious phenomenon which deserves further understanding.
2.9. Arithmetic Painlevé VI. The classical Painlevé VI equation has an elegant
geometric description due to Manin [56]. In [31] an arithmetic analogue of thisequation has been introduced and studied. We present below the main result in[31]. Recall our framework in 1.2.10 above.
Theorem 2.22. [31] Let E be an elliptic curve over R which does not have alift of Frobenius and let ψ ∈ O(J 2 (E)) be δ-character of order 2 attached to an∗
invertible 1-form ω on E. Consider the symplectic form η = ω ∧ φpω on J 1 (E).
Let Y = E\E[2] and let r ∈ O(Y ). Then the following hold:1) f = ψ − φ(r) defines a generalized canonical δ-flow on J 1 (Y ) which is Hamiltonian with respect to η.
2) There exists a canonical 1-form ν on X such that dν = η; in particular η isexact and if  is the associated Euler-Lagrange form then p is closed.
Note that f in the Theorem does not define a δ-flow on J 1 (E) (equivalently,
J (E) has no lift of Frobenius, equivalently the projection J 1 (J 1 (E)) → J 1 (E)does not have a section):
1

Theorem 2.23. [32]. Let E be an elliptic curve over R which does not have a liftof Frobenius. Then J 1 (E) does not have a lift of Frobenius.
3. Problems
Problem 3.1. Study the arithmetic jet spaces J n (X) of curves X (and more general varieties) with bad reduction.
This could be applied, in particular, to tackle Mazur’s question [59] about bounding the torsion points on curves unifromly in terms of the genus; in other words
replacing C(g, p) by C(g) in Theorem 2.3. Our proof in [12] is based on the studyof the arithmetic jet space of J 1 (X) at a prime p of good reduction. A study of thearithmetic jet space of curves at primes of bad reduction might lead to dropping
the dependence of C(g, p) on p. Evidence that arithmetic jet spaces can be handled
in the case of bad reduction comes in particular from the recent paper [27].
Problem 3.2. Study the δ-modular forms that vanish on arithmetically interesting
Zariski dense subsets of Shimura varieties (such as CM loci or individual non-CM
isogeny classes). Compute δ-invariants of higher dimensional correspondences.
This could be applied to extend results in [19], e.g. Theorem 2.14 above. A
deeper study of differential modular forms may allow one, for instance, to replace
S-isogeny class with the full isogeny class. The arguments might then be extended

DIFFERENTIAL CALCULUS WITH INTEGERS

29

to higher dimensional contexts and to the global field rather than the local fieldsituation. That such a deeper study is possible is shown by papers like [25], forinstance. For the higher dimensional case the theory in [3] might have to be further
developed to match the one dimensional theory in [15, 16]. In a related direction one
might attempt to use the methods in [19] to tackle Pink’s conjectures in [63]. In [19]it was shown that behind finiteness theorems in diophantine geometry one can havereciprocity maps that are somehow inherited from δ-geometry (and that provide
effective bounds); a similar picture might hold for (cases of) Pink’s conjecture. The
first case to look at for such reciprocity maps would be in the case of the intersection
between a multisection X of an abelian (or semiabelian) scheme G → S over a
curve S with the set of torsion points lying in special (CM or otherwise) fibers;
more general situations, in which torsion points are replaced by division points ofa group generated by finitely many sections, can be considered. Results of André,
Ribet, and Bertrand are pertinent to this question.
Problem 3.3. Compare the δ-geometric approach to quotient spaces with the approach via non-commutative geometry.
The quotients X/σ for the correspondences appearing in Theorem 2.6 do notexist, of course, in usual algebraic geometry. As Theorem 2.6 shows these quotients
exist, however, and are interesting in δ-geometry. Remarkably such quotients also
exist and are interesting in non-commutative geometry [60]. More precisely the 3cases (spherical, flat, hyperbolic) of Theorem 2.6 are closely related to the following
3 classes of examples studied in non-commutative geometry:
P1 (R), non-commutative boundary of the classical modular curve;1) (spherical) SL
2 (Z)
1

S
2) (flat) he2πiτi (θ ∈ R\Q): non-commutative elliptic curves;
3) (hyperbolic) Non-commutative space Shnc containing the classical Shimuravariety Sh (2-dimensional analogue of Bost-Connes systems).
It would be interesting to understand why these 3 classes appear in both contexts(δ-geometry and non-commutative geometry); also one would like to see whether
there is a connection, in the case of these 3 classes, between the 2 contexts.
Note that non-commutative geometry can also tackle the dynamics of rational functions that are not necessarily post-critically finite of Euler characteristic
zero. It is conceivable that some post-critically finite polynomials of non-zero Euler characteristic possess δ-invariants for some particular primes (with respect tothe anticanonical bundle or other bundles). A good start would be to investigate
the δ-invariants of σ(x) = x2 − 1. Another good start would be to investigate
δ-invariants of post-critically finite polynomials with Euler characteristic zero thatare congruent modulo special primes to post-critically finite polynomials with Euler
characteristic zero.

Problem 3.4. Study the de Rham cohomology of arithmetic jet spaces. Find arithmetic analogues of Kähler differentials Ω and D-modules. Find an object that is toD what O1 is to Sym(Ω).
The study of de Rham cohomology of arithmetic jet spaces was started in [7]where it is shown that the de Rham cohomology of J n (X) carries information
about the arithmetic of X. The de Rham computations in [7] are probably shadowsof more general phenomena which deserve being understood. Also the de Rhamsetting could be replaced by an overconvergent one; overconvergence is known to30

ALEXANDRU BUIUM

give an improved picture of the de Rham story and, on the other hand, as alreadymentioned, it was proved in [27] that most of the remarkable δ-functions occurring in
the theory possess a remarkable overconvergence property in the “δ-variables” calledδ-overconvergence. Finally one is tempted to try to relate the de Rham cohomology
of arithmetic jet spaces J n (X) to the de Rham-Witt complex of X in characteristic p
and in mixed characteristic. Note further that since the arithmetic jet space J 1 (X)is an analogue of the (physical) tangent bundle T (X) of X it follows that the sheaf
O1 is an arithmetic analogue of the sheaf Sym(ΩX/R ), symmetric algebra on theKähler differentials. But there is no obvious arithmetic analogue of the sheaf ΩX/R
itself. Also there is no obvious arithmetic analogue of the sheaf DX of differential
operators on X and of DX -modules. The absence of immediate analogues of Ω andD is of course related to the intrinsic non-linearity of p-derivations. It would beinteresting to search for such analogues. It is on the other hand conceivable that
there is a sheaf in the arithmetic theory that is to D what O1 is to Sym(Ω). Recall
that the associated graded algebra of D is canonically isomorphic to the algebraof functions on the (physical) cotangent bundle O(T ∗ (X)) (and not on the tangent
bundle); this looks like a discrepancy but actually the arithmetic jet space J 1 (X)
has a sort of intrinsic self-duality (cf. [16]) that is missing in the classical algebrogeometric case where the tangent bundle T (X) and the cotangent bundle T ∗ (X)and not naturally dual (unless, say, a symplectic structure is given).
The δ-overconvergence property mentioned in Problem 3.4 may hold the key to:
Problem 3.5. Define and study the/a maximal space of δ-modular forms on which
Atkin’s U operator can be defined.
Indeed the Hecke operators T (n) with p 6 |n are defined on δ-modular forms andhave a rich theory in this context [15]. In contrast to this T (p) and hence U are still
mysterious in the theory of δ-modular forms. An step in understanding U was takenin [26] where the theory mod p for series of order 1 was given a rather definitive
treatment. However the theory in characteristic zero seems elusive at this point.
There are two paths towards such a theory so far. One path is via δ-symmetry[24, 26]; this is a characteristic 0 analogue of the concept of δ-p-symmetry mod p
discussed above. Another path is via δ-overconvergence [27]. The two paths seemto lead into different directions and this discrepancy needs to be better understood.
Assuming that a good theory of U is achieved, this might lead to a Hida-liketheory of families of differential modular forms, including Galois representations
attached to such forms. It is conceivable that families in this context are not power
series but Witt vectors. Part of the quest for a U theory of δ-modular forms is toseek a δ-analogue of Eisentein series. It is conceivable that the rings On (X1 (N ))contain functions that do not vanish at the cusps and are eigenvectors of the Hecke
operators; such functions could be viewed as “δ-Eisenstein” forms of weight zero.
Problem 3.6. Interpret information contained in the arithmetic jet spaces J n (X)as an arithmetic Kodaira-Spencer “class” of X.
Indeed some of these arithmetic Kodaira-Spencer classes (e.g in the case of elliptic
curves or, more generally, abelian schemes) were studied in [15, 16] and lead tointeresting δ-modular forms. For general schemes (e.g for curves of higher genus)
these classes were explored in Dupuy’s thesis [40]. They are non-abelian cohomologyb d (in theclasses with values in the sheaf of automorphisms of p-adic affine spaces A
case of curves d = 1). These classes arise from comparing the local trivializations

DIFFERENTIAL CALCULUS WITH INTEGERS

31

of arithmetic jet spaces. In this more general case these classes may hold the key toa “deformation theory over the field with one element”. On the other hand Dupuy
proved in [40] that if X is a smooth projective curve of genus ≥ 2 over R thenJ 1 (X) is a torsor for some line bundle over X; this is rather surprising in view ofthe high non-linearity of the theory. One should say that the line bundle in question
is still mysterious and deserves further investigation.
Problem 3.7. Further develop the partial differential theory in [20, 21, 22].
Indeed in spite of the extensive work done in [20, 21, 22] the arithmetic partial
differential theory is still in its infancy. The elliptic case of that theory [20] (which,
we recall, involves operators δp1 and δp2 corresponding to two primes p1 and p2 ) is
directly related to the study of the de Rham cohomology of arithmetic jet spaces[7]; indeed one of the main results in [7] shows that the arithmetic Laplacians in[20] are formal primitives (both p1 -adically and p2 -adically) of global 1-forms on
the arithmetic jet spaces (these forms being not exact, although formally exact, andhence closed). By the way analogues of these results in [7] probably exist in the caseof modular curves; in the one prime case a beginning of such a study was undertakenin [16], where some of the main δ-modular forms of the theory were shown to satisfy
some remarkable systems of Pfaff equations. The hyperbolic/parabolic case of thetheory [21, 22] (which, we recall, involves a p-derivation δp with respect to a prime
p and a usual derivation operator δq ) could be further developed as follows. One
could start by “specializing” the variable q in δq to elements π in arbitrarily ramified
extensions of Zp . This might push the theory in the “arbitrarily ramified direction”
which would be extremely desirable for arithmetic-geometric applications. Indeed
our ordinary arithmetic differential theory is, at present, a non-ramified (or at
most “boundedly ramified”) theory. A further idea along these lines would be
to use the solutions of the arithmetic partial differential equations in [21, 22] tolet points “flow” on varieties defined over number fields. Some of the solutionsin [21, 22] have interesting arithmetic features (some look like hybrids between
quantum exponentials and Artin-Hasse exponentials, for instance) so the “flows”
defined by them might have arithmetic consequences. The challenge is to find (if at
all possible) “special values” of these solutions that are algebraic. One should alsomention that the arithmetic hyperbolic and parabolic equations in [21, 22] have, in
special cases, well defined “indices” that seem to carry arithmetic information; thechallenge would be to make the index machinery work in general situations and to
study the variation of indices in families.
Problem 3.8. Find an arithmetic analogue of Sato hyperfunction solutions of both“ordinary” and “partial” arithmetic differential equations.
Indeed Sato’s hyperfunctions, in their simplest incarnation, are pairs (f (x), g(x))
of functions on the unit disk (corresponding to the distribution f (x) − g(x−1 ))dgdf(x), −x2 dx(x)).) One
modulo (c, c), c a constant. The derivative of a pair is then ( dxcould then try to consider, in the arithmetic case, pairs (P, Q) of points of algebraic
groups with values in δ-rings modulo an appropriate equivalence relation and with
an appropriate analogue of differentiation with respect to p; this framework could
be the correct one for “non-analytic” solutions of the equations in [20, 21, 22].
Problem 3.9. Construct p-adic measures from δ-modular forms.

32

ALEXANDRU BUIUM

Indeed one of the main ideas in Katz’s approach to p-adic interpolation [50] wasto lift some of the remarkable Zp -valued (p-adic) measures of the theory to Wvalued measures where W is the ring of (generalized) p-adic modular forms. One
can hope that some of these W-valued measures of Katz can be further lifted tomeasures with values in the p-adic completion of the ring of δ-modular functions
M ∞ . Indeed recall from Theorem 2.11 that there is a canonical homomorphism
M ∞ → W whose image is p-adically dense, hence the “lifting” problem makes∞ -valued measures could then be evaluated at various ellipticdsense. These lifted M
curves defined over δ-rings to obtain new Zp -valued measures (and hence new padic interpolation results) in the same way in which Katz evaluated his measuresat special elliptic curves. Another related idea would be to interpret the solutionsin Zp [[q]] of the arithmetic partial differential equations in Problem 3.7 above as
measures (via Iwasawa’s representation of measures as power series). There is adiscrepancy in this approach in that the derivation of interest in Iwasawa’s theoryddwhereas the derivation of interest in Problem 3.7 is q dq
; neverthelessis (1 + q) dq
one should pursue this idea and understand the discrepancy.
Problem 3.10. Find arithmetic analogues of classical theorems in the theory ofdifferential algebraic groups and further develop the δ-Galois theory in [35]. Also
further develop the arithmetic analogue of the theory of symmetric spaces in [34],especially on the Riemannian side.
Indeed the theory of groups defined by (usual) differential equations (“differential
algebraic groups”) is by now a classical subject: it goes back to Lie and Cartan
and underwent a new development, from a rather new angle, through the workof Cassidy and Kolchin [36, 52]. It is tempting to seek an arithmetic analogueof this theory: one would like to understand, for instance, the structure of all
subgroups of GLn (R) that are defined by arithmetic differential equations. The
paper [14] proves an arithmetic analogue of Cassidy’s theorem about Zariski dense
subgroups of simple algebraic groups over differential fields; in [14] the case ofZariski dense mod p groups is considered. But Zariski dense groups such as GLn (Zp )are definitely extremely interesting (and lead to interesting Galois theoretic results
such as in [16], Chapter 5). So a generalization of [14] to the case of Zariski dense(rather than Zariski dense mod p) groups, together with a generalization of the
Galois theoretic results in [16], would be very desirable. For instance it would beinteresting to classify all δ-subgroups of the multiplicative group Gm (R) = R×(or more generally of GL2 (R)) and find the invariants of such groups acting on
0
00
\
R{x}
(p) (where R{x} = R[x, x , x , ...]). Also remark that, as in the case of usual
derivations, there are interesting (“unexpected”) homomorphisms in δ-geometry
between GL1 = Gm and GL2 , for instancea aψ∗ (a)Gm (R) → GL2 (R), a 7→0awhere ψ∗ is a δ-character, i.e. ψ ∈ Xn (Gm ). Cf. [43] for interesting developments
into this subject. The main open problem in the δ-Galois theory of GLn [35]seems at this point to decide if the δ-Galois groups always contain a subgroupof the diagonal matrices as a subgroup of finite index. Other problems are: toestablish a Galois correspondence; to understand the relation (already hinted at in[35]) between δ-Galois groups and Galois problems arising from the dynamics on
Pn ; to generalize the theory by replacing GLn with an arbitrary reductive group.

DIFFERENTIAL CALCULUS WITH INTEGERS

33

With regards to the analogue in [34] of the theory of symmetric spaces the main
open problems are to find analogues of the Riemannian metrics and curvature. An
analogue of Lie brackets can, by the way, be developed; it would be interesting touse it to develop an “arithmetic Riemannian theory”.
Problem 3.11. Compose some of the basic δ-functions of the theory in [16] (e.g.
the δ-modular forms on Shimura curves) with p-adic uniformization maps (e.g. withDrinfeld’s uniformization map of Shimura curves).
Indeed this might shed a new light (coming from the analytic world) on δgeometry. These composed maps would belong to a “δ-rigid geometry” which has
yet to be developed in case these maps are interesting enough to require it. By theway it is not at all clear that the functor that attaches to a formal p-adic scheme
its arithmetic jet space can be prolonged to a functor in the rigid category. The
problem (which seems to boil down to some quite non-trivial combinatorially flavored calculation) is to show that the arithmetic jet space functor sends blow-ups
with centers in the closed fiber into (some version of) blow-ups.
In the spirit of the last comments on Problem 3.11 one can ask:
Problem 3.12. Study the morphisms J n (X) → J n (Y ) induced by non-étale finite
flat covers of smooth schemes X → Y .
Indeed note that if X → Y is finite and étale then it is well known that
J n (X) → J n (Y ) are finite and étale; indeed J n (X) ' J n (Y ) ×Y X. But notethat if X → Y is only finite and flat then J n (X) → J n (Y ) is neither finite nor flatin general. One of the simplest examples which need to be investigated (some partial results are available [28], cf. Theorem 2.17) is that of the covers [pν ] : G → G ofsmooth group schemes (or formal groups) given by multiplication by pν . The geometry of the induced endomorphisms of the arithmetic jet spaces is highly complex
and mysterious. Understanding it might be, in particular, another path towards
introducing/understanding the Atkin operator U on δ-modular forms referred toin Problem 3.5. An obviously closely related problem is to understand the p-jet
spaces of p-divisible groups; cf. Theorem 2.17. Yet another example of interest is
the study of J n (X) → J n (X/Γ) for Γ a finite group acting on a smooth X; even thecase X = Y n with Y a curve and Σ = Sn the symmetric group acting naturally is
still completely mysterious. This latter case is related to the concept of δ-symmetry
mentioned in Problem 3.5 and appeared in an essential manner in the paper [24]:the failure of invariants to commute with formation of jet spaces (in the ramifiedcase) is directly responsible for the existence (and indeed abundance) of δ-functions
on smooth projective curves which do not arise from δ-characters of the Jacobian.
Problem 3.13. Understand which analytic functions X(Zp ) → Zp for smooth
schemes X/Zp are induced by δ-functions.
Indeed it was proved in [23] that a function f : Zp → Zp is analytic if, and only if,
there exists m such that f can be represented as f (x) = F (x, δx, . . . , δ m x), whereF ∈ Zp [x0 , x1 , ..., xm ]ˆ is a restricted power series with Zp -coefficients in m + 1
variables. This can be viewed as a “differential interpolation result”: indeed f (x)is given by a finite family of (unrelated) power series Fi (x) convergent on disjointballs Bi that cover Zp and the result says that one can find a single power series
F (x, δx, ..., δ m x) that equals Fi (x) on Bi for each i. One can ask for a generalizationof this by asking which analytic functions f : X(Zp ) → Zp defined on the Zp -points

34

ALEXANDRU BUIUM

of a smooth scheme X/Zp come from a δ-function f˜ ∈ On (X) (i.e. f = f˜∗ ); of course
such a f˜ cannot be, in general, unique. If X = A1 is the affine line then the resultin [23] says that any f comes from some f˜. This is probably still the case if X is any
affine smooth scheme. On the other hand this fails if X = P1 is the projective line
simply because there are no non-constant δ-functions in On (P1 ) [12] but, of course,
there are plenty of non-constant analytic functions P1 (Zp ) → Zp . There should
be a collection of cohomological obstructions to lifting f to some f˜ that shouldreflect the global geometry of X. This seems to us a rather fundamental questionin understanding the relation between p-adic analytic geometry and δ-geometry.
In the light of Theorem 2.17 one can ask:
Problem 3.14. Let α0 , ..., αν be elements in the algebraic closure of the fraction
field of R which are integral over R. Let Xi = Spec R[αi ] be viewed as a closed
subscheme of the line A1 over R. Compute/understand the arithmetic jet spaces
J n (∪νi=0 Xi ).
This is a problem “about the interaction” of algebraic numbers. Indeed the case
ν = 0 is clear; for instance if R[α] is totally ramified over R (and 6= R) then thearithmetic jet spaces are empty: J n (Spec R[α]) = ∅ for n ≥ 1. However, for ν ≥ 1,
an interesting new phenomenon occurs. Indeed if αi = ζpi (primitive pi -th rootof unity) then although J n (Spec R[ζpi ]) = ∅ for n ≥ 1 and i ≥ 1 we have that
J n (∪νi=0 Spec R[ζpi ]) = J n (µpν ) is non-empty and indeed extremely interesting; cf.
Theorem 2.17.
Problem 3.15. Find arithmetic analogues of Hamiltonian systems and of algebraically completely integrable systems. Find arithmetic analogues of the formal
pseudo-differential calculus.
This problem is motivated by the link (due to Fuchs and Manin [56]) between
the Painlevé VI equation (which has a Hamiltonian structure) and the Manin mapof an elliptic fibration. Painlevé VI possesses an arithmetic analogue whose study
was begun in [31]; this study is in its infancy and deserves further attention. Moregenerally there is an intriguing possibility that other physically relevant differential
equations (especially arising in Hamiltonian contexts, especially in the completely
integrable situation, both finite and infinite dimensional) have arithmetic analogues
carrying arithmetic significance. Finally it is conceivable that a meaningful arithmetic analogue of the formal pseudo-differential calculus in one variable [62]. p.
318 can be developed; in other words one should be able to bring into the picture
negative powers of the p-derivation δ : On (X) → On+1 (X) in the same way in which
formal pseudo-differential calculus brings into the picture the negative powers of
the total derivative operator acting on functions on jet spaces.
Problem 3.16. Find an arithmetic analogue of the differential groupoids/Lie pseudogroups in the work of Lie, Cartan, Malgrange [39].
Indeed recall that one can view the p-jet spaces J n (Σm ) of Σm = Spec Wm (R)and the jet maps induced by the comonad maps as an arithmetic analogue of the
Lie groupoid J n (R × R/R)∗ of the line; cf. [29] and Theorem 2.18 above for resultson this. It would be interesting to investigate subobjects of the system J n (Σm )that play the role of analogues of differential sub-groupoids and to find arithmetic
analogues of the differential invariants of diffeomorphism groups acting on natural
bundles arising from frame bundles. A theory of J n (Σm ), suitably extended to otherDIFFERENTIAL CALCULUS WITH INTEGERS

35

rings and to arithmetic analogues of poly-vector fields, could also be interpreted
as an arithmetic analogue of the theory of the deRham-Witt complex [44] because
it would replace the usual Kähler differentials Ω by constructions involving theoperators δp .
Problem 3.17. Compute On (T(N, κ)) where T(N, κ) is the Hecke Z-algebra attached to cusp forms on Γ1 (N ) of level κ.
This could lead to a way of associating differential modular forms f ] to classicalnewforms f of weight 6= 2. (Cf. Theorem 2.16.) This might also lead to a link
between δ-geometry and Galois representations. The problem may be related tothe study of double coset sets of GL2 (or more generally GLn ) and a link of this tothe study in [33, 34, 35] is plausible.
We end by stating two of the most puzzling concrete open problems of the theory.
Problem 3.18. Compute On (X) for an elliptic curve X.
Note that O1 (X) and Xn (X) have been computed (Theorem 2.2) and one expects
X (X) to “generate” On (X). In particular assume X is not a canonical lift and ψ
is a basis of X2 (X); is it true that On (X) = R[ψ, δψ, ..., δ n−1 ψ]ˆ? The analogue of
this in differential algebra is true; cf. [9].
nProblem 3.19. Compute Xn (G) for G an extension of an elliptic curve by Gm .
One expects that this module depends in an arithmetically interesting way on
the class of the extension. The problem is directly related to that of understanding
the cohomology of arithmetic jet spaces.
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Department of Mathematics and Statistics, University of New Mexico, Albuquerque,
NM 87131, USA
E-mail address: buium@math.unm.edu