arXiv:1312.5160v1 [math.NT] 18 Dec 2013

NUMBERS AS FUNCTIONS1
Yuri I. Manin
Max–Planck–Institut für Mathematik, Bonn, Germany
ABSTRACT. In this survey I discuss A. Buium’s theory of “differential equations in the p–adic direction” ([Bu05]) and its interrelations with “geometry overfields with one element”, on the background of various approaches to p–adic models
in theoretical physics (cf. [VlVoZe94], [ACG13]).
Introduction
One of the most beautiful (arguably, the most beautiful) mathematical formulas
is Euler’s identity
eπi = −1.
(0.1)√
It connects four numbers π = 3, 1415912 . . . , e = 2, 71828 . . . , i = −1, and −1
itself, and has a very strong physical flavor being the base of the universal principleof “interference of probability amplitudes” in quantum mechanics and quantum
field theory. The “−1” in the right hand side of (0.1) shows how two quantum
states with opposite phases may annihilate each other after superposition.
On the other hand, of these four numbers π, e, i, −1, only π looks as somethingsimilar to a “physical constant” in the sense that it can be (and was) measured,with a certain approximation.
Moreover, the traditional names of the respective classes of numbers, which we
nowadays tend to perceive as mathematical terms introduced by precise definitions
in courses of calculus, – irrational, transcendent, imaginary, negative, – in the courseof history conveyed the primeval bafflement of the rational mind, discovering thesenumbers but reluctant to accept them.
We may recall that at the time of their discovery these numbers had very different sources of justification: π in Euclidean geometry (which describes essentiallykinematics of solids in gravitational vacuum), −1 in commerce (“debt”), e in theearly history of computer science (Napier’s implementation of the discovery thata specific precomputation can facilitate everyday tasks of multiplication), i in theearly history of polynomial equations.
1 Based on talks at the International Workshop on p–adic methods for modelling of complex

systems, Bielefeld, April 15–19, 2013, and at Journées Arithmétiques, Grenoble, June 2–5, 2013.

1

2

When I was asked to deliver a talk at the Workshop on p–adic Methods for
Modelling of Complex Systems, I decided to present first a p–adic environment of
π and e.
Probably, the earliest “arithmetic” formula involving π is due to Euler (as wellas (0.1)):
π2 Y=(1 − p−2 )−1 .
(0.2)
6
p
However, it involves all primes p simultaneously, and in fact, can be best understoodas a fact from adèlic geometry.
of the simple–
Q As such, it looks as a generalisation∗
minded product formula v |a|v = 1 valid for all a ∈ Q , where v runs overall valuations of Q, p–adic ones and archimedean one. To be more precise, (0.2)
expresses the fact that the natural adelic measure of SL(2, AQ )/SL(2, Q) equals
1. For some more details, cf. [Ma89], where it was suggested that fundamental
quantum physics might be related to number theory via this adèlic philosophy,“democracy of all valuations”, and the exclusive use of real and complex numbers
in our standard formalisms is the matter of tradition, which we now try to overcome
by replacing “the first among equals” archimedean valuation by an arbitrary non–archimedean one.
Now we turn to e. Here, as the discoverer of p–adic numbers Kurt Hensel himself
remarked, we have a candidate for ep in each p–adic field, since the (archimedean)series for ep converges also p–adically:
p

e =

∞
X
pn

n=0

n!

.

(0.3)

Since the root of degree p of the right hand side of (0.3) understood as p–adic
number generates an extension of Qp of degree p, there can be no algebraic number
with such local components.
This argument looks tantalisingly close to a proof of transcendence of e, although,of course, it is not one. On the other hand, I do not know any adèlic formula
involving e in such a way as (0.2) involves π.
In this survey, I proceed with discussion consisting of three main parts.
A. I will describe a class of numbers (including transcendental ones) relevant for
Quantum Field Theory in the sense that they define the coefficients of perturbative

3

series for Feynmann’s path integrals. These numbers are called (numerical) periods,
they were introduced and studied in [KoZa01].
Roughly speaking, numerical periods are values at algebraic points of certain
multi–valued transcendental functions, naturally defined on various moduli spaces,
and also traditionally called (functions–)periods.
These functions–periods satisfy differential equations of Picard–Fuchs type, andsuch equations furnish main tools for studying them.
In the second part of this survey, I focus on the following program:
B. For a prime p, numerical periods also can be considered as solutions of “differential equations in the p–adic direction”.
The whole machinery of such differential equations was suggested and developedby Alexandru Buium, cf. his monograph [Bu05], and I briefly review it. I use thecatchword “numbers as functions” to name this analogy.
Alexandru Buium has convincingly shown that the right analog of the p–adic
derivation is (a natural generalization of) the Fermat quotient δp (a) := (a − ap )/p
initially defined for a ∈ Z. Unexpectedly, this formal idea had rich consequences:
Buium was able to construct analogs of classical jet spaces “in the p–adic direction”,
together with a theory of functions on these jet spaces, containing an incredible
amount of analogs of classical constructions traditionally requiring calculus.
Those numerical periods that were already treated by Buium include periodsof abelian varieties defined over number (or even p–adic) fields. (But the reader
should be aware that, in the absence of uniformization, this last statement only
very crudely describes a pretty complicated picture; see more details in the maintext.)
C. For Buium’s differential equations, “constants in the p–adic direction” turnout to be roots of unity and zero: Teichmüller’s representatives of residue classes
modulo p.
Until recently, algebraic geometry over such constants was motivated by very
different insights: for a more detailed survey cf. [Ma95], [Ma08]. It is known as“theory of the field F1 ”.
Briefly, this last field of inquiry is focused on the following goal: to make theanalogy between, say, Spec Z (or spectra of rings of algebraic integers) on the one
hand, and algebraic curves over finite fields, on the other hand, so elaborated and4

precise that one could use a version of the technique of André Weil, Alexander
Grothendieck and Pierre Deligne in order to approach Riemann’s conjecture for
Riemann’s zeta and similar arithmetic functions.
The solid bridge between F1 –geometry and arithmetic differential equations wasconstructed by James Borger: cf. [Bor11a,b], [Bor09], [BorBu09]. Roughly speaking, in order to define the p–adic derivative δp of elements of a commutative ringA, one needs a lift of the Frobenius map, that is an endomorphism a 7→ F (a),such that F (a) ≡ ap mod p. Borger remarked that a very natural system of such
lifts for all p simultaneously is encoded in the so called psi–structure or its slight
modification, lambda–structure, and then suggested to consider such a structure
as descent data on Spec A to F1 . A related notion of “cyclotomic coordinates” inF1 was independently suggested in [Ma08]. In particular, a ∈ A is a cyclotomic
co–ordinate (wrt a prime p) if F (a) = ap . I will return to these ideas in the lastpart of this survey.
Finally, I should mention that there exists a very well developed deep theory of“p–adic periods” for algebraic varieties defined over p–adic fields that replaced theclassic integration of differential forms over topological cycles with comparison ofalgebraic de Rham and étale cohomology theories: see [Fa88] and a recent contribution and brief survey [Be11]. Periods in this setting belong to a very big Fontaine’s
field BdR . The approach to periods via Buium’s p–adic geometry that we describe
in this survey has a very different flavour. It would certainly be important to findconnections between the two theories.
1. Periods
1.1. Numerical periods. M. Kontsevich and D. Zagier introduced an important subring P ⊂ C containing all algebraic numbers and a lot of numbers
important in physics (see [KoZa01]).
1.1.1. Definition. α ∈ P if and only if the real and imaginary parts of α arevalues of absolutely convergent integrals of functions in Q(x1 , . . . , xn ) over chainsin Rn given by polynomial (in)equalities with coefficients in Q.
1.1.2. Examples. a) All algebraic numbers are periods.
RRdxdy.
b) π =
x2 +y 2 ≤1c) Γ (p/q)q ∈ P.

5

It is not difficult to prove that periods form a subring of C. Feynman integrals(of a certain class) are periods. But it is still not known whether π −1 , e, or Euler’s
constant γ are periods (probably, not). There is a close connection between periods
and Grothendieck motives (see [KoZa01]), and 2πi corresponds to the Tate’s motive.
Since in the motivic formalism one formally inverts the Tate motive, it is also useful
to extend the period ring by (2πi)−1 .
d) The multiple ζ–values (Euler)ζ(n1 , ..., nm) =X

1

0<k1 <...<kmnm ,k1n1 ... kmni ≥ 1, nm > 1 .

(1.1)

are periods.
In order to see it, we reproduce the Leibniz and Kontsevich integral formula forthem.
Let n1 , . . . , nm be positive integers as in (1.1). Put n := n1 + · · · + nm , andε := (ε1 , ..., εn) where εi = 0 or 1, and εi = 1 precisely when i ∈ {1, n1 + 1, n1 +n2 + 1, . . . , n1 + · · · + nm−1 + 1}. Furthermore, putω(ε) :=

dtn
dt1∧ ... ∧
t1 − ε1tn − εnand∆0n := {(t01 , . . . , t0n ) ∈ Rn|

0 < t01 < · · · < t0n < 1}

Then we haveζ(n1 , . . . , nm ) = ζ(ε) = (−1)

m

Z

ω(ε).

∆0n

For further details, see [GoMa04], where the mixed motives associated with these
periods were identified: they are constructed using moduli spaces M 0,n and theircanonical stratifications.
1.2. Periods–functions. Sometimes we may introduce parameters in thedescription of elements of P sketched above and thus pass to the study of periods
as functions. To this end, it is first convenient to rewrite the definition in a moreformal algebraic–geometric framework as was already done in [KoZa01], sec. 4.1.
Consider a quadruple (V, D, ω, γ). Here V is a smooth algebraic variety of pure
dimension n, endowed with divisor D with normal crossings, n–form ω regular6

outside D, and a homology class γ ∈ HRn (V (C), D(C); Q). Moreover, (V, D, ω)
must be defined over Q, and the integral γ ω must converge. Then the set of suchintegrals coincides with the period ring P defined above.
It is now clear how to relativise this definition, replacing V by a relatively smoothmorphism f : V → S defined over Q, endowed with an appropriate S–family ofdata (D, ω, γ) having the necessary properties fiberwise.
Then we get interesting, generally transcendental functions on the base S, andeventually on moduli spaces/stacks, and these functions satisfy (versions of) classical Picard–Fuchs equations.
1.2.1. Example 1. Let S be the affine line with t–coordinate, and points
t = 0, 1 deleted. Over it, we have the family E of elliptic curves Et , that are
projective closures of the affine curve Et : Y 2 = X(X − 1)(X − t).
Here is the linear DE for the periods of the relative (over the base) 1–form dX/Y
along the closed fiberwise 1–cycles of Et :Lt ω := 4t(1 − t)d2 ωdω+ 4(1 − 2t)− ω = 0.
2
dt
dt

(1.2)

Example 2. Non–linear DE for the periods of dX/Y over relative 1–cycles withboundaries at sections P := (X(t), Y (t)) of finite order:µ(P ) = 0,

(1.3)

where


X ′ (t)Y ′ (t)
d
Y (t)
′
2t(t
−
1)
+
2t(t
−
1)X
(t)
−
.
µ(P ) :=
2(X(t) − t)2dt
Y (t)Y (t)2(1.4)

Notice that µ defined by (1.3) and extended to the function on the set of L–
points of the generic fiber Et with values in any differential extension L of Q(t) is“a differential character”:µ(P + Q) = µ(P ) + µ(Q)To explain (and prove) these results, it suffices to notice thatZ Pµ(P ) = Lt
dX/Y
∞

(1.5)

7

becauseLt (dX/Y ) = d

Y.
(X − t)2

1.3. Perturbative Feynman integrals. Here I will briefly describe the heuristic origin of a set of numerical periods (and periods–functions) indexed by labeledgraphs relevant for quantum field theory, following [Ma09], sec. 1. For a more focussed study of (some) of the integrals appearing in this way see [MüWZa12] and[W13].
A Feynman path integral is an heuristic expression of the form
R S(ϕ)
e
D(ϕ)
RP
S
(ϕ)
0
e
D(ϕ)
P

(1.6)

or, more generally, a similar heuristic expression for correlation functions.
Here the integration domain P stands for a functional space of classical fields ϕ
on a space–time manifold M . Space–time may be endowed with a fixed Minkovski
or Euclidean metric. In models of quantum gravity metric is one of the fields.
Fields may be scalar functions, tensors of various ranks, sections of vector bundles,
connections.
S : P → C is a functional of classical action: generally S(ϕ) is expressedas an integral over M of a local
R density on M which is called Lagrangian. In
our notation (1.6) S(ϕ) = − M L(ϕ(x))dx. Lagrangian density may depend onderivatives, include distributions etc.

Usually S(ϕ) is represented as the sum of a quadratic part S0 (ϕ) (Lagrangian of
free fields) and remaining terms which are interpreted as interaction and treated
perturbatively.
R
Finally, the integration measure D(ϕ) and the integral itself P should be considered as simply a part of the total expression (1.6) expressing the idea of “summing
the quantum probability amplitudes over all classical trajectories”.
To explain the appearance and combinatorics of Feynman graphs, we consider
a toy model, in which P is replaced by a finite–dimensional real space. We endow
it with a basis indexed by a finite set of “colors” A, and an Euclidean metric g
encoded by the symmetric tensor (g ab ), a, b ∈ A. We put (g ab ) = (gab )−1 .

8The action functional S(ϕ) will be a formal series in linear coordinates on P,(ϕ ), of the formaS(ϕ) = S0 (ϕ) + S1 (ϕ),

S1 (ϕ) :=

∞
X
1
k=1

k!

S0 (ϕ) := −X

1X
gab ϕa ϕb ,2a,bCa1 ,...,ak ϕa1 . . . ϕak(1.7)

a1 ,...,ak ∈A

where (Ca1 ,...,an ) are certain symmetric tensors. If these tensors vanish for all
sufficiently large ranks n, S(ϕ) becomes a polynomial and can be considered as a
genuine function on P. Below we will treat (gab ) and (Ca1 ,...,an ) as independent
formal variables, “formal coordinates on the space of theories”.
Now we can express the toy version of (1.6) as a series over (isomorphism classesof) graphs.
Here a graph τ consists of two finite sets, edges Eτ and vertices Vτ , and theincidence map sending Eτ to the set of unordered pairs of vertices. Each vertex is
supposed to be incident to at least one edge. There is one empty graph.
The formula for (1.6) including one more formal parameter λ (“Planck’s constant”) looks as follows:
R λ−1 S(ϕ)
X λ−χ(τ )eD(ϕ)
R P −1=w(τ )λ S0 (ϕ) D(ϕ)
|Aut
τ
|
e
P
τ ∈Γ

(1.8)

In the right hand side of (1.8), the summation is taken over (representatives of)
all isomorphism classes of all finite graphs τ . The weight w(τ ) of such a graph isdetermined by the action functional (1.2) as follows:w(τ ) :=XY

u: Fτ →A e∈Eτg u(∂e)Y

Cu(Fτ (v)) .

(1.9)

v∈Vτ

Here Fτ is the set of flags, or “half–edges” of τ . Each edge e consists of a pair offlags denoted ∂e, and each vertex v determines the set of flags incident to it denotedFτ (v). Finally, χ(τ ) is the Euler characteristic of τ .

9The passage of the left hand side of (1.8) to the right hand side is by definition
the result of term–wise integration of the formal series which can be obtained fromthe Taylor series of the exponent in the integrand. Concretely
Z

−1

eλ

S(ϕ)

D(ϕ) =

P

Z

−1

eλ

S0 (ϕ)P−1eλS0 (ϕ)

P

N=1

N!

∞
X

k1 ,...,kN =1

1
k1 ! . . . kN !

N!

N=1

Z
∞
X
λ−N

1+

∞
X
λ−N S1 (ϕ)NYY

dϕa :=

a

dϕa +

a

X

N
Y

aj ∈A,1≤j≤ki i=1(i)!

Ca(i) ,...,a(i)
1

ki

Z

−1

eλ
P

S0 (ϕ)

N
Y
i,j

(i)

ϕaj

Y

dϕa .

a

(1.10)

This definition makes sense if the right hand side of (1.10) is understood as aformal series of infinitely many independent weighted variables Ca1 ,...,ak , weightof Ca1 ,...,ak being k. In fact, the Gaussian integrals in the coefficients uniformly
converge, and one can use the so called Wick’s lemma.
The last remark is that periods appearing in concrete models of quantum field
theories are weights (1.9), in which the summation over maps u : Fτ → A is replacedby the integration over some continuous variables such as positions/momenta/coloursof particles moving along the edges of the respective Feymann graph: cf. [W13],[MüWZa12] and references therein.
2. Arithmetic differential equations
2.1. Analogies between p–adic numbers and formal series. Combining the lessons of previous examples we suggest now that in order to see “p–adic
properties” of numerical periods, transcendental numbers important for physics,
one could try to design a theory of “derivations in p–adic direction” and interpretnumerical periods as solutions of differential equations in the p–adic direction.
Below we present basics of such a theory due to A. Buium. We start with thefollowing table of analogies. On the formal series side, we consider rings of the formk[[t]] where k is a field of characteristics zero. On the p–adic side, we consider the
maximal unramified extension R of Zp .

10

p − ADICSP OW ER SERIESPP

ai ti ∈ k[[t]] =: L

Field of constants: ai ∈ k

εi pi ∈ R := Zunp

Monoid: εi ∈ µ∞ ∪ {0}(Teichmüller representatives)δp (∗) := Φ(∗)−∗
p

Derivation: d/dtp(Φ := lif t of F robenius)
Polynomial Diff. Operators (PDO):
D ∈ L[T0 , T1 , . . . , Tn ]p-adic PDO:
Dp ∈ R[T0 , T1 , . . . , Tn ](p–adic completion!)—————————————————————————Action of PDO: f 7→ D(f, f ′ , . . . f (n) ) or Dp (f, δp f, . . . , δpn f )—————————————————————————The Frobenius lift Φ : RP→ R involved
P pini the definition of the p–adic derivativeiδp is given explicitly as Φ( εi p ) := εi p .

2.2. Examples and applications. Here we give a sample of interesting p–adic
differential operators.
2.2.1. Example 1: p–adic logarithmic derivative. It is an analog of the map
Gm (L) → Ga (L) :f 7→ f ′ /f

(2.1)

where a point x ∈ Gm (L) is represented by the value f ∈ L∗ at x of a fixed algebraic
character t of Gm such that Gm = Spec [t, t−1 ]. Similarly, its p–adic version is thedifferential character
Gm (R) → Ga (R) :p2
pa 7→ δp a · a−p − (δp a · a−p )2 + (δp a · a−p )3 − . . .
2
3

(2.2)

11

Example 2: Quadratic reciprocity symbol:
 
p−1
a
=a 2
p

1+

∞
X

k=1

(2k − 2)!
(−1)k−1 2k−1
(δp a)k a−pk
2(k − 1)!k!

!

.

Example 3: a p–adic analog of the differential character µ of the group ofsections of a generic elliptic curve:d2dµ(P ) = (4t(1 − t) 2 + 4(1 − 2t) − 1)
dt
dt

Z P

dX
∞ Y

as a non–linear p–adic DO acting upon coordinates of P .
Such analogs were constructed in [Bu95] also for abelian varieties of arbitrary
dimension and called δp –differential characters ψ(P ). More precisely, let E be an
elliptic curve over R. Then there exist a differential additive map ψ : E(R) → R+of order 2 (as in the geometric case) or 1 (as for Gm ).
A character of order 2 exists if E has a good reduction and is not the canonicallift of its reduction in the sense of Serre–Tate: cf. additional discussion in 4.4 below.
A character of order 1 exists if either E has good ordinary reduction and is the
canonical lift, or E has a bad multiplicative reduction.
Using these multiplicative characters, A. Buium and the author constructed in[BuMa13] “Painlevé VI equations with p–adic time.”
2.3. General formalism of p–derivations. In the commutative algebra,
given a ring A and an A–module N , a derivation of A with values in N is any
additive map ∂ : A → N such that ∂(ab) = b∂a + a∂b. Equivalently, the map
A → A × N : a 7→ (a, ∂a) is a ring homomorphism, where A × N is endowedwith the structure of commutative ring with componentwise addition, inheriting
multiplication from A on A × {0} and having {0} × N as an ideal of square zero.

Similarly, in arithmetic geometry Buium defines a p–derivation of A with values
in an A–algebra B, f : A → B, as a map δp : A → B such that the map
A → B × B : a 7→ (f (a), δp (a)) is a ring homomorphism A → W2 (B) where W2 (B)is the ring of p–typical Witt vectors of length 2. Here Witt vectors of the form
(0, b) form the ideal of square zero only if pB = {0}.

12Making this definition explicit, we get δp (1) = 0, and the following versions ofadditivety and Leibniz’s formula:δp (x + y) = δp (x) + δp (y) + Cp (x, y),(2.3)

δp (xy) = f (x)p · δp (y) + f (y)p · δp (x) + p · δp (x) · δp (y),

(2.4)

where
Cp (X, Y ) :=X p + Y p − (X + Y )p
∈ Z[X, Y ].
p

(2.5)

In particular, this implies that for any p–derivation δp : A → B the respective map
φp : A → B defined by φp (a) := f (a)p + pδp (a) is a ring homomorphism satisfying
φp (x) ≡ f (x)p mod p, that is “a lift of the Frobenius map applied to f ”.
Conversely, having such a lift of Frobenius, we can uniquely reconstruct therespective derivation δp under the condition that B has no p–torsion:δp (a) :=φp (a) − f (a)ppgeneralising the definition given in 2.1 for A = B = R and identical morphism.
Working with p–derivations A → A with respect to the identity map A → A
and keeping p fixed, we may call (A, δ) a δ–ring. Morphisms of δ–rings are algebramorphisms compatible with their p–derivations.
2.4. p–jet spaces. Let A be an R–algebra. A prolongation sequence for A
consists of a family of p–adically complete R–algebras Ai , i ≥ 0, where A0 = Ab is
the p–adic completion of A, and of maps ϕi , δi : Ai → Ai+1 satisfying the following
conditions:a) ϕi are ring homomorphisms, each δi is a p–derivation with respect to ϕi ,compatible with δ on R.
b) δi ◦ ϕi−1 = ϕi ◦ δi−1 for all i ≥ 1.

Prolongation sequences form a category with evident morphisms, ring homomorphisms fi : Ai → B i commuting with ϕi and δi , and in its subcategory with fixed
A0 there exists an initial element, defined up to unique isomorphism (cf. [Bu05],
Chapter 3). It can be called the universal prolongation sequence.

13

In the geometric language, if X = Spec A, the formal spectrum of the i–th ring
A in the universal prolongation sequence is denoted J i (X) and called the i–th p–jet
space of X. Conversely, Ai = O(J i (X)), the ring of global functions.
i

The geometric morphisms (of formal schemes over Z) corresponding to φi aredenoted φi : J i (X) → J 0 (X) =: Xb(formal p–adic completion of X).

This construction is compatible with localisation so that it can be applied to thenon–necessarily affine schemes: cf. [Bu05], Chapter 3.
3. An arithmetically global version of Buium’s calculus
and lambda–rings

3.1. Introduction. p–adic numbers were considered in sec. 2 above as analogs
of formal functions/local germs of functions of one variable.
In this section, we discuss the following question: does there exist a (more) global
version of “arithmetic functions”, elements of a ring A, admitting p–adic derivations
δp with respect to several, eventually all primes p?
An obvious example is Z:
δp (m) =m − mp.
p

Generally, we need “lifts of Frobenii”: such ring endomorphisms Φp : A → A
that Φ(a) ≡ ap mod p. Then we may put
Φp (a) − apδp (a) =.
p
A general framework for a coherent system of such lifts is given by the following
definition:
3.2. Definition. A system of psi–operations on a commutative unitary ring A
is a family of ring endomorphisms ψ k : A → A, k ≥ 1, such that:ψ 1 = idA ,ψ p x ≡ xp mod pAψ k ψ r = ψ kr ,f or all primes p.

14

Another important structure is introduced by the following definition:
3.3. Definition. A system of lambda–operations on a commutative unitary ring
A is a family of additive group endomorphisms λk : A → A, k ≥ 0, such that
λ0 (x) = 1, λ1 = idA ,λn (x + y) =X

λi (x)λj (y).

i+j=nThese structures are related in the following way:
3.4. Proposition. (a) If A has no additive torsion, then any system of psi–
operations defines a unique system of lambda–operations satisfying the compatibility
relations:X(−1)k+1 kλk (x) =(−1)j+1 λi (x)ψ j (x).
i+j=k, j≥1(b) Generally, any system of lambda–operations defines a unique system of psi–
operations satisfying the same compatibility relations.
Briefly, such a ring, together with psi’s and lambda’s, is called a lambda–ring.
3.5. Example: a Grothendieck ring. Let R = a commutative unitary ring.

Denote by A = AR the Grothendieck K0 –group of the additive category, consisting of pairs (P, ϕ), where P is a projective R–module of finite type, ϕ : P → P
an endomorphism. Denote by [(P, ϕ)] ∈ A the class of (P, ϕ).

The ring structure on A is induced by the tensor product: [(P, ϕ)][(Q, ψ)] :=[(P ⊗ Q, ϕ ⊗ ψ].
The lambda–operations on A are defined by λk [(P, ϕ)] := [(Λk P, Λn ϕ)].

3.6. Example: the big Witt ring W (R). Again, let R = a commutative
unitary ring.
Define the additive group of W (R) as the multiplicative group 1 + T R[[T ]].

The multiplication ∗ in W (R) is defined on elements (1 − at) as (1 − aT ) ∗ (1 −bT ) := 1 − abT , and then extended to the whole W (R) by distributivity, continuity
in the (T )–adic topology, and functoriality in R.

15

Similarly, lambda–operations in W (R) are defined by λk (1 − aT ) := 0 for k ≥ 2,
and then extended by addition formulas (Def. 3.3) and continuity.
4. Roots of unity as constants:
geometries over “fields of characteristic 1”
4.1. Early history. In the paper [T57], J. Tits noticed that some basic numerical invariants related to the geometry of classical groups over finite fields Fq
have well–defined values for q = 1, and these values admit suggestive combinatorialinterpretations.
For example, if q = pk , p a prime, k ≥ 1, thencard (An (Fq ) \ {0})qn − 1==: [n]q ,card Gm (Fq )
q−1
 
n
j
n
,
card Gr (n, j)(Fq ) = card {P (Fq ) ⊂ P (Fq )} =:j qcard Pn−1 (Fq ) =and the q = 1 values of the right hand sides are cardinalities of the setsPn−1 (F1 ):= a finite set P of cardinality n,
Gr (n, j)(F1) := the set of subsets of P of cardinality j.
Tits suggested a program: make sense of algebraic geometry over “a field of
characteristic one” so that the “projective geometry” above becomes a special caseof the geometry of Chevalley groups and their homogeneous spaces.
The first implementation of Tits’ program was achieved only in 2008 by A. Connes
and C. Consani, cf. [CC11], after the foundational work by C. Soulé [So04]. However, they required F12 as a definition field.
Earlier, in an unpublished manuscript [KaS], M. Kapranov and A. Smirnov introduced fields F1n on their own right.
They defined F1n as the monoid {0} ∪ µn , where µn is the set of roots of unityof order n. Moreover, they defined a a vector space over F1n as a pointed set (V, 0)with an action of µn free on V \ {0}. The group GL(V ), by definition, consists of
permutations of V compatible with action of µn . Kapranov and Smirnov defined
the determinant map det : GL(V ) → µn and proved a beautiful formula for the
power residue symbol.

16

Namely, if q = pk ≡ 1 mod n and µn is embedded in F∗q , Fq becomes a vector
space over F1n , and the power residue symbol


a
Fq



n

q−1

:= a n ∈ µn

is the determinant of the multiplication by a in F1n –geometry.
Cf. also [Sm92], [Sm94].
As we noticed in sec. 2, constants with respect to Buium’s derivation δp in
R := Zun
p are roots of unity (of degree prime to p) completed by 0.
Therefore, in the context of the differential geometry “in the p–adic direction”
an independent project of Algebraic Geometry “over roots of unity”, or “in characteristic 1”, or else “over fields F1 , F1n , F1∞ ” acquires a new motivation. Moreover,it becomes enriched with new insights: whereas at the first stage schemes in characteristic 1 were constructed by glueing “spectra of commutative monoids”, now they
could be conceived as Z–schemes endowed with lambda–structure considered as
descent data: see [Bor11a,b], [Bor09]. Here is a brief survey of Borger’s philosophy,
showing that his schemes form a natural habitat for p–adic differential geometries
as well.
4.2. Borger’s philosophy. The category of affine F1 –schemes Af f1 can be
defined as the opposite category of rings endowed with lambda–structures, (A, ΛA ),and compatible morphisms. The forgetful functor to the usual category of affine
schemes Af f1 → Af f : (A, Λ) 7→ A is interpreted as the functor of base extension∗ 7→ ∗ ⊗F1 Z.
Thus, a lambda–structure on a ring A is a descent data on Spec A to F1 .
In particular, W (Z) must be considered as (a completion of?) Z ⊗F1 Z.

More generally, using general topos theory, Borger globalizes this construction,
constructing a natural algebraic geometry of λ–schemes, which should be thoughtof as a lifted algebraic geometry over F1 .
Just as all of usual algebraic geometry is contained in the big étale topos of Z,
λ–algebraic geometry is contained in a big topos, which should be thought of as
the big étale topos over F1 . There is a map of topoi from the big etale topos over
Z to the one over F1 .

17

Schemes of finite type over F1 (in this sense, as in most other approaches) arevery rigid, combinatorial objects. They are essentially quotients of toric varieties
by toric equivalence relations.
Non–finite–type schemes over F1 are more interesting. The big de Rham–Wittcohomology of X “is” the de Rham cohomology of X “viewed as an F1 –scheme”.
It should contain the full information of the motive of X and is probably a concrete
universal Weil cohomology theory.
The Weil restriction of scalars from Z to F1 exists and is an arithmetically global
version of Buium’s p–jet space.
In conclusion, we briefly mention some remaining challenges.
4.3. Euler factors at infinity and F1 –geometry. In [Ma95], I suggested that
there should exist a category of F1 –motives visible through the q = 1 point countof F1 –schemes. Predictions about such a point count were justified in Soulé’s geometry, cf. [So04]. In particular the zetas of non–negative powers of the “Lefschetz(dual Tate) motive” L must be:Z(L×n , s) =

s+n
.
2π

This provides a conjectural bridge between F1 –geometry and geometry of Spec Z
at the archimedean infinity, that is, Arakelov geometry: a Γ–factor of classical zetas,
e.g.,
Y s+n
ΓC (s) := [(2π)−s Γ(s)]−1 =2πn≥0(regularized product) looks like F1 –zeta of the dualized inf–dim projective spaceover F1 .
However, this phenomenon remains an isolated observation, and the archimedean
prime still remains “first among equals” breaking the democracy of all valuations.
4.4. Other geometries “under Spec Z”. In the traditional algebraic geometry,the special role of Spec Z is related to the fact that it is the final object of thecategory of schemes. Since it is very far from being “a point–like object”, it seemednatural to imagine that Spec F1 , being “really point–like”, will replace it. However,the belief that in an extended algebraic geometry there should necessarily exist a
final object, is unfounded. Already in the simplest category of Deligne–Mumford

18

stacks over a field k, admitting quotients with respect to the trivial action of any
finite group G, there is no final object, because we have non–trivial morphisms
Spec k → Spec k/G.
This led several authors to the contemplation of more general geometries lying“under Spec Z” but not necessarily at the bottom of the unfathomable abyss: cf.
the Toën–Vaquié project [TV05].
For example, in the Borger–Buium’s framework we may consider schemes forwhich Frobenius lifts are given only for some subsets of primes, eventually one primep, such as the Serre–Tate canonical liftings of Abelian varieties in characteristic p:
cf. [Katz81].
More precisely, for the simplest case of elliptic curves, denote by M the p–adiccompletion of the moduli stack of elliptic curves without supersingular locus. One
can define Frobenius lift on this stack: it sends an elliptic curve to its quotient by
its canonical subgroup. The latter is defined as the unique closed sub–groupscheme
whose Cartier dual is the étale lift to Zp of the Cartier dual of the kernel of Frobenius
on the fiber modulo p. This endomorphism also lifts to a natural endomorphism of
the universal elliptic curve. So James Borger suggests to say that M “descends tothe p–typical F1 ”, and the same can be said about the universal elliptic curve overit. The p–adic elliptic curves with Frobenius lift are called canonical liftings.
Notice that if we replace the p–adic direction by the functional one, we would
simply speak about families of elliptic curves with constant absolute invariants. But
p–adic absolute invariants of canonical liftings are by no means “constants” in thenaive sense, discussed in sec. 2, that is they are not Teichmüller representatives: cf.
a recent paper by Finotti, ”Coordinates of the j–invariant of the canonical lifting”,
posted at http://www.math.utk.edu/ finotti/ , and [Er13].
A better understanding of this discrepancy presents an interesting challenge forthe p–adic differential geometry.
Acknowledgements. Collaboration with A. Buium on [BuMa13] helped me muchin conceiving this survey. J. Borger generously explained me some of his constructions and motivations. Igor Volovich stimulated the final writing by inviting me
to give a talk at the International Workshop on p–adic methods for modelling ofcomplex systems, Bielefeld, April 15–19, 2013. I am grateful to them all.

19

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