Stories of the arithmetic infinite - Yves André - Paris Sorbonne, CNRS - UniPD
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Stories of the arithmetic infinite
Yves André - Paris Sorbonne, CNRS
Il punto di vista aritmetico, Venezia 12/11/2021
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoI. Some apories of the potential infinite.
I.1. Aristotle: distinction between the potential infinite and the
actual infinite (Metaphysics IX, Physics III).
Potential infinite (infinite in capacity): when a collection of things
has no point where it ends.
Actual infinite (infinite in effect): when an infinite collection of things
is circumscribed, exists as a closed totality - banished by Aristotle.
Geometry at Euclide’s time was a NOT the “science of abstract
space" (this is a modern viewpoint), but the study of finite
configurations which can be extended ad libitum, but remain finite.
In medieval times, the actual infinite became a “metaphysical
infinite", which had earthly manifestations only under rare and
strict conditions (cf. the story of Tempier’s condemnations).
Infinitum actu non datur.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoIn order to appropriate and domesticate the infinite,
mathematicians gradually
- distinguished between infinite and absolute,
- distinguished between infinitely small and infinitely large,
- invented a notation and a calculus of the infinitely small before
creating a concept of the continuum (and later of the actual
infinite); meanwhile, constructed new geometric spaces that
included the horizont (following the painters of the Renaissance).
More radical than Aristotle, Zeno of Elea, following his master
Parmenides, denied both motion and plurality.
In anachronical notation, Zeno’s most famous paradox of
dichotomy reads
1 1 1 ?
1+ + + + ··· = 2
2 4 8
- it took more than two millenaries to understand the · · · , and to logoslides
conclude the equality = 2.
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoI.2. In the XVIIIth c., Euler considered some infinite sums of
integers, such as:
?
(i ) 1 + 2 + 4 + 8 + ··· = ∞
?
(ii ) 1 + 2 + 3 + 4 + ··· = ∞
?
(iii ) 0! − 1! + 2! − 3! + 4! + · · · = ±∞, (n! = 1 · 2 · · · n),
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoand proposed alternative strange-looking answers:
(i ) 1 + 2 + 4 + 8 + · · · = −1
1
(ii ) 1 + 2 + 3 + 4 + ··· = −
12
∞
e −t
Z
(iii ) 0! − 1! + 2! − 3! + 4! + · · · = dt .
0 1+t
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoCase (i ) is based on the counting method by generating series:
1
1 + 2x + 4x 2 + 8x 3 + · · · =
1 − 2x
(then set x = 1).
Case (iii ) uses generating series again:
ŷ = 0!x − 1!x 2 + 2!x 3 − 3!x 4 + 4!x 5 + · · · ,
which satisfies the differential equation x 2 ŷ 0 + ŷ = x .
ŷ is the asymptotic expansion (Euler writes “evolutio") of the
R ∞ −t /x
solution y = 0 e1+t dt in the complex plane deprived from the
negative real half-line.
(De seriebus divergentibus 1746-1760; contains five other
justifications of (iii )). logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoFor (ii ), with the same method,
x
x + 2x 2 + 3x 3 + 4x 4 + · · · =
(1 − x )2
gives ∞ at x = 1. To get a finite answer, Euler used a finer type of
an n−s instead of an x n .
P P
generating series:
For an = 1, this corresponds to the Riemann zeta function
ζ(s) = n−s (known to Euler for s an integer).
P
Formula (ii ) which deals formally with ζ(−1) is then obtained by
relating ζ(−1) to
1 1 π2
ζ(2) = 1 + + + · · · = .
22 32 6
(Remarques sur un beau rapport entre les séries des puissances
tant directes que réciproques, 1749).
logoslides
This is much deeper than (i ) or (iii ): a full proof, starting with
making good sense of ζ(−1), waited 120 years (Riemann).
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoMorals of the story:
- Euler not only distinguished perfectly between convergent and
divergent series, but he had a working knowledge of asymptotic
expansions, well ahead of his time, and a phenomenal intuition.
- There are different ways of resumming divergent series, relevant
in different context. (Euler also considered
1 + 1 + 1 + ···
for instance, explained “the cause of great dissent among
mathematicians of whom some deny and others affirm that such a
sum can be found", but prefered to consider this sum as ∞ rather
than to interpret it artificially, without context, as ζ(0) = − 21 ).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmetico- Divergence is even more unwanted in Physics as in Mathematics.
A typical case is Quantum Field Theory, which systematically
produces some unwanted infinities. Systematic ways have been
found to resum these divergent series (indexed by Feynman
diagrams): renormalization techniques - of which Euler’s
techniques are toy models. Astoundly, the results of these formal
manipulations turn out to be incredibly close to the experimental
data.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoI.3. Another way to give sense to sums like 1 + 2 + 4 + 8 + · · · or
0! − 1! + 2! − 3! + 4! + · · · is via p-adic analysis (Hensel 1897).
What is it?
p: prime number.
Expansion in basis p : n = a0 + a1 p + · · · ak pk ∈ N, 0 6 ai < p.
Invert p: a−` p−` + · · · + ak pk ∈ N[ p1 ].
“Completions" (in two completely different ways):
`→∞ R (expansion of real numbers in base p),
k →∞ Qp (field of p-adic numbers).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoFor instance
- 1 + 2 + 4 + 8 + · · · does converge 2-adically to −1.
- 0! − 1! + 2! − 3! + 4! + · · · can be considered as a p-adic
number for every prime number p, but nothing is known about
these numbers.
- There is also a p-adic story “related to" 1 + 2 + 3 + 4 + · · · ...
Nowadays, p-adic numbers pervade Number Theory (see later).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoSome apories of the actual infinite.
II. Some apories of the actual infinite.
II.1. Here is a solitary game, invented by the logician Goodstein,
which is played with numbers. It is based on hereditary base-b
notation: one writes n in base b, then writes all exponents in base
b, and so on...
For instance, in hereditary base-2 notation,
2
25 = 22 + 22+1 + 1.
Now, one picks an integer n > 0, writes is in hereditary base-2.
2
One changes all 2’s in 3’s (for instance 25 = 22 + 22+1 + 1
3
becomes 33 + 33+1 + 1 = 7625597485069),
and one substracts 1.
Next step: one writes this number in hereditary base-3, changes all
logoslides
3’s in 4’s, substracts 1, and so on.
The game ends if one eventually gets 0.
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoFor instance, starting from n = 2, one gets
21 31 − 1 = 2 2−1=1 0: the game ends at the third
step.
Starting from n = 3, one gets
3 = 21 + 1 31 + 1 − 1 = 3 41 − 1 = 3 2 1 0: the
game ends at the fifth step.
Starting from n = 4, one gets
4 = 22 33 − 1 = 26 = 2.32 + 2.31 + 2 2.42 + 2.41 + 1 =
2 1
41 2.5 + 2.5 = 60 83 . . ., and the game ends only at the
402653211
far far away step 3.(2 − 1).
For bigger n’s, this worsens up to the point that no known system
of notation for big numbers is able to express the growth of the
sequence. Still, Goodstein proved that the game always ends. The
mouse which nibbles 1 at each step finally wins against the huge
jumps which arise from changing base b to base b + 1!
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoGoodstein’s proof uses the infinite, in fact the smallest countable
ordinal ω = N: he observes that writing the m-th term G(n)m of the
sequence (starting from n) in hereditary base m + 1, and changing
all m + 1 into ω , one obtains a decreasing function in m.
For instance, G(4)1 = 4 = 22 ω ω , G(4)2 = 26 =
2 1
2.3 + 2.3 + 2 2ω + 2ω + 2 < ω ω .
2
This sequence thus reaches 0, and so does the original sequence.
Isn’t there a more standard proof, by induction?
NO (Kirby, Paris): whereas the step at which the game ends is
computable (say, by a Turing machine), the theory of finite
arithmetic (as devised by Peano) is unable to prove that the game
actually ends.
The game is a reflection in the finite world of the axiom of infinity.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoII.2. The presence of the infinite behind the finite is not limited to
such strange examples. The standard technique of generating
series to enumerate a hierarchy of finite configurations Cn uses
studies the series y = ∞ n
P
1 |Cn |x , the functional or differential
equations it satisfies. Even if one does not know how to solve the
equation, asymptotic analysis helps to control the behaviour of y at
∞ and get estimates for |Cn |.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoExample. If Cn is the set of partitions of n
(for n = 4, C4 = {(1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, 4}),
it turns out that
X∞
y = 1/( (−1k )x k (3k −1)/2 ),
−∞
and √
|Cn | ∼ (4 3 · n)−1 exp(π (2n/3))
p
(Hardy, Ramanujan).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoII.3. Back to the actual infinite: a few words about those famous
large cardinals.
Starting from ℵ0 = |N| (the smallest infinite cardinal), one forms
the ladder
ℵ 2ℵ0
ℵ0 , 2ℵ0 , 22 0 , 22 ,...
Two questions:
1) at the bottom of the ladder, are there intermediate cardinals
between the first two rungs (i.e. between the countable and the
continuum); more generally, between consecutive rungs?
2) And are there any cardinals above the top of the ladder?
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoQuestion (1) is undecided (Cantor was in favour of NO). The usual
axomatic of set theory ZFC is too weak to decide (Gödel, Cohen).
This does not settle or discard the issue, but raises instead the
problem of completing ZFC in order to decide it.
Recent advances in set theory (Woodin...) seem to show that a
precise answer to Question (2) is a key to Question (1).
The top of the ladder impacts on the bottom: the existence of large
cardinals (with some properties, e.g. with respect to partitions)
impacts the complexity of subsets of the continuum (Cantor’s
starting point).
Moral of the story: higher infinities shed light on lower ones, and
conversely, via a dialectic between existence and complexity.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoIII. The ascent to the absolute (Lautman)
[A vista toward more contemporary mathematics, and a glimpse of
a new spring in Number Theory]
III.1. let P ∈ Q[x ] be an irreducible polynomial, with roots
α1 , . . . , αn ∈ C. In modern terms, Galois (∼ 1830) attached to P
the group of automorphisms of the field Q(αi )i =1,...,n viewed as a
group of substitutions of α1 , . . . , αn , and establishes a
correspondence between subfields and subgroups.
One may also consider the field of all algebraic numbers: Q̄ ⊂ C.
The group of automorphisms of Q̄ is the absolute Galois group of
Q, denoted GQ . There are “projections" GQ → Aut (Q(αi )) for
every P, and GQ is the inverse limit of this system of projections,
whence a natural topology: GQ is a compact (profinite) group.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoA single object, GQ therefore encodes the properties of all
algebraic numbers at the same time. Furthermore, one can
perform new constructions on this object: one can study GQ
through its “continous linear representations" and their
“deformations", and thereby get a lot of information about GQ
(cf. Wiles’ proof of Fermat).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoStill...
- one does not know how to describe/construct any element
beyond the two obvious elements: identity id and complex
conjugation c,
- one does not know the structure of GQ .
Q ⊂ Q(µ∞ ) := Q(exp(2i π/n))n∈N ) ⊂ Q̄.
Then GQ is an extension of the abelian group Aut (Q(µ∞ )) ∼ = Ẑ∗
by Gal (Q̄/Q(µ∞ )).
Shafarevich’s conjecture: Gal (Q̄/Q(µ∞ )) is a free (profinite)
group.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoIII.2. One may replace the base field Q by other fields:
- GR = {id, c} (since any polynomial with real coefficients has all its
roots in C)
- GQp : unlike GQ , one can construct many elements, and in fact
topological generators, and the structure of this topological group
is “known", but very complicated (Winberg, Jannsen...)
Bijection N ↔ Fp [x ] : a0 + a1 p + · · · + ak pk 7→ a0 + a1 x + · · · ak x k
(does not preserve algebraic structures: on the RHS, addition is
performed without carrying).
Then invert p: a−` p−` + · · · + ak pk ∈ N[ p1 ] ↔ Fp [x , x −1 ]
Completion: k → ∞ Qp ↔ Fp ((x )).
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoAs in Shafaravich’s conjecture, one may consider
Qp ⊂ Qp (µp∞ ) ⊂ Q̄p ,
or else
∞
Qp ⊂ Qp (p1/p ) ⊂ Q̄p ,
and complete p-adically:
∞ ∞
K := Qp (p1/p )ˆ ↔ K [ := Fp ((x 1/p )).
The bijection between these two fields still does not preserve
algebraic structure, but
GK ∼= GK [ ,
(Fontaine, Wintenberger): the theory of algebraic equations over K
is equivalent to the theory of algebraic equations over K [ (which is
simpler: additions are carried out without carrying). logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoThis is the starting point of Scholze’s perfectoid geometry (2010; in
his language, K and its tilt K [ are perfectoid fields), who
established a tilting equivalence between perfectoid spaces over K
and perfectoid spaces over K [ .
This was the beginning of a new spring in Number Theory - not
only toward the future of the theory, but also toward solving old
questions from the 60’s in commutative algebra and algebraic
geometry.
2
Moral of the story: introducing deep ramification (the p1/p , p1/p ,
etc.), instead of superficially complicating the situation, simplifies it
dramatically.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoIII.3. A different (and final) question: could one extend the absolute
Galois theory of algebraic numbers to some transcendental
numbers?
Back to Euler:
x sin x
Πn∈Zr0 (1 − )= ∈ Q[[x ]],
nπ x
which suggests that the non-zero integral multiple of π may be
considered as conjugates of π .
If one insists to have a Galois group which permutes transitively
the conjugates, one is forced to include all non-zero rational
multiple of π as well. Whence a tentative answer: the Galois group
is Q× and permutes the conjugates Q× .π .
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoBeyond zeta values, Euler also introduced multiple zeta values
n1−s1 . . . nk−sk
X
ζ(s) =
n1 >...>nk
(in special cases), and discovered many algebraic relations
between these numbers.
X
Zs = Q.ζ(s).
s1 +...+sk =s
These and other relations (discovered later) imply
dimQ Zs 6 ds
where ds are the Taylor coefficients of (1 − x 2 − x 3 )−1 , given by a
Fibonacci-like recursion (Goncharov, Terasoma). One conjectures
equality (Zagier). logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoThe natural framework to discuss and prove this is the theory of
motives: this was dreamed by Grothendieck in the 60’s,
transformed into a useful fiction by Deligne in the mid 70’s, and
then into a full-fledged theory by Voevodsky and others in the 90’s.
To motives, one knows how to attach motivic Galois groups (a far
reaching extension of Galois theory to systems of polynomials
equations in several variables). For the motives attached to
multiple zeta values, which are called mixed Tate motives over Z,
their absolute Galois group - named the cosmic Galois group by
Cartier because this group also controls renormalization theory of
quantum fields - has been determined by Brown: it is an extension
of Q∗ by a free (prounipotent) group.
It would then follow from a deep conjecture in transcendental
number theory (also due to Grothendieck) that this cosmic Galois
group is the symmetry group of the multiple zeta values.
logoslides
Yves André - Paris Sorbonne, CNRS Storie dell’infinito aritmeticoYou can also read
