Lectures on Arithmetic Noncommutative Geometry Matilde Marcolli

145 Pages·2012·0.97 MB·English

Lectures on Arithmetic
Noncommutative Geometry
Matilde Marcolli

v
And indeed there will be time
To wonder \Do I dare?" and, \Do I dare?"
Time to turn back and descend the stair.
...
Do I dare
Disturb the Universe?
...
For I have known them all already, known them all;
Have known the evenings, mornings, afternoons,
I have measured out my life with co(cid:11)ee spoons.
...
I should have been a pair of ragged claws
Scuttling across the (cid:13)oors of silent seas.
...
No! I am not Prince Hamlet, nor was meant to be;
Am an attendant lord, one that will do
To swell a progress, start a scene or two
...
At times, indeed, almost ridiculous{
Almost, at times, the Fool.
...
We have lingered in the chambers of the sea
By sea-girls wreathed with seaweed red and brown
Till human voices wake us, and we drown.
(T.S. Eliot, \The Love Song of J. Alfred Prufrock")

Contents
Preface ix
Chapter 1. Ouverture 1
1. The NCG dictionary 3
2. Noncommutative spaces 4
3. Spectral triples 6
4. Why noncommutative geometry? 12
Chapter 2. Noncommutative modular curves 15
1. Modular curves 15
2. The noncommutative boundary of modular curves 22
3. Limiting modular symbols 27
4. Hecke eigenforms 39
5. Selberg zeta function 41
6. The modular complex and K-theory of C(cid:3)-algebras 42
7. Intermezzo: Chaotic Cosmology 44
Chapter 3. Quantum statistical mechanics and Galois theory 51
1. Quantum Statistical Mechanics 53
2. The Bost{Connes system 56
3. Noncommutative Geometry and Hilbert’s 12th problem 61
4. The GL system 64
2
5. Quadratic (cid:12)elds 70
Chapter 4. Noncommutative geometry at arithmetic in(cid:12)nity 81
1. Schottky uniformization 81
2. Dynamics and noncommutative geometry 88
3. Arithmetic in(cid:12)nity: archimedean primes 93
4. Arakelov geometry and hyperbolic geometry 97
5. Intermezzo: Quantum gravity and black holes 100
6. Dual graph and noncommutative geometry 105
7. Arithmetic varieties and L{factors 109
8. Archimedean cohomology 115
Chapter 5. Vistas 125
Bibliography 131
vii

Preface
Noncommutative geometry nowadays looks as a vast building site.
Ontheonehand, practitioners ofnoncommutative geometry(orge-
ometries) already built up a large and swiftly growing body of exciting
mathematics, challenging traditional boundaries and subdivisions.
On the other hand, noncommutative geometry lacks common foun-
dations: formanyinterestingconstructionsof\noncommutativespaces"
we cannot even say for sure which of them lead to isomorphic spaces,
because they are not objects of an all{embracing category (like that of
locally ringed topological spaces in commutative geometry).
Matilde Marcolli’s lectures re(cid:13)ect this spirit of creative growth and
interdisciplinary research.
She starts Chapter 1 with a sketch of philosophy of noncommuta-
tive geometry a(cid:18) la Alain Connes. Brie(cid:13)y, Connes suggests imagining
C(cid:3){algebras as coordinate rings. He then supplies several bridges to
commutative geometry by his construction of \bad quotients" of com-
mutative spaces via crossed products and his treatment of noncom-
mutative Riemannian geometry. Finally, algebraic tools like K{theory
and cyclic cohomology serve to further enhance geometric intuition.
Marcolli then proceeds to explaining some recent developments
drawing upon her recent work with several collaborators. A common
thread in all of them is the study of various aspects of uniformization:
classical modular group, Schottky groups. The modular group acts
upon the complex half plane, partially compacti(cid:12)ed by cusps: rational
points of the boundary projective line. The action becomes \bad" at
irrational points, and here is where noncommutative geometry enters
the game. A wealth of classical number theory is encoded in the co-
e(cid:14)cients of modular forms, their Mellin transforms, Hecke operators
and modular symbols. Their counterparts living at the noncommuta-
tive boundary have only recently started to unravel themselves, and
Marcolli gives a beautiful overview of what is already understood in
Chapters 2 and 3.
ix
x PREFACE
Schottky uniformization provides a visualization of Arakelov’s ge-
ometry at arithmetic in(cid:12)nity, which serves as the main motivation of
Chapter 4.
Among the most tantalizing developments is the recurrent emer-
gence of patches of common ground for number theory and theoretical
physics.
In fact, one can present in this light the famous theorem of young
Gauss characterising regular polygons that can be constructed using
only ruler and compass. In his Tagebuch entry of March 30 he an-
nounced that a regular 17{gon has this property.
Somewhat modernizing his discovery, one can present it in the fol-
lowing way.
In the complex plane, roots of unity of degree n form vertices of
a regular n{gone. Hence it makes sense to imagine that we study
the ruler and compass constructions as well not in the Euclidean, but
in the complex plane. This has an unexpected consequence: we can
characterize the set of all points constructible in this way as the maxi-
mal Galois 2{extension of Q. It remains to calculate the Galois group
of Q(e2(cid:25)i=17): since it is cyclic of order 16, this root of unity is con-
structible. Moreover, the same is true for all p{gons where p is a prime
of the form 2n +1 but not for other primes.
A remarkable feature of this result is the appearance of a hidden
symmetry group. In fact, the de(cid:12)nitions of a regular n{gon and ruler
and compass constructions are initially formulated in terms of Eu-
clidean plane geometry and suggest that the relevant symmetry group
must be that of rigid rotations SO(2), eventually extended by re(cid:13)ec-
tions and shifts. This conclusion turns out to be totally misleading: in-
stead, one should rely upon Gal(Q=Q). The action of the latter group
upon roots of unity of degree n factors through the maximal abelian
quotient and is given by (cid:16) (cid:16)k; with k running over all k mod n
7!
with (k;n) = 1, whereas the action of the rotation group is given by
(cid:16) (cid:16) (cid:16) with (cid:16) running over all n{th roots. Thus, the Gal(Q=Q){
0 0
7!
symmetry does not conserve angles between vertices which seem to be
basic for the initial problem. Instead, it is compatible with addition
and multiplication of complex numbers, and this property proves to be
crucial.
With some stretch of imagination, one can recognize in the Eu-
clidean avatar of this picture a physics (cid:13)avor (putting it somewhat
pompously, it appeals to the kinematics of 2{dimensional rigid bodies
PREFACE xi
in gravitational vacuum), whereas the Galois avatar de(cid:12)nitely belongs
to number theory.
In the Marcolli lectures, stressing number theory, physics themes
pop up at the end of Chapter 2 (Chaotic Cosmology in general rela-
tivity), the beginning of Chapter 3 (formalism of quantum statistical
mechanics), and (cid:12)nally, sec. 5 of Chapter 4 where some models of
black holes in general relativity turn out to have the same mathe-
matical description as {adic (cid:12)bers of curves in Arakelov geometry.
1
The reemergence of Gauss’ Galois group Galab(Q=Q) in Bost{Connes
symmetry breaking, and of Gauss’ statistics of continued fractions in
the Chaotic Cosmology models, shows that connections with classical
mathematics are as strong as ever.
Hopefully, this lively exposition will attract young researchers and
incite them to engage themselves in exploration of the rich new terri-
tory.
Yuri I. Manin. Bonn, March 17, 2005.
CHAPTER 1
Ouverture
Noncommutative geometry, as developed by Connes starting in the
early ’80s ([16], [18], [21]), extends the tools of ordinary geometry to
treat spaces that are quotients, for which the usual \ring of functions",
de(cid:12)ned as functions invariant with respect to the equivalence relation,
is too small to capture the information on the \inner structure" of
pointsinthequotient space. Typically, forsuch spaces functions onthe
quotients are just constants, while a nontrivial ring of functions, which
remembers the structure of the equivalence relation, can be de(cid:12)ned
using a noncommutative algebra of coordinates, analogous to the non-
commuting variables of quantum mechanics. These \quantum spaces"
are de(cid:12)ned by extending the Gel’fand{Naimark correspondence
X loc.comp. Hausdor(cid:11) space C (X) abelian C(cid:3)-algebra
0
,
by dropping the commutativity hypothesis in the right hand side. The
correspondence then becomes a de(cid:12)nition of what is on the left hand
side: a noncommutative space.
Such quotients are abundant in nature. They arise, for instance,
from foliations. Several recent results also show that noncommuta-
tive spaces arise naturally in number theory and arithmetic geometry.
The (cid:12)rst instance of such connections between noncommutative geom-
etry and number theory emerged in the work of Bost and Connes [9],
which exhibits a very interesting noncommutative space with remark-
able arithmetic properties related to class (cid:12)eld theory. This reveals a
very useful dictionary that relates the phenomena of spontaneous sym-
metrybreaking inquantum statistical mechanics tothe mathematics of
Galois theory. This space can be viewed as the space of 1-dimensional
Q-lattices up to scale, modulo the equivalence relation of commensu-
rability (cf. [32]). This space is closely related to the noncommutative
space used by Connes toobtain aspectral realization ofthe zeros ofthe
Riemann zeta function, [23]. In fact, this is again the space of com-
mensurability classes of 1-dimensional Q-lattices, but with the scale
factor also taken into account.
More recently, other results that point to deep connections between
noncommutative geometry and number theory appeared in the work
1