Fundamental groups, prinipal bundles, and
Diophantine geometry
Minhyong Kim
14 Marh, 2007
Kings olloquium
1
Diophantine equation:
for
f (x) = 0
f (x1 ; x2 ; : : : ; xn ) 2 Z [x1 ; x2 ; : : : ; xn ℄
2
Can be onsidered in any number of dierent environments suh as
Z ; Z [1=62℄; Q ; Z [i℄; Q [i℄; : : : ; Q [i; ℄; : : : ; R ; C ; Q p ; C p ; : : :
The designation of the equation as Diophantine alls attention to
our primary fous on ontexts loser to the beginning of the list.
3
Notation: X for the equation thought of as a geometri objet in
various ways. X (R) for set of solutions in ring R.
4
Famous results:
(1)
xn + y n
=z
has only the obvious solutions in Z as long as n 3.
(2)
f (x; y ) = 0
for a generi f of degree at least 4 has only nitely many solutions
in Q (i; ; e).
5
n
Diophantine geometry has it origins in the use of elementary
oordinate geometry for desribing solution sets, or at least for
generating solutions.
Quadrati equation in two variables:
x2 + y 2
6
= 1:
Real solution set is a irle. Leads to idea of onsidering the
intersetions with all lines that pass through the spei point
( 1; 0). Equations
y = m(x + 1)
for various m
Substitution leads to the onstraint
x2 + (m(x + 1))2
or
=1
(1 + m2 )x2 + 2m2 x + m2
7
1 = 0:
One solution x = 1 is already rational.
Slope m is rational ) other solution is also rational.
Varying m, we an generate thereby all the other rational
solutions to the equation, e.g.,
( 99=101; 20=101)
orresponding to m = 10.
[$ Pythogorean triple 992 + 202 = 1012 ℄
8
An example of degree 3:
x3 + y 3
= 1729:
(9; 10) is a solution (Ramanujan).
Lines through it?
Unfortunately, the previous argument for the rationality of
intersetion points fails.
Can obtain one other solution, using the tangent line to the real
urve at the point (9; 10).
9
Equation of the tangent line,
81(x 9) + 100(y
10) = 0
or
= ( 81=100)x + 1729=100;
and substitute to obtain the equation
y
x3 + ((
81=100)x + 1729=100)3 = 1729:
We have arranged for x = 9 to be a double root, and hene, the
remaining root is fored to be rational.
10
Even by hand, you an (tediously) work out the resulting rational
point to be
( 42465969=468559; 24580=271):
Can ontinue to obtain innitely many rational solutions.
Key point is a natural group struture on the set of points.
Determined as follows: Fix one point 0 2 X (Q ) as the origin. For
any P , Q, there exists a rational funtion f with zeros exatly at P
and Q and one pole at 0. P + Q is dened to be the other pole.
Can add or simply double old solutions to get new ones.
11
Geometri tehniques of the same general avor an be made
onsiderably more sophistiated.
Compat smooth urve X , dened by equation
F (z0 ; z1 ; z2 ) = 0
in projetive spae.
Can try to generalize the previous disussion in a formal way by
dening
P iX
= Z [X ℄=(geometri equivalene relation R)
P = Q , there exists a rational funtion whose zeros
are exatly the fP g and whose poles are the fQ g.
R:
i
i
i
i
i
i
12
This relation is quite ompliated in general. For degree three
equations, redues to relation between three points on the urve.
Aounted for by the topology of a torus:
X (C )
= C =
where C is a lattie.
For higher degree equations, sum of two points will no longer be on
the urve. No group law:
X (C ): Riemann surfae of higher genus.
Heneforward, assume X is a urve of genus 2.
13
But there is another geometri struture underlying this
onstrution.
P i = J Z
where
J = Z [X ℄0 =R
X
X
X
Z [X ℄0
and
= fn [P ℄ j n = 0g
P
P
= H 0 (X (C ); (C ) ) =H1 (X (C ); Z )
Many other desriptions and onstrutions.
JX (C )
X
14
Weil gave a purely algebrai onstrution of J as a projetive
variety:
J Sym (X )
In partiular,
X dened over Q ) J dened over Q .
Finally, if b 2 X (C ), then get a map
X
g
X
X
ib : X,!JX
dened over Q that sends any other point x to [x℄ [b℄. Albanese
map.
15
In partiular,
X (Q ),! JX (Q )
and one might attempt to study the struture of X (Q ) using
J (Q ). Weil's main motivation for algebrai onstrution.
In fat, J (Q ) is a nitely-generated abelian group. Frequently
innite, again beause of group struture. But points of J are
usually not points of X . Cannot be used to generate points on X .
X
X
X
16
Mordell's onjeture: X has at most nitely many rational points.
Proved in 80's by Faltings.
From our perspetive, an arithmeti manifestation of
inompatibility of group law on J with ompliated topology of
X . Weil had attempted in his thesis to implement this idea diretly
to prove Mordell's onjeture (without suess).
X
17
Remark: Problem is the intrinsially abelian nature of the ategory
of motives reeting the properties of homology. So, even in the
best of possible worlds (i.e., where all onjetures are theorems),
the ategory of motives misses out on fundamental objets of
arithmeti, i.e., sets
X (Q ):
Might attempt to replae J by a more ompliate objet.
X
18
Weil 1938: `Generalization of abelian funtions'.
`A paper about geometry disguised as a paper about analysis whose
motivation is arithmeti' (Serre).
Stresses importane of developing `non-abelian mathematis with a
key role for non-abelian fundamental groups.
Clearly motivated by the Mordell onjeture.
19
Reason for emphasizing non-abelian 1 : Classial onstrution of
J is homologial.
Paper established rst theorems relating fundamental groups and
vetor bundles on urves.
In addition to previous desriptions, reall that J over C an also
be thought of as
-the spae of unitary haraters (S 1 -valued) of 1 (X (C ));
-spae of line bundles of degree zero on X (C ).
X
X
20
Weil's generalized this to vetor bundles, leading eventually to
work Narasimhan-Seshadri, Donaldson, Simpson, et., referred to
as non-abelian Hodge theory.
For example, the theorem of N-S says that there is an equivalene
between moduli of irreduible unitary representations of 1 and
that of stable vetor bundles of degree zero on X (C ).
21
From view of arithmeti, the point of suh theorems is to
`algebraize' data of 1 , thereby leading to an arithmeti objet
dened over Q , with potential for arithmeti appliations. That is,
theory of vetor bundles is a kind of theory of fundamental groups
over Q .
However loss of Albanese map:
x 7! OX ((x)
(b))
No way to assoiate a vetor bundle to a point.
22
However, one needn't algebraize geometrially.
Arithmeti topology
gives another way to dene fundamental groups over Q :
In fat, Grothendiek's theory of the etale fundamental group leads
to a theory of non-abelian Albanese maps.
23
Basi idea:
ina
(x) := [1 (X ; b; x)℄
b
where the image runs over a lassifying spae (similar to lassifying
spae of mixed Hodge strutures).
In fat, previous abelian Albanese map an be viewed as
x 7! [1 (X ; b; x)=1 (X ; b)(3) ℄
(quotient modulo a level of the desending entral series).
24
1 (X ; b; x)
is a torsor or a prinipal bundle for 1 (X; b). Have an
ation by omposition
1 (X ; b; x) 1 (X ; b)!1 (X ; b; x)
and the hoie of an path p 2 1 (X ; b; x) determines a bijetion
1 (X ; b) ' 1 (X ; b; x)
l 7! p Æ l
25
1 (X ; b; x)
is a prinipal bundle over a point, and hene, trivial.
Grothendiek's theories allow us to enrih points in various ways.
Example:
Spe(Q )
Funtion-theoreti enrihment of a point.
26
Topologial enrihment: The etale topology.
Spaes like Spe(Q ) or Spe(Z ) are endowed now with very
non-trivial topologies that go beyond sheme theory.
In general, a Grothendiek topology on an objet T allows open
sets to be ertain maps with range T from domains that are not
neessarily subsets of T .
27
For example, an onsider the overing spae topology on a
topologial spae. Leads to nothing essentially new.
In algebrai geometry, there are many maps that behave formally
like loal homeomorphisms without atually being so.
Etale
maps between shemes: proper nite-to-one maps with
surjetive tangent map.
28
On C z
7! z2
is etale.
sheme-theoretially,
Spe(C [t; t 1 ℄)!Spe(C [t; t 1 ℄)
orresponding to
C [t; t
1
℄
C [t; t
t2
t
29
1
℄
Same for
Spe(Q [t; t 1 ℄)!Spe(Q [t; t 1 ℄)
Main point, in fat, is that we an refer to suh overing maps for
spaes with very sparse olletions of points.
30
The onneted etale overings of Spe(Q ) are maps
Spe(F )!Spe(Q );
where F is a nite (separable) eld extension of Q .
Pro-nite sheaves on Spe(Q ) anonially identied with pro-nite
sets arrying ontinuous ation of
G = Gal(Q =Q ))
31
For Spe(Z ), one an onstrut an open overing using the two
maps
Spe(Z [i℄[1=2℄)!Spe(Z )
and
p
Spe(Z [(1 + 7)=2℄[1=7℄)!Spe(Z ):
32
Sheaf (o)homology for these theories rather well-known, with
numerous appliations: Weil onjetures, Faltings' theorem, Wiles
theorem, et.
Grothendiek's exoti topologies also lead to interesting homotopy
groups.
33
variety dened over Q and b 2 X (Q ).
Reall the universal overing spae
X
(X~ (C ); ~b)!(X (C ); b)
whih is a prinipal 1 (X ; b)-bundle over X (C ). This speializes to
the 1 (X ; b; x):
~ ' 1 (X ; b)
X
and
~ ' 1 (X ; b; x)
X
via lifting of paths.
b
x
34
~ (C )!X (C )
X
an be approximated by nite overs
Xi (C )! X (C )
Example:
exp(2i()) : C !C is approximated by
() : C !C Note that the approximating system is dened over Q .
n
35
Even in general, there is a system
~ et
X
= fX g
i
that an be dened over Q , and viewed as an arithmeti universal
overing spae.
(Warning: In this assertion, we are using the rationality of the
base-point.)
36
The ber X~ now onsists of systems of algebrai (Q ) points, that
are ated on by G = Gal(Q =Q ). Still has the struture of a
pro-nite group, the etale fundamental group
et
b
b):
1et (X;
For any other point x 2 X (Q ),
; b; x) := X~ xet
1et (X
also onsists of algebrai points, and is a pro-nite prinipal bundle
b).
for 1 (X;
et
37
The group and prinipal bundle arry ompatible Galois ations
reeting the fat that there are sheaves
~ )
b (X
et
and
~ et )
x (X
on Spe(Q ) underlying these sets.
[Reall:
b; x : Spe(Q )! X
℄
38
Thus we get an arithmeti Albanese map
b))
X (Q )! H 1 (G; 1et (X;
; b; x)℄
x 7! [1et (X
where the target is a lassifying spae for prinipal
; b)-bundles on the etale topology of Spe(Q ).
1 (X
For urves of genus 1, this map is injetive!
et
39
In this form, a bit diÆult to study, beause geometry has been
entirely removed. Only pronite topology remains.
Can reinsert geometry at the level of `oeÆients' for the
non-abelian ohomology by replaing the fundamental groups by
suitable algebrai ompletions.
Geometry of the spae vs. geometry of oeÆients
is an important theme in arithmeti geometry.
40
nitely-generated disrete group. C [ ℄ group algebra.
: C [ ℄!C [ ℄ C [ ℄
indued by
for 2 . Then, in fat,
= ff
i.e.,
7! 2 C [ ℄ : (f ) = f f g
is the set of `group-like elements' in the Hopf algebra C [ ℄.
41
But an onsider the formally ompleted group algebra
C [[
℄℄ := lim C [ ℄=I
n
where I C [ ℄ is the augmentation ideal.
Then we get more group-like elements, e.g.,
exp( log( ))
for 2 and 2 C .
The unipotent ompletion
in C [[ ℄℄.
u
of
is the set of group-like elements
42
Thereby arrive at a more strutured objet:
is a pro-algebrai group.
:= =( )( ) is an algebrai group for eah n.
u
u
n
u
u
n
43
Can do this to the fundamental group:
1 (X (C ); b)! 1u (X (C ); b)
getting the group generated by the holonomy of all unipotent
onnetions on X (C ).
Also have `unipotent prinipal bundle of paths'
1u (X (C ); b; x)
44
Can arry out same onstrution for the etale fundamental group
and the ompleted group ring
b)℄℄
Q p [[1et (X;
to arrive at the Q -pro-unipotent etale fundamental group:
p
u;Q p
1
b)
(X;
45
The previous lassifying spae gets replaed by
u;Q p
Hf1 (G; 1
b))
(X;
whih then has the struture of a pro-algebrai variety (being a
moduli spae of prinipal bundles for a pro-algebrai group).
There are nite-dimensional quotients
u;Q p
Hf1 (G; [1
b)℄ )
(X;
n
obtained by onsidering quotients modulo the desending entral
series.
46
They t into a tower:
b)℄4 )
(X;
u;Q p
b)℄3 )
(X;
-
u;Q p
Hf1 (G; [1
-
..
.
..
.
Hf1 (G; [1
?
?
?
- H (G; [ (X; b)℄ )
u;Q p
Hf1 (G; [1
X (Q )
1
f
u;Q p
1
b)℄2 )
(X;
1
rening the map at the bottom (whih has a lassial interpretation
in Kummer theory).
47
End up with a diagram:
X (Q )
?
u;Q p
Hf1 ( ; [1
- X (Q
-H(
b)℄ )
(X;
1
n
f
p
?
u;Q p
p ; [1
)
b)℄ )
(X;
n
involving a loal version of the lassifying spae on the lower right
hand orner, with G = Gal(Q =Q ).
Vertial maps are all of the form
p
p
p
x 7! [1
u;Q p
(X ; b; x)℄
obtained from the previous one by pushing out prinipal bundles.
48
The loal lassifying spae
Hf1 (
p
u;Q p
; [1
b)℄ )
(X;
n
has the struture of a p-adi symmetri spae, and the map
X (Q p )!Hf1 (
u;Q p
p ; [1
b)℄ )
(X;
n
is obtained by solving p-adi dierential equations (for parallel
transport).
49
Theorem 0.1
Let X be a urve and suppose
b)℄n ) < dimHf1 ( p ; [1u;Q p (X;
b)℄n )
dimHf1 ( ; [1u;Q p (X;
for some n. Then X (Q ) is nite.
Theorem is intimately related to non-abelian nature of the
fundamental groups and the orresponding non-linearity of the
lassifying spaes.
50
Idea of proof: (1)
X (Q p )!Hf1 (
u;Q p
p ; [1
b)℄ )
(X;
n
is nite-to-one: a purely analyti fat;
(2)
Q
b)℄ )
Im(X (Q )) H 1 ( ; [1 p (X;
is a dense analyti urve: Fat about p-adi transendental
funtions;
(3)
Q
b)℄ )) H 1 ( ; [1 Q p (X;
b)℄ )
Im(H 1 ( ; [1 p (X;
lies inside a proper losed subvariety.
u;
p
f
n
p
u;
f
u;
n
f
51
p
n
(2) and (3) imply that
Im(X (Q p )) \ Im(Hf (G; [1
u;Q p
b)℄ )) H 1 ( ; [1 Q p (X;
b)℄ )
(X;
u;
n
is nite.
Together with (1), get niteness of X (Q ).
52
f
p
n
Can use the theorem to prove niteness of integral points for
hyperboli urves of genus zero and ertain kinds of hyperboli
urves of genus one, e.g.,
y2
= x3 + k
The dimension hypothesis for general urves follows from `general
struture theory of mixed motives', i.e.,
Standard motivi onjetures ) Faltings' theorem.
53
Related to non-abelian extensions of the onjetures of Birh and
Swinnerton-Dyer. Proofs are an extension of:
Non-vanishing of L-funtion ) niteness of rational points
that ours for ellipti urves.
In fat,
Finiteness of zeros of p-adi L-funtion ) niteness of integral
points
in the ase of CM ellipti urves.
54