Download WHAT IS A GLOBAL FIELD? A global field K is either • a finite

WHAT IS A GLOBAL FIELD?
JEAN-BENOIT BOST
A global field K is either
• a finite degree extension field of Q, i.e.,
K = Q(α) = Q[x]/(P (x))
d
d−1
where P (x) = x + a1 x
+ · · · + ad ∈ Q[x] \ Q is an irreducible polynomial of
degree d such that P (α) = 0;
or
• a finite degree extension field of Fp (T ), where p is a prime number, Fp = Z/pZ,
and
P Fp (T ) :=
P , Q ∈ Fp [T ] , Q 6= 0 .
Q
As [K : Fp (T )] < +∞,
Fp ⊆ {α ∈ K | α algebraic over Fp } =: k
is a finite field extension Fp ⊆ Fq . We will call k the field of constants of K.
There is an element t ∈ K \ k such that K is a finite separable extension of k(t).
Since t is transcendental over k, the field k(t) is isomorphic to k(T ), the field of
rational functions over k. Thus, we have an isomorphism
K = k(t)[α] ∼
= k(T )[X]/(P (T , X))
where P (T , X) ∈ k[T , X] is an absolutely irreducible polynomial such that degX P ≥
∂P
6= 0.
1 and ∂X
Two natural questions:
Why is the above definition relevant?
Why is it meaningful to consider number fields and function fields on the same
footing?
There must be some analogies between number fields and function fields!
In the following, I will give a historical survey on the subject. Here is an outline:
(1) The work of Dedekind, Weber, ... , Grothendieck;
(2) The work of Hensel, ... , Tate, Langlands;
(3) Algebraic varieties over global fields.
However, nothing concerning Galois cohomology will be touched in this very
incomplete survey.
1. Dedekind, Weber, ... , Grothendieck: Algebra
A global field K is a field isomorphic to the field of rational functions κ(C) of
some integral scheme C of finite type over Z which has Krull dimension 1. The
nonempty open subsets of C are characterized as subsets of the form C S
\ F , where
F is a finite set of closed points of C. Choose a finite open cover C = 1≤i≤n Ui ,
Date: Seminar talk at THU. M.S.C on July 27, 2010.
1
2
JEAN-BENOIT BOST
where each Ui = Spec Ai is affine. Since the Ui ’s are of finite type Z as C is, for
each i,
Ai ∼
= Z[x1 , · · · , xv ]/Ii
for some prime ideal Ii of Z[x1 , . . . , xv ]. For each i, the field of fractions of Ai is K
and since dim Ai = 1,
{ maximal ideals of Ai } = { non-zero prime ideals of Ai } .
The canonical map C → Spec Z gives natural homomorphisms Z → Ai , ∀ i.
Consider the fiber square
CQ −−−−→
C




y
y
Spec Q −−−−→ Spec Z
C → Spec Z being a morphism between integral schemes of dimension 1, either the
generic fiber CQ of C → Spec Z is nonempty, which means the structural morphism
C → Spec Z is dominant; or the the image of C → Spec Z consists of a single closed
point of Spec Z. The generic point of C lies in each nonempty Ui . So the first
case happens if and only if the natural map Z → Ai is injective for every i, or
∼
equivalently, Ai ⊗ Q 6= 0 for every i. In this case, one proves that Ai ⊗ Q −→ K.
The field K is finitely generated as a Q-algebra, so K is a number field. In the
second case, the morphism C → Spec Z factors as
C −→ Spec Fp −→ Spec Z
for some prime number p, or equivalently, ker(Z → Ai ) = pZ for each i. In this
case, p = 0 in K so that K has characteristic p > 0 and is a function field.
Actually, the “curve” C may assumed to be regular. Then in the decomposition
[
C=
Ui , Ui = Spec Ai
1≤i≤n
each Ai is Dedekind domain which is finitely generated as a Z-algebra. Further,
there is a canonical “maximal” regular C: in the number field case, call
OK := {α ∈ K | ∃ P ∈ Z[X] \ Z a monic polynomial such that , P (α) = 0}
the ring of integers of K. Then the canonical regular model for K is C =
Spec OK . In the function field case, C is a smooth projective, geometrically irreducible curve over k ∼
= Fq . For example, if K = Fq (T ), C = P1Fq ; if K =
2
Fq (x)[Y ]/(Y − P (x)), where P ∈ Fq [x] has degree 3, gcd(P, P 0 ) = 1 and the
characteristic p is 6= 2, then C is an elliptic curve (with affine equation y 2 = f (x)).
Historical comments. (1) Dedekind and Weber are the first to apply “arithmetic”
approaches to the theory of algebraic functions in one variable. By a “function field
in one variable over C”, we mean a finite degree extension field K of C(T ). Such a
field has the form
K∼
= C(T )[X]/(P (T, X)) ,
where P ∈ C[T , X] is irreducible of degree deg P ≥ 1. Elements R(t, x) of such a
field K are rational functions on the algebraic curve
C := {(t , x) ∈ C2 | P (t, x) = 0} ⊆ A2 (C)
WHAT IS A GLOBAL FIELD?
3
After removing singularities of C, there exists a bianalytic isomorphism onto the
e an .
complement of a finite subset in a compact Riemann surface C
C \ {singular points}
bianalytic iso.
/C
e an \ a finite subset
A1 (C)
P1 (C)
e an is deduced from C by deleting a finite subset of singular
The Riemann surface C
points, and then by “adding” a finite set of points.
e an ) of
By Riemann’s work, the function field K is isomorphic to the field M(C
e an :
meromorphic functions on the compact Riemann surface C
e an ) .
K∼
= M(C
Thus the field may be investigated by analytic methods, e.g. by using ∂, ∂∂ =
i∆ and Dirichlet principle, etc. The lack of rigorous analytic foundations was a
motivation for Dedekind and Weber to develop a purely algebraic approach.
According to Weil’s Rosetta Stone, there should be an enormous amount of
mathematics naturally divided into three parts:
• number fields, i.e., finite extension fields of Q;
• function fields in one variable over a finite field (ever studied by Artin and
Schmidt), i.e., K = Fq (t)[x] with an algebraic relation P (t, x) = 0;
• function fields in one variable over C (ever studied by Riemann), i.e., K =
C(t)[x] with an algebraic relation P (t, x) = 0;
each with its own framework and techniques and each written in its own language
in similar texts.
Historical comments. (2) From Kronecher’s viewpoint, basic objects of study of
arithmetic or algebraic geometry are (in modern languages) schemes of finite type
over Z. The objects of interest consist of an ideal I = (P1 , . . . , Pf ) of a polynomial
ring Z[X1 , . . . , XN ] and the quotient ring A = Z[X1 , . . . , XN ]/I. The ring A is an
integral domain if and only if I is a prime ideal. In this case, we write K for the
fraction field of A.
In the hierarchy by Krull (=Krnoecher=absolute) dimension, dim A = 0 if and
only if A = K. In this case, K is a finite field. dim A = 1 if and only if A 6= K and
any non-zero prime ideal of A is maximal. In this case, K is a global field.
Another discovery of Kronecker is that if we put
Np := #{(x1 , . . . , xN ) ∈ FN
p | P1 (x1 , . . . , xN ) = · · · = Pf (x1 , . . . , xN ) = 0} ,
then the series
X
p : prime
Np
ps
converges for s ∈ C, Re(s) > d := dim A and
X Np
= n · log(s − d)−1 + O(1)
ps
as s → d+ ,
p : prime
where n is the number of d-dimensional irreducible components of Spec A.
4
JEAN-BENOIT BOST
More generally, we may consider solutions of the system P1 = · · · = Pf = 0
in any finite field Fq , which corresponds to maximal ideals m of A such that q =
Nm := #(A/m).
After Euler, Riemann, Dedekind, Artin, Hasse and Weil, one defines
Y
(1 − Nm )−1
ζX (s) =
m∈X0
for X = Spec A. One has
log ζX (s) =
X Np
p
ps
+ error terms .
The function ζSpec Z is the usual zeta function in analytic number theory.
Some most important problems concerning the zeta functions involves the study
of analytic continuation of ζX , of zeros and poles of ζX , etc.
When X is defined over a finite field Fp , great contributions have be made by
Dwork, Grothendieck and Deligne. In the case dim X = 1, there are remarkable
works by Hecke and Schmidt. When dim X = 2 and XQ is an elliptic curve, we
have the famous theorem of Wiles. In the case where XQ is a Shimura variety, a
lot of problems remain to be interesting topics for further study.
2. Hensel, ... , Tate, Langlands: Analysis and Representation Theory
An absolute value on a field K is a map
| · | : K → R+
such that the following properties hold for all x, y ∈ K
(1) |x + y| ≤ |x| + |y|;
(2) |xy| = |x| · |y|;
(3) |x| = 0 ⇐⇒ x = 0.
An absolute value | · | on K is called ultimetric if |x + y| ≤ max{|x| , |y|} for
all x, y ∈ K.
Given an absolute value | · |, d|·| (x, y) := |x − y| defines a distance on K. The
absolute value | · | is called nontrivial if there exits some x ∈ K ∗ such that |x| =
6 1.
Two absolute values | · | and | · |0 are said to be equivalent if d|·| and d|·|0 define the
same topology on K. This condition is equivalent to saying that there is a positive
real number α ∈ R∗+ such that | · |0 = | · |α .
The set of places of K, denoted by V (K), is the set of all non-trivial absolute
values on K modulo the above equivalence relation. By convention, when K is
a Riemannian field (i.e., function field of an algebraic curve over C) or a global
function field (i.e., a finite extension of Fp (T )), we restrict ourselves to places represented by absolute values that are trivial on the field of constants C or k = Fq .
Given v = [| · |] ∈ V (K), define
Kv := the completion of K with respect to d|·| .
Then Kv is an extension field of K, and | · | extends to an absolute value on Kv .
e an ), there is a bijection
For a Riemannian field K ∼
= M(C
∼
e an −→
C
V (K) ;
P 7−→ [| · |P ]
WHAT IS A GLOBAL FIELD?
5
e an ), the value |f |P is defined as
where for f ∈ M(C
|f |P := exp(−vP (f )) ,
with vP (f ) := the valuation of f at P .
For a global function field K, say the function field of a curve C over a finite
field k = Fq . Let C0 be the set of closed points of C and let
[
C(Fq ) :=
C(Fqn ) .
n≥1
We may identify C0 with the set of orbits of the Galois action of Gal(Fq /Fq ) ∼
= Ẑ
on the set C(Fq ). Give P ∈ C0 corresponding to the orbit of a point x ∈ C(Fq ),
we say the field of definition of x is Fqn if κ(P ) = Fqn . Set NP := |κ(P )| and
defined an absolute valuation | · |P on K by
−vP (f )
|f |P := NP
,
where vP denotes the valuation at P (or at x). Then there is a bijection
∼
C0 −→ V (K) ;
P 7−→ [| · |P ]
For a number field K, the ring of integers OK is a Dedekind domain. Every
non-zero ideal I of OK has a unique factorization into products of non-zero prime
ideals:
Y
I=
pvp (I)
p6=0
where every vp (I) ≥ 0 and vp (I) = 0 for almost all p. For each p ∈ (Spec OK )0 ,
the residue field Fp := OK /p is a finite field with prime field Fp if p is the prime
number such that p ∩ Z = pZ. Set Np := #Fp = p[Fp :Fp ] and define a function
vp : K → Z ∪ {+∞} by


if x = 0
+∞ ,
vp (x) := vp (xOK ) ,
if x ∈ OK \ {0}


vp (a) − vp (b) ,
if x = a/b with a, b ∈ OK , b 6= 0 .
−vp (x)
Then x 7→ |x|p := Np
(Spec OK )
a
defines an absolute value on K. There is a bijection
∼
({σ : K ,→ C}/complex conjugation) −→ V (K)
which sends a p ∈ (Spec OK ) to the place [| · |p ] and sends a complex embedding
σ : K → C to the place [|σ(·)|]. If K = Q, for each prime number p, one has
|pn a/b|p = p−n ;
∀ a, b , n ∈ Z , b 6= 0 such that p - ab .
The absolute value corresponding to the unique embedding σ : Q → C is the usual
archimedean absolute value on Q.
From the above, we see that there is a uniform way of recovering the points of C
from the field K. Actually, the analogy between number fields and global function
fields become more satisfactory when one takes the archimedean places [|σ(σ)|] into
account.
6
JEAN-BENOIT BOST
As a basic example for analogies among the three kinds of fields mentioned
before, we remark that there is a product formula for each of the three cases. For
e an ),
a Riemannian field, we have for each f ∈ M(C
Y
X
|f |P = 1 .
vP (f ) = 0 , or equivalently ,
ean
P ∈C
ean
P ∈C
∗
For a global function field, we have for each f ∈ K ,
X
vP (f ) log NP = 0 , or equivalently ,
Y
|f |P = 1 .
P ∈C0
P ∈C0
For a number field, we have for each x ∈ K ∗ ,
X
X
vp (x) log Np +
log |σ(x)| = 0 ,
p
σ:K,→C
or equivalently,
Y
|f |v = 1 ,
P ∈V (K)
where | · |v = | · |p if v = [| · |p ]; | · |v = |σ(·)| if v = [σ(·)] for some σ such that
σ(K) ⊆ R; and | · |v = |σ(·)|2 if v = [σ(·)] for some σ such that σ(K) * R.
Another example is the analogous description of completions. For a Riemannian
e an and if z is a local analytic
field, if v is a place corresponding to a point P ∈ C
coordinate at P , then
∼
Kv −→ C[[T ]][T −1 ] ;
z 7−→ T
and the ring of integers Ov := {x ∈ K | | · |P ≤ 1} is
∼
Ov −→ C[[T ]] .
For a global function field, if v is a place corresponding to a P ∈ C0 , FP = κ(P )
and z is a local parameter at P , then
∼
Kv −→ FP [[T ]][T −1 ] ;
z 7−→ T
and
∼
Ov −→ FP [[T ]] .
For a number field, if v is an archimedean place, associated to a complex embedding
∼
∼
σ : K → C, then Kv −→ R if σ(K) ⊆ R and Kv −→ C if σ(K) * R. If v is a
non-archimedean place, associated to a non-zero prime ideal OK with p ∩ Z = pZ,
then Kv is a finite field extension of QP and Ov is the integral closure of Zp in Kv .
Note that if K is a global field, then every completion Kv is a locally compact
field and Ov is an open compact subring of Kv . A locally compact field with nondiscrete topology is called a local field . In fact, all local fields are obtained in this
way, i.e., completions of global fields.
To study global fields, we may use some analytic tools. For example, there exists
a Haar measure λv on each (Kv , +) such that
λv (xE) = |x|v λv (E) ,
for all x ∈ Kv and E ⊆ Kv . The ring of adèles of the global field K is defined as
Y
AK := {(xv ) ∈
Kv | xv ∈ Ov for almost all v}
v∈V (K)
WHAT IS A GLOBAL FIELD?
7
with coordinate-wise addition and multiplication. This is a locally compact ring and
K is a discrete cocompact subring of AK via the diagonal embedding x 7→ (v 7→ x).
N.B.: There is an Haar measure λ = ⊗v λv on AK such that λv (Ov ) = 1 for
almost all v. One recovers the product formula by observing that multiplication by
any x ∈ K ∗ preserves the Haar measure λ. As a fact, any field K equipped with
absolute values having the above property is actually a global field.
The group of idèles of the global field K is the group JK := A∗K is the group
of invertible elements in the ring AK equipped with the subspace topology induced
from the product topology on AK × AK via the injection
JK = A∗K −→ AK × AK ;
x 7→ (x, x−1 ) .
The multiplicative group JK is a locally compact group. Define the subgroup JK1
by the exact sequence
1 −→ JK1 −→ JK −→ R∗+ −→ 1
where the map JK → R∗+ is given by
(xv ) 7−→
Y
|xv |v .
v
By the product formula, K ∗ is a subgroup of JK1 . In fact, K ∗ is a cocompact
subgroup of JK1 . This is a fancy reformulation of Dirichlet’s theorems: the class
group
Cl(K) := {non-zero fractional ideals of OK }/ ∼ ,
where ∼ is the equivalence relation defined by
I ∼ J ⇐⇒ I = λJ ,
for some λ ∈ K ∗ ,
∗
is a finitely generated abelian group of rank #{archimedean places}−
is finite and OK
1.
Actually, JK /K ∗ contains a lot of deep non-trivial information. The group of
characters
(JK /K ∗ )∧ := {χ : JK → U (1) a continous group homomorphism | χ|K ∗ = 1}
may be identified with the group of Hecke’s Größencharaktere. Analysis on JK /K ∗
can give information about the L-function.
More generally, by the work of Gelfand and Langlands, for G = GLN , the group
G(AK )1 := {g ∈ GLN (AK ) | | det(g)| = 1}
contains G(K) = GLN (K) as a discrete subgroup of finite covolume. The group
G(AK )1 acts unitarily on L1 (G(AK )1 /G(K)). This representation is supposed
to encode deep informations concerning the global field K, e.g, the Galois group
Gal(K/K), motives over K and their Hasse-Weil zeta functions.
3. Varieties over global fields
Let K = κ(C) be a field as in Weil’s Rosetta Stone and let X be an algebraic
variety over K. One may find a model X of X over C relevant for studying X over
K, i.e., a flat morphism X → C with generic fiber X/K.
8
JEAN-BENOIT BOST
Examples. (1) The number field case. If X is a subvariety of AN
Q with ideal
IX = {P ∈ Q[x1 , . . . , xN ] | P |X = 0}. Then I := IX ∩ Z[x1 , . . . , xN ] defines a
model X ⊆ AN
Z .
(2) The Riemannian field case. If Xis a smooth projective curve over C(C), one
may find a smooth projective complex surface fibered over C.
In the geometric cases (i.e, the cases of a global function field or a Riemannian
field), C is a smooth projective curve over k = Fq or C, one may use some extra
tools available when doing geometry over k.
Example 1. The Riemanniann field case. Let π : X → C be a surjective morphism
from a smooth projective surface X /C to a smooth projective curve C/C with
generic fiber X/K. Let X(K) be the set of K-rational points of X and let X (C)
be the set of regular (holomorphic) sections of π. We may identify X(K) with the
set of meromorphic sections of π so that X (C) is a subset of X(K). Since X is
projective, we have X (C) = X(K). Thus, X(K) may be investigated using the
geometry of complex surfaces.
Mordell conjecture for function fields over C (Manin): if the genus of X is
≥ 2, then X(K) is finite.
An analogous but simpler situation is an abelian scheme:
A
−−−−→ A



π
y
y
Spec K −−−−→ C
Now there is a natural bijection between A(K) with the set A (C) of analytic
sections of π : A → C. In this situation, there is an analytic proof of Mordell-Weil
conjecture. We sketch the ideas of proof as follows:
There is an sequence of abelian sheaves over C an :
0 −→ Γ −→ LieA −→ A −→ 0
which gives on fibers an exact sequence
0 −→ Z2g −→ C2g −→ C2g /(Zg + ΩZg ) −→ 0
where g is the genus of the curve C. Taking cohomology gives an exact sequence
H 0 (C , LieA ) −→ A (C) −→ H 1 (C , Γ) .
By a “curvature” argument, one can prove that “A has no fixed points”. Together
with “some positivity of the Hodge bundle”, this yields H 0 (C , LieA ) = 0. It
follows that A (C) → H 1 (C , Γ) is an injection. The topoligcal Euler-Poincaré
characteristic χtop (C , Γ) is
χtop (C , Γ) = 2gχtop (C) = 2g(2g − 2) = 4g(g − 1) .
So H (C , Γ) is a finitely generated Z-module of rank 4g(g − 1). Hence, A(K) ∼
=
A (C) is finite generated as a Z-module, of rank at most 4g(g − 1).
1
N.B.: A typical “Hodge theoretic” argument usually consists of a “sign of curvature” argument and a relation between coherent cohomology groups and Betti/topological
cohomology groups.
WHAT IS A GLOBAL FIELD?
9
Example. 2. The global function field case. Let K = Fq (C), i.e., C is a model of
the function field K. Put
Y
(1 − NP−s )−1 .
ζK (s) := ζC (s) =
P ∈C0
Artin and Schmidt proved that ζK (s) is a rational function of q −s . In fact, one has
ζK (s) =
where P (X) =
Q
1≤i≤2g (1
P (q −s )
(1 − q −s )(1 − q 1−s )
− ai X) ∈ Z[X]. Moreover,
#C(Fqn ) = q n + 1 −
2g
X
ani .
i=1
The Riemann Hypothesis for ζK is equivalently to each of the following statements:
√
√
(1) |ai | = q for each i; (2) |#C(Fqn ) − q n − 1| ≤ 2g g n for all n ≥ 1.
We will give a sketch of the proof of the Hasse-Weil estimate:
√
|#C(Fq ) − q − 1| ≤ 2g q .
Consider the projective surface S := C ×Fq C and use
• intersection theory on a projective surface: There is an intersection symmetric
bilinear form h· , ·i defined for curves on S, which has the following properties: if
C1 and C2 meet properly, then
hC1 , C2 i = #(C1 ∩ C2 )
and hC1 , C2 i is invariant when C1 or C2 moves in an algebraic family. The
Hodge index theorem (Castelnuovo-Severi conjecture) states that for curve classes
C1 , . . . , Cn on S the matrix of intersection numbers (hCi , Cj i) has signature
(− , − , . . . , − , 0, . . . , 0)
or (+ , − , . . . , − , 0 , . . . , 0) .
P
Consequently, if there are ni ∈ R such that i, j ni nj hCi , Cj i > 0, then
(−1)n+1 det(hCi , Cj i) ≥ 0 .
• Frobenius morphisms: Over a field of characteristic p > 0, one has (a + b)p =
ap + bp . So there is a Frobenius morphism
F = Fq : C/Fq −→ C/Fq
given by the coordinate expression
(x0 , . . . , xN ) 7−→ (xq0 , . . . , xqN )
when C is embedded in a projective space PN
Fq . We may then identify
C(Fq ) = C(Fq )F .
Note that F = Fq is a morphism of degree q from C to itself.
Sketch of proof of Hasse-Weil estimate. We consider the curve classes
C1 = V := {P } × C ; C2 = H := C × {P } ;
C3 = ∆ := the diagonal of CFq ×Fq CFq
C4 = G := the graph of the Frobenius morphism F
10
JEAN-BENOIT BOST
over SFq = CFq ×Fq CFq . Write

0
1

1
1
N := #C(Fq ). The intersection matrix hCi , Cj i is

1
1
1

0
1
q
.

1 2 − 2g
N
q
N
q(2 − 2g)
Since hC1 + C2 , C1 + C2 i = hV + H , V + Hi = 2 > 0, by Hodge index theorem,
det(hCi , Cj i) = (N − (q + 1))2 − 4g 2 q ≤ 0 ,
whence the Hasse-Weil estimate.
A natural question is how to transfer these arguments to the number field case.
Tools we have now in hand include:
• Coherent cohomology coming from heights, Diophantine geometry and Arakelov
theory;
• (p-adic) Hodge theory coming from heights, Arakelov theory, étale cohomology,
etc.
But what else?