Let F be a finite field, let q be the cardinality ofF, and let m be a positive
integer dividing q−1. Thus, the field F contains a primitive m-th root of
unity. Letµm(F) be the group ofm-th roots of unity inF. LetKbe a function
field with exact field of constants F and let S be the set of places of K. For
every place v ∈ S letdv be the degree of the place v, that is, the degree of the residue fieldOv/Pv of the completionKv ofK atv overF. Moreover, we
denote the group of ideles of K byJ and the groupQ
vUv of unit ideles by
U. Let :J →J/Jm be the canonical projection of J ontoJ/Jm. Given a subgroupAofJ we will writeAfor the group (A·J)/Jm. Since the fieldKis understood, we will often drop any subscriptK when it is used in formulas.
Definition 4.55 (Group of divisors). LetF be a finite field and letK be a
function field with exact field of constants F. Thegroup of divisors DivK of
K is the free abelian group on the set of places ofK. We have the degree map
deg : Div→Z, X v∈S nvv7→ X v∈S nvdv.
It is a surjective map [Corollary V.1.11 in [68]] and its kernel is the group Div0 of divisors ofK of degree 0. There is a natural continuous map
div :J →Div, j= (jv)v 7→div(j) =
X
v
(ordvjv)·v,
where ordv denotes the normalized valuation vKv on Kv. This is a com- mon notation for function fields and their completions. We denote the integer deg(div(j)) = P
v(ordvjv)dv by degj and call it the degree of j. Hence, we get the short exact sequence
1−→U −→J−→div Div−→0.
Definition 4.56(Group of principal divisors). LetFbe a finite field, letKbe a function field with exact field of constantsF, let JK be the group of ideles of
4.5. Function fields
K, and let DivK be the group of divisors ofK. Thegroup of principal divisors PrK ofKis the image ofK∗⊂JK in DivK under the map div : JK →DivK. Composing the map div : J →Div with the degree map deg : Div →Z we
obtain a map J→Z, which gives rise to the split short exact sequences
1−→J0−→J −→Z−→0
and
1−→J0−→J −→
Z/mZ−→0, (4.57)
where J0 is the group of ideles ofK of degree zero.
Theorem 4.58. Let Fbe a finite field, let q be the cardinality ofF, let mbe a positive integer dividing q−1, and letµm(F) be the group ofm-th roots of
unity in F. Let K be a function field with exact field of constants F, let J be the group of ideles ofK, and let U be the group of unit ideles ofK. Then the subgroup U of J/Jm equals its own annihilator in J/Jm with respect to the
norm-residue symbol
(·,·) :J/Jm×J/Jm→µm(F).
Proof. Theorem 3.89 implies that for every tame non-Archimedean place v
the groupUv/Uvmequals its own annihilator inKv∗/(Kv∗)mwith respect to the local norm-residue symbol. Since all places are tame non-Archimedean, we get the identityU =U⊥.
Theorem 4.59. Let Fbe a finite field, let q be the cardinality ofF, let mbe
a positive integer dividing q−1, and letµm(F) be the group ofm-th roots of
unity in F. Let K be a function field with exact field of constants F, let J be
the group of ideles ofK, and let U be the group of unit ideles ofK. Then the inverse of the norm-residue symbol induces a perfect pairing of finite abelian groups
(·,·)−1: (K∗∩U)×J /(K∗·U)→µm(
F)
that is characterized by (a), and its domain has the property (b).
(a) For each pair(f, j)∈(K∗∩(U·Jm))×J such that the divisorsdiv(f)
anddiv(j)have disjoint supports one has
f , j·(K∗·U) (·,·)−1 7−−−−→(f, j)−1=f(div(j))q−m1, where f(div(j)) =Y v N(Ov/Pv)/F(f nv modPv)if div(j) =X v nvv.
Chapter 4. The global norm-residue symbol
(b) Any pair of elements in (K∗ ∩ U) × J /(K∗ · U) is of the form
f , j·(K∗·U)
withf andj as in (a).
Proof. By Lemma 4.24 the group K∗ is a discrete subgroup of J. Since the group U is a compact subgroup of J, by Lemma 4.44 the subgroup K∗·U
is closed. Hence, the group K∗·U is a closed subgroup ofJ/Jm. Taking its annihilator and using Theorem 4.52 and Theorem 4.58 we get
(K∗·U)⊥=K∗⊥∩U⊥=K∗∩U .
The groupK∗∩Uis finite, because it is discrete by Theorem 4.53 and compact. Theorem 4.39 implies that the induced pairing exists and is a perfect pairing of finite abelian groups.
Now we are going to prove the formula in (a). Let S be the set of places of K and for each v ∈ S let Kv be the completion of K at v. By Theo- rem 3.89 and Theorem 3.86 for each v ∈ S the extension Kv(fv1/m)/Kv is unramified. For each v ∈ S we denote by Frv the Frobenius element in the Galois group Gal(Kv(fv1/m)/Kv). By the weak approximation theorem for ev- ery f ∈K∗∩(U·Jm) the set of all j ∈J such that div(f) and div(j) have disjoint supports maps surjectively to J /(K∗·U). This proves (b). Since both the inverse of the norm-residue symbol and the formula give rise to a group homomorphismJ /(K∗·U)→µ
m(F) when the first argument is fixed, we can
assume that j∈J has divisor div(j) =w, wherew is a place ofKnot in the support of div(f). Letdwbe the degree of the placew. The computation
(f, j)−1= (j, f) = Y v∈S (j, f)v =Y v∈S Frordv(j) v (f 1 m v )/f 1 m v = = Frw(f 1 m w )/f 1 m w = f qdw−1 m w modPw = f qdw−1 q−1 q−1 m w modPw = = (N(Ow/Pw)/F(f modPw)) q−1 m =f(div(j)) q−1 m ,
where the last equality in the first line follows from (b) in Theorem 3.80, gives the desired result.
Corollary 4.60. For allb∈J andζ∈F∗ one has
(b, ζ) =ζqm−1
degb
.
Proof. Let div(b) = P
vnvv. Hence, we have degb = Pvnvdv. Using Theo- rem 4.59 we get (b, ζ) = (ζ, b)−1= Y v N(Ov/Pv)/Fζ nv !qm−1 =ζPvnvdv q−1 m =ζq−m1 degb .
4.5. Function fields
Corollary 4.61. The norm-residue symbol induces a perfect pairing
(·,·) :J /J0×
F∗→µm.
Proof. It follows from Corollary 4.60 and the short exact sequence 4.57 that the annihilator of F∗ inJ isJ0.
Raising to the (q−1)/m-th power and the degree map give the isomorphisms
F∗ ∼→µm and J /J0 ∼
→Z/mZ,
respectively. Using these two isomorphisms, from the perfect pairing of Corol- lary 4.61 we get the perfect pairing
(·,·) :Z/mZ×µm→µm, (a, ζ)→ζa.
The explicit description follows from Corollary 4.60.
Corollary 4.62. The norm-residue symbol induces the perfect pairing
(·,·)−1: (K∗∩U)/
F∗×J0/(K∗·U)→µm(F). (4.63)
Proof. Since the annihilator of K∗∩U is K∗·U and by Corollary 4.61 the annihilator of J0 is
F∗, the pairing induced by the norm-residue symbol is
perfect.
Definition 4.64 (Divisor class groups). Let F be a finite field, let K be a
function field with exact field of constantsF, let DivK and Div0K be the group of divisors of K and the group of degree zero divisors ofK, respectively, and let PrKbe the group of principal divisors ofK. Thedivisor class groupPicKof
K is the group DivK/PrK and thedivisor class group of degree zero divisors
Pic0k is the group Div
0
K/PrK.
The surjective map K∗∩U →Pic0[m] that maps an elementa=ujm, with
a ∈K∗, u∈U, j ∈J, to div(j) is well-defined by Lemma 4.51 and fits into the short exact sequence
1−→F∗−→K∗∩U −→Pic0[m]−→0.
Hence, we have an isomorphism (K∗∩U)/
Chapter 4. The global norm-residue symbol
Furthermore, from the exact sequence
1−→U −→J0 div−→Div0−→0,
where Div0 is the group of divisors of degree zero, we get an isomorphism
J0/(K∗·U)∼= Pic0
/mPic0. (4.66)
Corollary 4.67. The norm-residue symbol induces the perfect pairing
{·,·}: Pic0[m]×Pic0/mPic0→µm(F),
{[D],[E] modmPic0} 7→f(E)qm−1,
where D and E are divisors with disjoint supports and f ∈ K∗ has divisor
mD.
Proof. Combining isomorphisms 4.65 and 4.66 with the perfect pairing 4.63 gives the perfect pairing{·,·}: Pic0[m]×Pic0/mPic0→µm(F).
Consider elements [D]∈Pic0[m] and [E]∈Pic0 represented by divisorsD
and E = P
vnvv of K with disjoint supports. Let f be a nonzero rational function with divisor mD. By isomorphism 4.65 we can writef as uim with
u∈U andi∈J. Hence, we have div(i) =D. Moreover, by isomorphism 4.66 we can choosej∈J0 with div(j) =E. Using Theorem 4.59 we get
{[D],[E] modmPic0}= (f, j)−1=f(div(j))q−m1 =f(E) q−1