Título iv infracciones en particular
4.2 importancia de la propuesta
We present in a more explicit way the quadratic norm-residue symbol in the fieldQ2of 2-adic rationals. For unramified extensions ofQ2we state and prove
3.10. The field of two-adic rationals and its unramified extensions
Lemma 3.106.
Lemma 3.102. Let Q2 be the field of 2-adic rationals, let Z2 be its ring of
integers, let U be group of units of Z2, and for each n∈Z≥0 letU(n) be its n-th higher unit group. Then there is the equality
Z∗2 2
=U(3).
Proof. By Theorem 3.37 we have the equalities
Z∗2=U =U (1).
Hence, if a is an element in Z∗2, we can write a = 1 + 2x with x∈ Z2. By
squaring we obtain
a2= 1 + 4x(1 +x)≡1 mod 8.
Since by definition we have 1 + 8Z2=U(3), we geta2∈U(3)and therefore the
inclusion Z∗2 2
⊆U(3). Theorem 3.71 gives [Z∗2 : Z∗2 2
] = 4. Since U(3) has also index 4 in Z∗2, we obtain the equality Z∗2
2
=U(3).
Theorem 3.103. Let Q2 be the field of2-adic rationals and for everya∈Q∗2
leta be the residue classamodQ∗2 2
of ainQ∗2/Q∗2 2
. Then the natural map
h2i × h−1i × h5i→∼ Q∗2/Q
∗
2 2
is a group isomorphism and each of the groupsh2i,h−1i, andh5ihas order2. Proof. The split sequence of Theorem 3.37 gives the isomorphism
Q∗2∼=h2i ×Z∗2.
By squaring and applying Lemma 3.102 we obtain the isomorphism
Q∗2 2∼
=h4i ×U(3).
Taking the quotients we get the desired isomorphism. Since by definition we have 1+8Z2=U(3), each element in{2,−1,5}has nontrivial image inQ∗2/Q∗2
2
and therefore each of the groupsh2i,h−1i, andh5ihas order 2.
Theorem 3.104. Let Q2 be the field of2-adic rationals and for everya∈Q∗2
leta be the residue classamodQ∗2 2
of ainQ∗2/Q∗2 2
. Then the table
(·,·) 2 −1 5 2 1 1 −1
−1 1 −1 1
Chapter 3. The local norm-residue symbol
gives an explicit description of the perfect pairing
(·,·) :Q∗2/Q∗22×Q∗
2/Q∗2 2
→ {±1}
induced by the quadratic norm-residue symbol
(·,·)Q2,2:Q∗2×Q∗2→ {±1}.
Proof. Using bilinearity of Theorem 3.69 and the relations of Theorem 3.74 we obtain the equalities
1 = (1−2,2) = (−1,2), 1 = (1−5,5) = (4,5)(−1,5) = (22,5)(−1,5) = (2,5)2(−1,5) = (−1,5), 1 = (−2,2) = (−1,2)(2,2) = (2,2), 1 = (−5,5) = (−1,5)(5,5) = (5,5), 1 = (−1,2)(2,−1) = (2,−1), 1 = (−1,5)(5,−1) = (5,−1).
Since by Theorem 3.70 the pairing induced by the norm-residue symbol is perfect, we get
(2,5) = (5,2) = (−1,−1) =−1.
Corollary 3.105. Let Q2 be the field of2-adic rationals and let U be the unit
group of the ring of integers of Q2. Then the annihilator U⊥ in U of U with
respect to the norm-residue symbol (·,·)Q2,2 :Q2×Q2 → {±1} is the second
higher unit group U(2) of the ring of integers of Q2 and the perfect pairing
(·,·) :U/U⊥×U/U⊥→µ2
induced by the norm-residue symbol is given by
fora, b∈Z∗2 (a, b) =
(
−1 ifa≡b≡3 mod 4,
1 otherwise.
Proof. This follows from Theorem 3.104.
3.10. The field of two-adic rationals and its unramified extensions
we show what each piece is canonically equal to.
Q∗2/Q∗2 2 2 U/U2 |2|−1=2 (U/U2)⊥ 2 1 h2i ×(Z/8Z)∗ Z/2Z (Z/8Z)∗ (Z/4Z)∗ h5i Z/2Z 1
Lemma 3.106. LetF be a finite unramified extension of the fieldQ2of2-adic
rationals, let OF be the ring of integers of F, let P be the maximal ideal of
OF, and let (·,·)F :F∗×F∗→ {±1} be the quadratic norm-residue symbol of
F. Then for alla, b∈ OF one has
(1 + 2a,1 + 2b)F = (−1)TrP((a+P)(b+P))
with TrP denoting the trace map from OF/P toZ/2Z.
Proof. LetUFbe the group of units of the ring of integers ofF. Letd∈1+4OF and let δ be an element in an algebraic closure of F with δ2 = d. Since by
Theorem 3.95 the extensionF(δ)/F is unramified, Theorem 3.86 implies that for all c ∈ UF we have (c, d) = 1. Hence, we may assume a, b ∈ UF and therefore we have
1 + 2b
1−4ab ∈F
∗\ {1}.
By Theorem 3.74 for eachc∈F∗\ {1} one has (1−c, c)F = 1. The equality
−2b(1 + 2a) 1−4ab , 1 + 2b 1−4ab F = 1
follows. Since the norm-residue symbol is an antisymmetric bilinear map and we have just proved that for all c ∈ UF we have (c,1−4ab) = 1, we obtain the equality
(1 + 2a,1 + 2b)F = (1−4ab,2)F.
Let (·,·)Q2 : Q∗2×Q∗2 → {±1} be the quadratic norm-residue symbol of Q2
and denote the norm map fromF toQ2by NF /Q2. Theorem 3.100 implies the
equality
(1−4ab,2)F = NF /Q2(1−4ab),2
Chapter 3. The local norm-residue symbol
Since we have the congruence
NF /Q2(1−4ab)≡1−4 TrP((a+P)(b+P)) mod 8,
CHAPTER
4
The global norm-residue symbol
We extend the theory of the norm-residue symbol to global fields and groups of ideles. As a corollary of Theorem 4.59 we obtain a new proof [Corollary 4.67] of the fact that the Tate pairing is a perfect pairing [Theorem 4.69]. Moreover, Theorem 4.86 gives a pairing similar to the Tate pairing in the case of number fields.
4.1
Global fields
The reference for this section is Chapter II in [9] by Cassels and Fr¨ohlich. Note that they call ‘valuation’ what we will call ‘absolute value’.
Definition 4.1 (Number field). A number field is a finite field extension of the field Qof rational numbers.
Definition 4.2 (Function field). Afunction field is a finite field extension of the field F(t) of rational functions in one variabletover a finite fieldF.
Definition 4.3 (Global field). Aglobal field is a field that is either a number field or a function field.
Definition 4.4 (Absolute value). An absolute value | · | on a field K is a function
| · |:K→R≥0
Chapter 4. The global norm-residue symbol
(a) For all a∈Kone has: |a|= 0 if and only ifa= 0. (b) For alla, b∈Kone has|ab|=|a| · |b|.
(c) There exists C ∈ R such that for all a ∈ K with |a| ≤ 1 one has
|1 +a| ≤C.
Remark 4.5. In many books (c) of Definition 4.4 is replaced by the triangle inequality|a+b| ≤ |a|+|b|for alla, b∈K. Our choice is inspired by the desire to call the normalized absolute value on the field of complex numbers, which is the square of the ordinary absolute value, an absolute value.
Definition 4.6 (Trivial absolute value). Thetrivial absolute value on a field
K is the unique functionK→ {0,1} that is an absolute value onK.
Definition 4.7 (Equivalent absolute values). Two absolute values
| · |1:K→R≥0 and | · |2 : K → R≥0 on a field K are equivalent if there
exists C∈R>0 such that for alla∈K one has
|a|1=|a|C2.
Remark 4.8. LetKbe a field. An absolute value| · |onKinduces a topology onKthat is generated by all sets of the form{x∈K:|x−a|< d}witha∈K
andd∈R>0.
Theorem 4.9. Let K be a field. Two absolute values on K are equivalent if and only if they induce the same topology on K.
Proof. See Section 4 of Chapter II in [9] by Cassels and Fr¨ohlich.
Definition 4.10 (Place). Aplace vof a global fieldKis an equivalence class of nontrivial absolute values on K.
Definition 4.11 (Non-Archimedean and Archimedean places). A place v of a global field K is non-Archimedean if the completion Kv of K at v is a non-Archimedean local field. Otherwise, it is Archimedean.
For a place v of a global fieldK we denote the normalized absolute value on the completion Kv ofK atv by| · |v:Kv→R≥0.
Theorem 4.12. Let K be a function field. Then all places of K are non- Archimedean.
Proof. See Section 1 of Chapter III in [18] by Fr¨ohlich and Taylor.
Theorem 4.13. Let K be a number field, let F be either R or C, let
| · |F :F →R≥0 be the ordinary absolute value on F, and let σ : K ,→ F
be a field embedding. Then the function | · |: K →R≥0, a7→ |σ(a)|F, is an