9:43 a.m. March 1, 2014
Essays on metaplectic representations
Local quadratic extensions
Bill Casselman
University of British Columbia
cass@math.ubc.ca
Suppose F to be a p-adic field. In this essay I’ll prove:
Theorem. If E/F is a separable quadratic extension with norm N : E → F then F × /N E × has order two.
In addition, I’ll discuss the structure of the quotient and its relation to the ramification of E/F . The exposition
will be elementary and essentially self-contaned.
This is a simple case of the fundamental theorem of local class field theory, which for any abelian extension
E/F constructs an explicit isomorphism of F × /N E × with the Galois group. There will be nothing new in
my treatment, which has been basically an exercise in reading §V.3 of [Serre:1968]. Serre in fact deals with
the slighly more complicated case in which E/F is a cyclic extension of prime degree.
The proof to be presented does not do that, but it does say something about a relationship between the Galois
group and norms. In all proofs of full local reciprocity that I am aware of, the case of prime cyclic extensions
is an important step.
The Theorem can be supplemented by a converse: given any subgroup H of F × of index two, there exists a
separable quadratic extension E/F with N E × = H . It is unique up to isomorphism. I have already shown
why this is true if the residue characteristic is odd. I’ll prove this in some later version of this essay when
the characteristic of F itself is other than 2. I am not aware of any simple proof in the case that F does have
characteristic 2.
If E/F is generated by a root of x2 + ax + b, there are elegant formulas in all cases (to be found in Chapter
XIV of [Serre:1968]) for deciding whether or not a given x in F × lies in N E × or not. The proofs I know start
with the general theorems of local class field theory.
Contents
1.
2.
3.
4.
Introduction
The different
Norms and the filtration of units
References
1. Introduction
Let
oF = integers in F
pF = maximal ideal of oF
̟F = a generator of pF
UF = o×
F (the units of oF )
FF = oF /pF .
Local quadratic extensions
2
n
Also, let ord be the map from F × to Z taking u̟F
(u a unit) to n. And for each n ≥ 0 let
UFn = {u ∈ UF | u ≡n 1} .
Here ≡n means congruence modulo pn .
The quotients associated to the filtration of unit groups have simple structures:
1.1. Lemma. The projection UF → oF /pF induces an isomorphism of UF0 /UF1 with F×
F , and for n ≥ 1 the
map
x 7−→ 1 + x̟Fn
induces an isomorphism of F with UFn /UFn+1 .
The second isomorphism is not at all canonical, since for n ≥ 1 it depends on a choice of ̟F . The canonical
isomorphism is with pn /pn+1 , which may be identified with ⊗n (p/p2 ).
Fix the quadratic extension E/F , and let the conjugation in E be x 7→ x. Let
N = the norm map from E to F : x 7→ xx
TR =
the trace from E to F : x 7→ x + x .
I’ll first finish with the easy cases of the Theorem. In the first, the extension E/F is the unique unramified
extension of F . This means that ̟E = ̟F and that FE is a quadratic extension of FF . I leave it as an exercise
×
×
to prove that in this case N o×
E = oF (a consequence of Hensel’s Lemma), and that N E is the union of all
× 2n
subsets oF ̟F for n in Z.
Now suppose E/F to be ramified. The second, still simple, case is that in which the residue field characteristic
is odd. There are two possibilities for E , which are obtained by adjoining a square root of either ̟F or ε̟F
for some non-square unit ε. It is then easy to see (again, say, by Hensel’s Lemma) that N o×
E consists of those
units in oF whose image in FF is a square. If we choose v ̟F to be the norm of one of the square roots, the
n
subgroup N E × then contains this as well as all uv n ̟F
with u in this group. Again, it clearly has index two
×
in F .
So we are reduced to the third case, in which E/F is ramified and the residue field characteristic is 2. This is
by far the most interesting. In this case E has as F -basis elements 1, ̟E . The second satisfies an Eisenstein
equation
x2 − ax + b = 0 .
with a in p and b = u̟F for some unit u. I may as well choose ̟F to be b. With this assumption
̟E + ̟E = a,
̟E ̟ E = ̟F .
REDUCTION TO UNIT GROUPS.
1.2. Lemma. The inclusion of UF /N UE in K × /N L× is an isomorphism.
Proof. This can be proved directly, but following Serre I exhibit it as a consequence of a well known result
that will be applied several times. I leave its proof as an exercise:
1.3. Lemma. (The Snake Lemma) Given two exact sequences in a commutative diagram
0 −→
0 −→
A
1 −→
yf1
B1 −→
A
2 −→
yf2
B2 −→
A
3 −→ 0
yf3
B3 −→ 0
Local quadratic extensions
3
the following sequence is exact:
0 −→ Ker(f1 ) −→ Ker(f2 ) −→ Ker(f3 ) −→ Coker(f1 ) −→ Coker(f2 ) −→ Coker(f3 ) −→ 0 .
In our case, the diagram is
ord
×
0 −→ U
−→
E −→ E

yN
yN
0 −→ UF
−→ F ×
ord
−→
Z −→

yI
Z
0
−→ 0 .
The outer map is the identity on Z since the norm of ̟E is ̟F . This concludes the proof of Lemma 1.2.
2. The different
So now we must prove that UF /N UE has order two. Proving this will use the congruence filtrations {U n }
of the unit groups and maps induced by the norm on associated quotients.
The different ϑE/F is by definition the smallest ideal d of oE such that
basic facts about it.
TR(d−1 )
⊆ oF . I recall a couple of
2.1. Proposition. The different ϑE/F is the ideal of oE generated by ̟E − ̟ E .
Proof. Recall that ̟E is a root of the polynomial f (T ) = T 2 − aT + ̟F , and that 1, ̟E form a basis of oE .
Since
f ′ (̟E ) = 2̟E − a = 2̟E − (̟E + ̟E ) = ̟E − ̟E ,
the Lemma follows from the equations
1
=0
f ′ (̟E )
̟E
= 1.
f ′ (̟E )
TR
TR
These equations are a special case of a more general result, which I include here because its proof is more
illuminating:
2.2. Lemma. Suppose
f (T ) =
n
X
ci T i =
i=0
Then
X
i
xki
=
f ′ (xi )
n
Y
(T − xi ) .
i=1
0 if 0 ≤ k < n − 1
1 if k = n − 1.
This follows from the familiar partial fraction decomposition
X 1
1
1
=
·
′ (x ) T − x
f (T )
f
i
i
i
since the left hand side may be expressed as
1
1
a1
1
P
·
=
+
+
·
·
·
n−j
Tn
Tn
T n+1
j cj /T
4
Local quadratic extensions
and the right as
X
i
1
X 1
x2i
1
1
xi
1
+
+
·
·
·
.
·
·
=
·
+
f ′ (xi ) T 1 − xi /T
f ′ (xi ) T
T2 T3
i
This concludes the proof of Proposition 2.1.
From now on, let m be the order of ϑE/F , so that ϑE/F = pm
E . The previous result says that
m = ord(̟E − ̟E ) .
This and the previous Proposition tell us how to calculate the different in terms of the trace of ̟E .
2.3. Corollary. If ̟E satisfies the Eisenstein equation x − ax + ̟F = 0 then ϑE/F is generated by 2̟E − a.
d
e
So that if 2 = u̟F
and a = v̟F
with units u, v then
m = min 2d + 1, 2e .
2.4. Corollary. If σ 6= 1 in G(E/F ) then
̟E
m−1
= 1 + x̟E
̟E
with x 6≡1 0.
Explicitly,
a − ̟E
̟E
=
= 1 − (2 − a/̟E ) .
̟E
̟E
2.5. Lemma. For n ≥ 0
TR
m+⌊l/2⌋
pm+ℓ
= pF
E
.
Here ⌊x⌋ is the greatest integer n ≤ x. For example, ⌊1/2⌋ = 0, ⌊−1/2⌋ = −1.
−m−1
Proof. The characterization of ϑE/F tells us that TR p−m
= oF and TR pE
= p−1
E
F . The trace is F -linear,
m+ℓ
2m+2ℓ
so that upon multiplying by pF
= pE
we get
TR
TR
m+2ℓ
pE
= pm+ℓ
F
m+2ℓ−1
pE
= pm+ℓ−1
.
F
This proves the Lemma, since ⌊(2ℓ)/2⌋ = ℓ, ⌊(2ℓ − 1)/2⌋ = ℓ − 1.
Following this, for all ℓ we have
(2.6 )
for some unit τm+ℓ−1 .
m+2ℓ−1
TR(x̟E
) = τm+ℓ−1 x̟Fm+ℓ−1
Local quadratic extensions
5
3. Norms and the filtration of units
How does the norm map behave with respect to the congruence filtration of UE ? The basic formula is very
k
k
simple. Suppose k > 0. Every element in UE
is of the form 1 + x̟E
, and
k
N (1 + x̟k ) = 1 + TR(x̟E
) + x2 ̟Fk .
(3.1 )
According to Lemma 2.5, this implies immediately that
(3.2 )
N UEm+ℓ ⊆ UFr

if ℓ < 0
m + ℓ
if ℓ = 0
with r = min m + ⌊ℓ/2⌋, m + ℓ = m

m + ⌊ℓ/2⌋ if ℓ > 0.
3.3. Proposition. Suppose k ≥ 1.
(a) If k < m − 1, the map induced by the norm on from UEk /UEk+1 to UFk /UFk+1 takes
k
1 + x̟E
7−→ 1 + x2 ̟Fk .
(b) If k = m − 1, the map induced by the norm on from UEm−1 /UEm+1 to UEm−1 /UEm takes
m−1
1 + x̟E
7−→ 1 + (τm−1 x + x2 )̟Fm−1 .
(c) If k = m + 2ℓ − 1 with ℓ ≥ 1, the map induced by the norm on from UEm+2ℓ−1 /UEm+2ℓ+1 to
UEm+ℓ−1 /UEm+ℓ takes
m+2ℓ−1
1 + x̟E
7−→ 1 + τm+ℓ−1 x̟Fm+ℓ−1 .
From now on, the proof of the Theorem takes several small steps.
Step 1. The first is:
m−1
with UF /UFm−1 .
3.4. Lemma. The norm map induces an isomorphism of UE /UE
Proof. Using a standard argument about filtrations (amounting to an inductive application of Lemma 1.3),
k
this will follow from the claims (a) for k ≤ m the norm takes UE
to UFk and (b) for each k < m − 1 the
induced map
UEk /UEk+1 −→ UFk /UFk+1
is an isomorphism.
0
Since U 0 = o× and U 1 = 1 + p, both UE
/UE1 and UE0 /UE1 may be identified with the units in the common
×
residue field F . On this, the induced norm map takes x to x2 , which is an automorphism of the residue
field. This takes care of the case k = 0.
In the case 0 < k < m − 1, the claim follows from case (a) of the last Proposition, since x 7→ x2 is an
automorphism of F.
Step 2. Consider now the commutative diagram
0 −→ UEm−1 −→ 
UE
yN
yN
0 −→ UFm−1 −→ UF
−→ UE
/UEm−1 −→ 0
yNm−1
−→ UF /UFm−1 −→ 0 .
According to Lemma 3.4, the right-hand map is an isomorphism. Lemma 1.3 therefore implies:
3.5. Lemma. The map
UFm−1 /N UEm−1 −→ UF /N UE
Local quadratic extensions
6
induced by inclusions is an isomorphism.
m−1
Given this, in order to prove the Theorem it remains only to prove that UFm−1 /N UE
has order two.
Step 3. Next I’ll prove:
m
= UFm .
3.6. Lemma. We have N UE
Proof. For ℓ ≥ 0 we have
N UEm+2ℓ+1 ⊆ UFm+ℓ
N UEm+2ℓ+2 ⊆ UFm+ℓ+1 .
In order to prove Lemma 3.5, it will suffice to show that for each ℓ ≥ 0 the induced map
UEm+2ℓ+1 /UEm+2ℓ+2 −→ UFm+ℓ /UFm+ℓ+1
is surjective. This is case (c) of Proposition 3.3.
Step 4. I shall apply Lemma 1.3 one more time. Consider the commutative diagram
m
0 −→ U
E
yN
0 −→ UFm
m−1
−→ U
−→ UEm−1/UEm −→ 0
E
yN
yNm−1
−→ UFm−1 −→ UFm−1 /UFm −→ 0
Given the previous Lemma, Lemma 1.3 implies that
3.7. Lemma. The homomorphism induced by projections
UFm−1 /N UEm−1 −→ Coker(Nm−1 )
is an isomorphism.
m−1
Step 5. It remains now to examine the map from UE
/UEm to UFm−1 /UFm induced by the norm map. This
is case (b) of Proposition 3.3. Modulo 1 + pm
F we have
m−1
m−1
N (1 + x̟E
) = 1 + (τ + x2 )̟E
)
with τ = τm−1 . The map on F taking x to τ x + x2 has cokernel of order two, because
τ x + x2 = τ 2 (x/τ ) + (x/τ )2
and the map x 7→ x + x2 is an F2 -linear map from F to itself with kernel the image of F2 in F. So the image
m−1
of UE
in UFm−1 /UFm has order two.
This concludes the proof of the Theorem.
The proof shows that:
3.8. Lemma. All the arrows
Coker(Nm−1 ) ←− UFm−1 /N UEm−1 −→ UF /N UE −→ F × /N E ×
are isomorphisms.
Remark. I have remarked that although this proof does not give a direct isomorphism of the Galois group
G(E/F ) with F × /N E × , there is an implicit relation with the Galois group. This is because the kernel of the
induced norm map
UEm−1 /UEm → UFm−1 /UFm
Local quadratic extensions
7
m−1
is the image of G under the map σ 7→ σ(̟E )/̟E . Since the groups UE
/UEm and UFm−1 /UFm may both be
considered vector spaces over F2 , its cokernel and kernel must have the same size.
Remark. The more general result in which E/F is cyclic of prime order is not much more difficult to prove.
Also, much of the argument goes through when the residue field is assumed to be algebraically closed and
E/F of degree equal to the residue field characteristic. In that case the conclusion is that F × = N E × . This
leads to one of the fundamental components in local class field theory, that all elements of the Brauer group
of a local field split over its unramified extension.
Remark. (3.2) says that
N UEn ⊂ UFr
where
r = min ⌊(m + k)/2⌋, k .
Suppose we plot all the points in the discrete graph of the function k 7→ min ⌊(m + k)/2⌋, k , and then take
the convex hull of the union of these points and the x-axis. Its top is the graph of the function
x 7−→
x
if 0 ≤ x ≤ m − 1
(x + (m − 1))/2 if m − 1 ≤ x,
which is closely related to what Serre calls the Herbrand function ϕ(x). This observation suggests that in
general Serre’s function might be interpreted as a function whose graph is the convex hull of some interesting
set of points. This is indeed true, according to some ideas of John Tate. I’ll take this up elsewhere.
4. References
1. Jean-Pierre Serre, Corps locaux, Hermann, 1968.