Homework S2016
Contents
1 HW1, Due on Feb 5th
1
2 HW2, due on Feb 12th
2
3 HW3, due on Feb 19th
4
4 HW4, due on Feb 26th
6
5 HW5, due on Mar 4th
8
6 HW6, due on Mar 23th
10
7 HW7, due on April 14th
11
8 HW8, due on April 21th
13
9 Final exam
13
1
HW1, Due on Feb 5th
Some of the questions below need Newton polygon. Let K be a non-archimedean
local field (with valuation normalized such that a uniformizer has valuation
equal to one). Let f (X) = an X n + ... + a1 X + a0 ∈ K[X] be a polynomial
of degree n. The Newton polygon is defined to be the lower convex hull on
the (x, y)-plane of the set of points (i, v(ai )), i = 0, ..., n. Let λ1 , ...λr be the
slopes of the line segments of the Newton polygon and denote the lengths of
the corresponding line segments projected onto the x-axis by m1 , ..., mr (so
1
that m1 + ... + mr = n). Then there are precisely mi number of roots (in
K) of f (x) with valuation equal to −λi for every i = 1, 2, ..., r. You may use
this result freely.
1. Show that Qp has no nontrivial field automorphism (i.e., there is no
σ 6= id : Qp → Qp as an isomorphism of fields, where we don’t assume
that σ is continuous).
2. Let Qp be the algebraic closure of Qp and Cp its completion with respect
to the unique extension of absolute value of Qp .
P
n
0
(a) Consider the series ∞
n=1 ζn0 p where ζn0 = ζn is any primitive n 0
th root of unity if (n, p) = 1 and otherwise n = 1. Show that the
series converges in Cp but the limit is not inside Qp . In particular,
Qp is not complete.
(b) Show that Cp is algebraic closed. (Hint: Krasner lemma.)
3. Which of the following polynomials are irreducible over Q5 ?
X 3 + 5X + 125, X 3 + 5X + 25, X 3 + 5X 2 + 25,
X 4 + 5X 2 + 25, X 4 + 15X 2 + 25.
4. Consider the polynomial f (X) = X p − X − 1/p ∈ Qp [X].
(a) Show that f (X) is irreducible over Qp .
(b) Let α be a root of f and denote K = Qp (α) with integer ring OK
and maximal ideal mK . Show that for i = 0, 1, ..., p − 1, there
exists a unique element βi ∈ OK such that α + βi is a root of f (X)
and βi ≡ i( mod mK ). In particular, K/Qp is Galois.
2
HW2, due on Feb 12th
Let K be a non-archimedean local field, i.e., a finite extension of Qp (the
mixed characteristic case), or the field of Laurent series Fq ((π)). For a ring R
(with 1), recall that a (commutative one-dimensional) formal group F over
R is a power series F (X, Y ) such that
2
1. F (X, Y ) = X + Y + terms of degree at least 2.
2. F (X, F (Y, Z)) = F (F (X, Y ), Z).
3. There exists a unique iF ∈ X · R[[X]] such that F (X, iF (X)) = 0.
4. F (X, Y ) = F (Y, X).
Let R be an OK -algebra, with the structure ring homomorphism i : OK →
R. A formal group F over R is always assumed to be commutative, and
one-dimensional. A formal OK -module over an OK -algebra R is the data
(F, ι) where F is a formal group F with a ring homomorphism ι : OK →
EndR (F ) such that dι : OK → LieR (F ) ' R coincides with the structure
homomorphism i : OK → R.
1. “Local Hermite-Minkowski theorem”.
(a) For a fixed N , show that there are only finitely many unramified
field extensions of Qp of degree at most N .
(b) For a fixed N , there are only finitely many field extensions of Qp
of degree at most N . (Hint: Eisenstein polynomial and Krasner
lemma.)
2. Let F (X, Y ) be a formal group. Show that F (X, 0) = X and F (0, Y ) =
Y (namely, there is no terms like X m , m > 1). Deduce that the
condition on the existence of iF (X) in the definition of formal groups
follows from the others.
3. Let R be OK or a field of arbitrary characteristic p = 0 or a prime
number. Let Ga be the formal additive group over R. What is the
endomorphism algebra of Ga ?
4. Let R be a commutative Q-algebra and F be a formal group defined
over R. Show that there is a unique isomorphism, called the logarithm
of F
logF : F → Ga
such that logF (X) ≡ X ( mod deg ≥ 2). In particular, any formal
group over a Q-algebra is isomorphic to the formal additive group. If
3
F = Gm is the formal multiplicative group, then
∞
i
X
i−1 X
= log(1 + X).
logF (X) =
(−1)
i
i=1
5. Now R = OK . Set
−1
−2
q
f (X) = X + π X + π X
q2
+ .... =
∞
X
i
π −i X q .
i=0
−1
Show that F (X, Y ) := f (f (X) + f (Y )) lies in R[[X, Y ]] and defines
a formal OK -module with OK -action given by
[a]F (x) = f −1 (a · f (X)),
a ∈ OK ,
with logFK = f , where FK denotes the base change of F from R = OK
to F .
6. Let R = OK , and f ∈ Fπ a Lubin-Tate power series. Let (Ff , ι) be the
Lubin-Tate formal OK -module over R, with ι : OK → EndR (Ff ).
(a) Show that EndR (Ff , ι) ' OK .
(b) Let K/L be an unramified quadratic extension. Consider the formal OL module (Ff , ι|OL ). Base change to R = k (viewed as an
OL -algebra). Compute the endomorphism of Endk (Ff , ι|OL ).
3
HW3, due on Feb 19th
The first four questions are concerned with the Lubin-Tate formal module
over a local field of positive characteristic. Let K be the field of Laurent
series Fq ((π)) and fix a separable closure of K. Let f be the Lubin-Tate power
series
f (T ) = πT + T q .
The Lubin-Tate formal OK -module (F, ι) over OK is easy to describe as the
formal additive group F (X, Y ) = X + Y and ι : OK → End(F ) characterized
by
[a]f (X) = aX, a ∈ Fq
and
[π]f (X) = πX + X q .
4
1. Let t1 be a nonzero root of f (T ) = 0, and tn+1 a root of f (T )−tn = 0 for
n > 2. Let Kπ,n = K(tn ) and Kπ = ∪n Kπ,n the Lubin-Tate extension.
b π be the completion of Kπ . Show that the sequence
Let L = K
2
tq1 , tq2 , ...
is a Cauchy sequence, hence converges in L (but not in Kπ ).
2. Denote by t the limit of the sequence above. Show that tq
L, for all n > 1.
−n
exists in
×
3. Lubin–Tate theory asserts that φK : OK
' Gal(Kπ /K) where an ele×
ment a ∈ OK acts on Kπ by φK (a)(tn )P
= [a]f (tn ).1 This action extends
×
i
continuously to L. Show that, if a = ∞
i=0 ai π ∈ OK , ai ∈ Fq , then
φK (a)(t) =
∞
X
i
ai t q .
i=0
−∞
−1
−n
−∞
4. Let Fq [tq ] denote the ring Fq [t, tq , ..., tq , ...]. Let Fq [[tq ]] denote
−∞
the completion of the ring Fq [tq ] with respect to the t-adic topology:
Fq [[tq
−∞
]] := lim Fq [tq
n
−∞
]/(tn ).
P
−∞
Concretely, Fq [[tq ]] consists of i∈I ai ti where I is a subset of Z[1/q]≥0 ,
and for every C > 0 there are only finitely many nonzero ai such that
i < N.
−∞
−∞
Show that OL /(t) = Fq [tq ]/(t), and OL = Fq [[tq ]]. Deduce from
this that L is a perfect field (i.e., the Frobenius is an isomorphism).
(This is an example of a perfectoid field in characteristic p).
The following two questions are on group cohomology. They are in the
texts of Milne, but you should try them yourselves before looking there (if you
have-not seen them before).
1
We invert the usual local Artin map such that the uniformizers are sent to the geometric Frobenius, i.e., the automorphism of K ur = Fq ((π)) given by the geometric Frobenius
on Fq sending α ∈ Fq 7→ α1/q .
5
1. Let G be a cyclic group with P
a generator σ. Denote by N mG the norm
map: M → M sending m to g∈M gm. Show that
H 1 (G, M ) '
Ker(N mG : M → M )
.
(σ − 1)M
2. Hilbert 90. Let L/K be a finite Galois extension with Galois group G
so that G acts on L× . Show that H 1 (G, L× ) = 0.
4
HW4, due on Feb 26th
1. Let G be a finite group and Gp any p-Sylow subgroup. Consider the
restriction maps Resp : H r (G, M ) → H r (Gp , M ). For x ∈ H r (G, M ),
show that x = 0 if and only if Resp (x) = 0 for all primes p.
2. Let H be a subgroup of G and M an H-module. Let cIndG
H M be the
“compact”-induction, i.e. cIndG
M
=
Z[G]
⊗
M
.
Show
Shapiro’s
Z[H]
H
Lemma for homology:
Hi (G, cIndG
H M ) = Hi (H, M )
for all i ≥ 0.
3. Let G = P SL(2, Z). Recall that G has a free subgroup H ∼
= F2 of index
6, generated by
1 2
1 0
,
0 1
2 1
You do not need to show this–however, use this to show that for every
G-module A, and for every x ∈ H i (G, A), i ≥ 2, we have 6x = 0. (Hint:
the augmentation ideal for a free group is a free module.)
b Let g = Z,
b the profinite group (the inverse
4. (Cohomology of g = Z)
limit limn Z/nZ). Let M be a g-module (with continuous action, i.e.,
the stabilizer of every m ∈ M is an open subgroup of g, equivalently
M = ∪H M H is the union of invariants of open subgroups of g). Recall
from Milne notes that the cohomology of a pro-finite group G acting
continuously on M can be computed as the inductive limit, via the
inflations, over all open normal subgroups H :
H i (G, M ) = lim H i (G/H, M H ).
H
6
(a) Denote by σ the topological generator 1 in g. Let M 0 be the set
of elements annihilated by 1 + σ + σ 2 + ... + σ n for some n ∈ N.
Show that
M0
.
H 1 (g, M ) =
(σ − 1)M
In particular if M is torsion, we have H 1 (g, M ) =
M
.
(σ−1)M
(b) If M is divisible (multiplication by n is surjective for all n ∈ N)
or finite torsion, then
H 2 (g, M ) = 0.
(c) Show that H i (g, Q) = 0, for all i ≥ 0. Here Q has the trivial g
action.
Below are some questions that are not required to hand in.
1. (Transfer/Verlagerung) Suppose that H ⊂ G is a subgroup, not necesarily of finite index, and let A be a G module. Recall that the inclusion i : H ,→ G induces natural maps i∗ : H i (G, A) → H i (H, A) and
i∗ : Hi (H, A) → Hi (G, A). We often denote the map i∗ by Res (restriction), since is corresponds to restriction of cocycles. Now suppose that
H ⊂ G is a finite index subgroup, with [G : H] = n. As shown in class,
the map
NG/H : H 0 (H, A) → H 0 (G, A)
X
a 7→
sa
s∈G/H
is well-defined, and extends cohomologically to a map, which we denote
by CoRes (corestriction)
CoRes : H q (H, A) → H q (G, A)
In addition, it was shown in class that
CoRes ◦ Res = n
There is a corresponding construction in homology: set
0
NG/H
: H0 (G, A) → H0 (H, A)
X
a 7→
s−1 a
s∈G/H
7
Check that this is well-defined and show it extends to all homology
groups Hq . What is the analogue of CoRes ◦ Res = n in this context?
Prove the corresponding identity. Finally, note that, taking A = Z and
q = 1, this contruction gives a map
V : Gab → H ab
What is this map explicitly?
2. Let G be a finite cyclic group. Describe the 2-periodicity of Tate cohomology for cyclic groups in terms of cup product with an element of
H 2 (G, Z) ' H 1 (G, Q/Z).
5
HW5, due on Mar 4th
Recall that the Brauer group of a field K is defined as
×
BrK := H 2 (Gal(K/K), K ) =
lim
H 2 (L/K, L× )
L/K,f inite Galois
where K is a separable closure of K. We say that an element α ∈ BK is split
in an extension L of K if it is in the kernel of Res : BrK → BrL .
1. Let k be a field of characteristic p, and let K be k 1/p . Show that an
element α ∈ Brk is split by K if and only if pα = 0. Deduce that the
p-primary component of Brk is trivial if k is perfect (i.e., the p-th power
map is bijective).
2. Let K/k be a field extension of degree n. Let α ∈ Brk be split by K.
Show that nα = 0. (Hing: treat the separable and purely inseparable
cases separately.)
The following questions aim to prove that the maps φL/K can be glued to
form a map φK : K × → Gal(K ab /K).
1. Let G be a finite group. We know that there is a natural isomorphism
H −2 (G, Z) ' Gab .
8
Identify H −1 (G, Q/Z) with
sider the cup product
1
Z/Z,
|G|
viewed as a subgroup of Q/Z. Con-
b −2 (G, Z) ⊗ H 1 (G, Q/Z) → H
b −1 (G, Q/Z) ' 1 Z/Z.
∪:H
|G|
Show that for s ∈ Gab resp. χ ∈ Hom(G, Q/Z), viewed as elements in
b −2 (G, Z) resp. H 1 (G, Q/Z), we have
H
s ∪ χ = χ(s).
2. Let L/K be a Galois extension of non-archimedean local fields. Consider the (norm residue) isomorphism
b 0 (G, L× ) = K × /NL× → H
b −2 (G, Z) = Gab
φL/K : H
defined by inverting the cup product with the fundamental class uL/K ∈
H 2 (G, L× ) = BrL/K . Let α ∈ K × and α its class in K × /NL× =
b 0 (G, L× ). Now take a character χ ∈ H 1 (G, Q/Z) = Hom(G, Q/Z)
H
and denote by δχ ∈ H 2 (G, Z) its image under the connecting map
δ : H 1 (G, Q/Z) = Hom(G, Q/Z) → H 2 (G, Z)
arising from the short exact sequence
0 → Z → Q → Q/Z → 0.
Show that
χ(φL/K (α)) = invK (α ∪ δχ) ∈ Q/Z.
3. Deduce from the last question that if we have a tower of abelian extension non-archimedean local fields
K ⊂ L0 ⊂ L,
then for α ∈ K × , we have
φL/K (α)|L0 = φL0 /K (α).
Here we extend φL/K (resp. φL0 /K ) as maps from K × to Gal(L/K)
(resp. Gal(L0 /K)).
9
6
HW6, due on Mar 23th
Below K is a finite extension of Qp , ΓK = Gal(K/K), M is a finite ΓK module, H i (M ) = H i (K, M ) = H i (ΓK , M ). The local Tate duality asserts
that the cup product induces a perfect pairing between two finite abelian
groups
∪ : H i (K, M ) × H i (K, M ∗ ) → Q/Z,
where M ∗ = Hom(M, µ∞ ), µ∞ = ∪n µn . The Euler–Poincaré characteristic
is defined for any finite Gal(K/K)-module M :
χ(M ) =
h0 (M )h2 (M )
,
h1 (M )
hi = #H i (M ).
Then we have a formula
χ(M ) =
1
,
#OK /nOK
n = #M.
1. Applying Tate’s local duality theorem to the module Z/nZ we have a
pairing
Hom(GK , Z/nZ) × K × /(K × )n → Q/Z.
When K contains a primitive n-th root of unity ζ, this pairing gives
rise to the Hilbert symbol (How?)
( , ) : K × /(K × )n × K × /(K × )n → Q/Z
Confirm that this agrees with the more commonly given definition of
the Hilbert symbol, which we recall below. [The usual definition of the
Hilbert symbol is as follows. Given a, b ∈ K, we identify a ∈ K × /(K × )n
1
1
1
with the character χ
a ∈ Hom(GK , n Z/Z) = H (K, n Z/Z) defined by
√
fixing an n-th root n a of a, then defining for σ ∈ GK , χa (σ) by
√
σ( m a)
√
= ζ (nχa (σ)) .
m
a
Then we set, for a, b ∈ K × /(K × )n
×
(a, b) = b ∪ δχa ∈ H 2 (K, K ) = Q/Z
where δ is the connecting map in cohomology coming from the exact
sequence of trivial GK modules 0 → Z → Q → Q/Z → 0. ]
10
2. Let G be a finite group, and suppose V and V 0 are finitely generated
Zp [G]-modules such that V ⊗ Qp = V 0 ⊗ Qp . Show that one has the
following identity in the Grothendieck group K(G):
[V /pV ] − [p V ] = [V 0 /pV 0 ] − [p V 0 ]
Here p V denotes the p-torsion in V . [Hint: Reduce to the case where
V ⊃ V 0 ⊃ pV , and use the exact sequence 0 →p V 0 →p V → V /V 0 →
V 0 /pV 0 → V /pV → V /V 0 → 0.]
3. How many elements does K × /(K × )2 have? How many different quadratic
extension does K have? In general, how many elements does K × /(K × )`
have for a prime `?
e has cohomomlgical dimension one, i.e.,
4. Show that the group G = Z
for all finite G-module M , we have H i (G, M ) = 0 for i > 1, and there
exists M such that H 1 (G, M ) 6= 0.
5. (Local Tate duality for archimedean local fields) Let K be an archimedean
local field, and G = Gal(K̄/K). For a finite G-module M , write
M ∗ = Hom(M, K̄ × ) as a G-module. Show that when K = R there is
a duality induced by cup product HTi (K, M ) × H 2−i (K, M ∗ ) → 21 Z/Z.
0
)#H 0 (K,M ∗ )
Write χ(M ) = #H (K,M
. Show that
H 1 (K,M )
T
χ(M ) = |m|K ,
m = #M,
where |m|K = m (resp. |m|K = m2 ) if K is real (resp. complex).
7
HW7, due on April 14th
1. Let m be a fixed positive integer. Let K be a number field such that
every rational prime p | m is unramified in K. Prove that K and Q(µm )
are linearly disjoint.
2. Let K be a number field. Let A× be the group of ideles and A×
f =
Q0
v<∞ Kv be the group of finite ideles. Give one example K 6= Q where
K × ,→ A×
f (the diagonal embedding) is a discrete subgroup and give
one counterexample. Justify your answer.
11
3. Using class field√
theory, prove that Q(ζ5 ) is the maximal finite abelian
extension of Q( 5) that is unramified away from 5 and ∞ and has
degree prime to 5.
4. Show that, for a number field K, the idéle class group CK = IK /K × of
K has subgroups of finite index that are not open.
5. (a) Prove that there is a neighborhood of 1 in C× that contains no
nontrivial subgroups. Deduce that the kernel of a continuous homomorphism G → C× for a profinite group G must be an open
(hence finite-index) subgroup.
×
(b) Let χ : A×
be a continuous homomorphism. Show that
K → C
for all but finitely many non-archimedean places v of K, we have
χ|Ov× = 1.
(c) A Hecke character for a global field K is a continuous homomor×
×
phism χ : A×
K → C such that χ(K ) = 1. (Equivalently, it is
×
a continuous homomorphism χ : AK /K × → C× ). Recall that we
have an absolute value map
|| · || :
/ R×
IK = A×
(xv )v∈ΣK
/
+
Q
v∈ΣL
|xv |v
whose kernel is denoted by I1K . Note that I1K /K × is compact
(why?). Show that every Hecke character χ can be written as
χ = χ0 · || · ||s where χ0 is a Hecke character with valued in the
unit circle, and s ∈ C. Show that the real part of s is uniquely
determined.
(d) (Global Langlands correspondence for GL(1)) Prove that the composition with the Artin map φK : IK /K × → Gab
K defines a bijection
between the set of Hecke characters of finite order and the set of
continuous homomorphisms GK → C× .
6. Let L/K be a cyclic extension of number fields, and let a ∈ K. If the
image of a in Kv is a norm from Lv for all places v, then it is a norm
from L. (Recall that here Lv denotes Lw for any one place w of L above
v.) Show that this fails for non-cyclic extension.
12
8
HW8, due on April 21th
1. Let K be a finite extension of Qp or R. Show that
×
H 3 (Gal(K/K), K ) = 0.
×
What is H 4 (Gal(K/K), K )?
2. Let K be a number field. Show that
H 3 (Gal(K/K), CK ) = 0
where CK = limL/K CL , for all finite extension L/K, and CL denotes the
idele class group of L. What is H 4 (Gal(K/K), CK )?
×
3. Let K be a number field. Show that H 3 (Gal(K/K), K ) = 0.
(Hint to (1),(2),(3): Class formation and Tate–Nakayama theorem. It is also
common to consider H i (Gal(K/K, M ) as the inductive limit H i (Gal(L/K, M L )
over all finite Galois L/K.)
9
Final exam
1. Let K be a finite extension of Qp . Let Kπ be the Lubin-Tate extension
of K associated to a uniformizer π. Let π1 , π2 be two uniformizers.
(a) Compute Kπ1 ∩ Kπ2 , and Kπ1 Kπ2 .
(b) Denote ΓL = Gal(K/L). Can ΓKπ1 and ΓKπ2 generate ΓK ?
2. (Easy Grunwald-Wang) Let K be a number field, m = 2t m0 , m0 odd,
and S a finite set of primes of K. Let α ∈ K satisfy α ∈ (Kv× )m for all
v∈
/ S. If K(ζ2t )/K is cyclic (which is certainly satisfied if t ≤ 2) then
α ∈ (k × )m . Show this by following this outline: First, reduce to the
case when m is a prime power. Second, if m = pr (p not necessarily 2),
r
show that if K(ζpr )/K is cyclic, then any α ∈ (Kv× )p for all v ∈
/ S must
r
0
be a global p -th power. Do so in the following way: call K = K(ζpr ),
13
r
and show that α = β p for some β ∈ K 0 using Kummer theory. Then,
by comparing the factorization over K
pr
x −α=
l
Y
fi (x)
i=1
and the factorization over K 0
r
p
Y
(x − βζpi r )
x −α=
pr
j=1
conclude that, choosing a root βi for each fi , there are intermediate
field K ⊂ K(βi ) ⊂ K 0 . Analyze the splitting of v in these fields for
various i and use that K 0 /K is cyclic to conclude. Finally, third, show
the theorem–by assumption it suffices to work with p 6= 2, then take
norms from K 0 to K to conclude.
3. (“Counterexample” to easy Grunwald-Wang) Show that 16 is an 8-th
power for R and all Qp , p 6= 2, but not an 8-th power in Q. Hint: Show
Q(ζ8 ) is unramified
√ √ at all odd p so for all odd p, Qp contains at least
one of 1 + i, 2, −2.
4. Compute the number of degree-p abelian extensions of a finite extension
K of Qp . Your answer will depend on the ramification degree e of K/Qp .
(Hint: you may need local Tate duality.)
5. Give an example of global field K and a Hecke character χ : IK → S1
such that χ|I1K has infinite order. Here I1K is the kernel of the absolute
value map
/ R×
|| · || :
IK = A×
+
(xv )v∈ΣK
/
Q
v∈ΣL
|xv |v
6. Let K be a finite extension of Qp , K∞ /K a totally ramified Γ =
Gal(K∞ /K) ' Zp -extension. We constructed continuous Kn -linear,
Γ-equivariant maps
b ∞ → Kn ,
πn : K
14
by πn (x) = p−m trKm+n /Kn (x) when x ∈ Kn+m , and extending by contib ∞ we have
nuity. Show that for x ∈ K
x = lim πn (x).
n
7. Let K 0 /K be a finite extension of number fields. Denote by HK , HK 0
the Hilbert Class Fields of K and K 0 respectively, i.e. the maximal
unramified extensions of each (at all places, including both finite and
infinite; recall that we say that an archimedean local field extension
F 0 /F is ramified only when F 0 = C and F = R).
(a) Show that the norm map ClK 0 → ClK by N[A] := [NK 0 /K A] is a
well-defined homomorphism bwteen ideal class groups. By checking on classes of prime ideals prove that the diagram
/
ClK 0
N
Gal(HK 0 /K 0 )
/
ClK
Gal(HK /K)
commutes, where the horizontal maps are the natural isomorphisms (described by Frobenius elements on classes of primes, i.e.,
induced by the Artin homomorphism) and the right vertical map is
the natural restriction map induced by the inclusion HK ,→ HK 0 .
(b) Let L be the maximal subextension of K 0 that is abelian and
everywhere (including infinite places) unramified over K. Show
that the norm map between ideal class groups is surjective (and
hence hK |hK 0 ) when L = K.
(c) Prove that if K 0 /K is totally ramified at some place (perhaps
archimedean in case [K 0 : K] = 2) then hK |hK 0 .
15