Download MATH882201 – Problem Set I (1) Let I be a directed set and {G i}i∈I

MATH882201 – Problem Set I
(1) Let I be a directed set and {Gi }i∈I an inverse system of finite groups with projection maps
φij : Gi → Gj for all i, jQ∈ I satisfying j ≤ i. Give each Gi the discrete topology and denote
by π the product π := i∈I Gi endowed with the product topology. Define
G := lim Gi := {(gi )i∈I | φij (gi ) = gj for all j ≤ i} ⊆ π
←−
i∈I
(a) Show that G is a closed subset of π.
(b) Give G the subspace topology. Show that G is compact and totally disconnected for
this topology.
(c) Prove that the natural projection maps φi : G → Gi are continuous, and that the
(open) subgroups Ki := ker φi for a basis of open neighborhoods of the identity.
(d) Show that a subgroup of G is open if and only if it is closed and of finite index.
(2) If L/K is a Galois extension, GK = Gal(L/K), and V has the discrete topology, then a
representation
ρ : GK → GLE (V )
is continuous if and only if it factors through Gal(F/K), for some finite Galois extension
F/K.
(3) Prove the above statement if V is instead a vector space over R or C, with the usual topology.
(Hint: first prove that there are open neighborhoods of the identity in GLn (R) and GLn (C)
which contain no non-trivial subgroups.)
(4) Let ρ : GQ → GLn (C) be a continuous representation.
(a) Prove that up to conjugation by an element of GLn (C), the representation ρ factors
through GLn (K) for some field K of finite degree over Q.
(b) Prove that we may take K above to be an abelian extension of Q.
(5) Let I ⊆ Gal(Qp /Qp ) be the inertia subgroup and W ⊆ I the wild inertia subgroup. Show
that there is a non-canonical isomorphism of topological groups
Y
I/W '
Z` .
`6=p
(6) Let ρ : GQ → GLn (Qp ) be a continuous representation. Show that for all ` 6= p, the image
under ρ of any wild inertia group W` at ` is finite. Is the same necessarily true of the image
of any I` ?
(7) Suppose T is a lattice in a finite dimensional Q` -vector space. Prove that a representation
GK −→ AutZ` (T ) is continuous if and only if GK −→ Aut(T /`n T ) is continuous for each n
(the topology on T /`n T is the discrete topology).
(8) Let K be a field and suppose A is a continuous GK -module with the discrete topology.
Prove that the following are equivalent:
(a) A is a continuous GK -module
(b) For every
S a ∈ A, the stabilizer of a in G is open.
(c) A = U AU , the union being over open subgroups of U ⊂ GK .
(9) If T ≈ Zn` is a lattice with a continuous action GK → AutZ` (T ), then the action of GK on
T ⊗Z` Q` given by g · t ⊗ z = gt ⊗ z is also continuous.
(10) Suppose that K is a p-adic field. Prove that µ`∞ (K(µ` )) is finite if and only if the image of
the cyclotomic character χ` : GK → Z×
` is open.
(11) Let t ∈ Z` be a topological generator so that the closure of tZ is Z` (we write the addition
on Z` mutliplicatively).
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(a) If M ∈ GLn (Q` ), show that t 7→ ρ(t) = M determines a continuous representation
ρ : Z` → GLn (Q` ) if and only if for any z ∈ Z` , the limit n7lim
M n exists in GLn (Q` ).
→z
n∈Z
(b) Show that this limit exists for all z ∈ Z` when


1 1
.. ..


.
.


M = Mn = 
 ∈ GLn (Q` ),
.

.. 1 
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(12) Let Vn be the n-dimensional continuous representation Z` → GLn (Q` ) determined by the
matrix Mn in exercise 11(b). Show that Vn ∼
= Symn−1 (V2 ).
(a) Show that Vn is the unique (up to isomorphism) unipotent and indecomposable representation of Z` .
(b) Show that Vn ∼
= Symn−1 (V2 ). (Hint: consider the minimal polynomial for the action of
t on Symn−1 (V2 ).)
(13) In class we showed that if K is a p-adic field with algebraically closed residue field, then
i
every V ∈ Repst
Q` (GK ) is a subquotient of some V (X)(j) for some smooth and projective
algebraic variety X/K and integers i and j. Prove this for V which is only potentially
semistable. Hint: you’ll want to use ‘Weil restriction’ and Frobenius reciprocity.
(14) (This question requires some knowledge of elliptic curves and the theory of complex multiplication.) Recall Faltings’ theorem: if K is a number field, and if A and A0 are abelian
varieties over K, then A and A0 are isogenous (over K) if and only if there is an isomorphism
of GK -representations V` (A) → V` (A0 ) (for any prime `). It is natural to try to refine this
criterion and ask whether A and A0 are isomorphic if and only if T` (A) ∼
= T` (A) for every
prime `.
(a) Show that this is false by considering an elliptic curve E/K satisfying EndK (E) ∼
= OL
for some imaginary quadratic field L with class number greater than 1.
(b) If A satisfies End(A) = Z, then is this refinement true?
(c) Let R be a ring and suppose E and E 0 are elliptic curves over a number field K with
R∼
= End(E) ∼
= End(E 0 ). Is it true that E ' E 0 if and only if there is an isomorphism
T` (E) ' T` (E 0 ) of (R ⊗Z Z` )[GK ]-modules for each prime `?
(d) If E and E 0 are elliptic curves over a number field K satisfying End(E) 6' End(E 0 ), is
it true that T` (E) 6' T` (E 0 ) for some prime `?
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