MATH882201 – Problem Set II
In what follows, K is a p-adic field and mK is the maximal ideal of OK .
(1) Let E be the Tate curve attached to some non-zero q ∈ mK , and recall the associated
infinite dimensional GK -representation B` = Q` [u]. Show that the monodromy operator
N : B` → B` (−1) mapping b(u) 7→ −b0 (u) ⊗ t−1 is GK -equivariant.
Similarly, show that if (D, N ) ∈ C (the category defined in class), then the map
N : B` ⊗ D → (B` ⊗ D)(−1)
b ⊗ d 7−→ N b ⊗ d + b ⊗ N d
is GK -equivariant.
(2) Suppose (D, N ) ∈ C and let V` : C → RepQ` (GK ) be the functor
(D, N ) 7→ ker (N : B` ⊗ D −→ (B` ⊗ D)(−1))
defined in class. Check that dimQ` D = dimQ` V` (D, N ).
(3) Prove that direct sums, tensor products, and subrepresentations of semistable `-adic representations are also semistable.
(4) Show that for any V ∈ RepQ` (GK ), there is a GK -equivariant injection V` (D` (V )) ,→ V .
(5) Let E/K be an elliptic curve. Then the arithmetic of elliptic curves gives a non-degenerate,
alternating, GK -equivariant pairing
V` (E) × V` (E) −→ Q` (1),
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called the Weil pairing. If E is a Tate curve, so that V` (E) is an explicit extension of Q`
by Q` (1), can you give an example of such a pairing? Can you describe the Weil pairing
explicitly in this case?
Show that every (D, N ) ∈ C arises as D` (V ) for some V ∈ RepQ` (GK ).
The construction of the functor D` = Dq` relied on a choice of Tate curve parameter q ∈ mK .
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What is the relationship between Dq` (V ) and Dq` (V ) for two different q, q 0 ∈ mK ? Are they
isomorphic in the category C ?
Assume the residue field F of K is finite. Then any (D, N ) ∈ C determines and is determined
by a pair (ρ, N ), where ρ is a continuous representation of the Weil group WK on the `-adic
vector space D, given the discrete topology. Ben (in his seminar) showed how to construct
from (ρ, N ) an `-adic representation V of GK . Is V isomorphic to V` (D, N )?
Do exercises 1.4.1-1.4.4 in Brinon-Conrad.
Let p be a prime, let n be an integer. Let q be the largest power of p such that q | n and
q 6= n. Show that the p-adic valuation of nq is 0 if n is not a power of p and 1 if n is a
power of p.
Let K be a p-adic field with finite residue field Fq , and let α ∈ GLn (CK ). Show that there
exists a continuous unramified linear (as opposed to semi-linear) CK -representation
GK
b
/ / GF ' Z
q
/ GLn (CK )
b to α if and only if all eigenvalues of the matrix α have absodefined by sending sending 1 ∈ Z
lute value 1. Use this to give an example of a continuous, n-dimensional GK -representation
with CK coefficients that does not factor through GLn (L) for any algebraic extension L/K.
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