REMARKS: preliminary readings, corrections, supplements, etc.
[04.01.2014] The list of fundamental problems at the end of class was meant to be incomplete anyway, but I wanted to state at least a couple of more problems in rep. theory of p-adic groups. Including the ones I already mentioned, here they are:
1. Classifying/constructing all irreducible admissible representations
- Subproblem: coarse classification - how to produce all irr. adm. reps from supercuspidal reps (for GL(n))
2. For two groups G and G', relate irr. reps of G(K) to those of G'(K).
- Langlands functoriality is an instance: we'll see two examples, which are Jacquet-Langlands and base change.
3. Classify by Galois data - local Langlands correspondence
4. Understand characters of irr. reps (e.g. explicit character formula)
5. Geometric rep. theory: Realize irred. reps or even a correspondence as in #2 or #3 above in the cohomology space of a suitable geometric object; use this to prove or reprove results (Example: To prove local Langlands for GL(n), people have used the etale cohomology of Lubin-Tate and Drinfeld spaces.)
[03.18.2014] I gave a classification of inertial types when n=2 into 4 cases: psi, psi_1 and psi_2 denote characters of I_K below.
(1) unramified up to a character: tau = [(psi+psi,0)]
(2) Steinberg: tau = [(psi+psi,N)], N is nonzero nilpotent
(3) split ramified: tau = [(psi_1+psi_2,0)], psi_1 \neq psi_2
(4) irreducible: tau = [(r_0,0)] is a restriction to I_K of an irreducible WD-rep
For the above list to be exhaustive and *mutuallly disjoint*, I have to impose the additional condition that tau is the restriction of *reducible* WD-rep in (3). This is the condition I missed in Today's class. (The main issue is that an irred. WD-rep may give rise to a reducible type of the form [(psi_1+psi_2,0)]. Also see Problem set 6. However one can show that such a type can't also come from a reducible WD-rep.) Then all the assertions I made about the irred. components of the generic fiber Spec(R^Box[1/ell]) are correct. Parts (i)-(iv) basically correspond to Thm 4.1.3 (1st assertion), Thm 4.1.5, Thm 4.1.3 (2nd assertion), and Thm 4.1.1 in [Pil], respectively.
[03.14.2014] I realized that I had abused notation in stating the local Euler-Poincare characteristic formula. I wrote chi(G_K,M)=dim_F(O_K/#M) =dim_F(M)dim_F(O_K/#F), where F is a finite field (over F_l). However O_K/#M and O_K/#F are obviously not F-vector spaces in general. To correct, dim_F(O_K/#M) should be log_{#F}(#(O_K/#M)) and likewise for dim_F(O_K/#F). Compare with [Mil-ADT] Theorem 2.8, where his definition of E-P characteristic is a little different. Apart from the different sign convention, Milne uses the actual cardinality rather than the dimension over F in the definition.
[02.27.2014] Today we reduced the proof of Carayol's lemma to the case A=F[e]/e^2 and B=F (F is the finite field over F_l) but didn't have time to finish. I could leave it there, but will spend the first 10 minutes or so next time (March 4) on the proof in that case, at least getting to the point where it's basically a routine check. It will also explain why we want to assume that \bar{\rho} is absolutely irreducible and not just Schur, though it should be possible to weaken the assumption in some special cases.
[02.25.2014] Today we had fantastic questions! Let me address them below.
1. Let me repeat that in stating the l-finiteness hypothesis Hyp(Gamma), we do restrict to *open* finite index subgroups Delta of Gamma. However the question still stands: "Is every finite index subgroup of a profinite group (automatically) open?"
It's not so easy to come up with counterexamples but they do exist. However when Gamma is topologically finitely generated, it is a big recent theorem by Nikolov and Segal that it is true! See the first page in introduction of [NS07] to see the history of the problem and further references. So we could dispense with the openness condition on Delta when Gamma is top. finitely generated after all.
2. When stating a lemma that every top. fin. generated Gamma satisfies Hyp(Gamma), we encountered the question "Is every open finite index subgroup Delta of Gamma also top. finitely generated?"
This sounds true but it's worth explaining why it's true. Let Gamma' be a dense finitely generated subgroup of Gamma. Then Delta \cap Gamma' is a dense subgroup of Delta and has finite index in Gamma'. So it suffices to invoke the fact that every finite index subgroup of a finitely generated group is again finitely generated. There's a nice online discussion of this fact.
3. For any two objects A and B of the category C_O (complete noetherian local O-algebras whose residue fields are F), is every O-algebra hom f:A->B automatically local (i.e. maps m_A into m_B)?
Right after class Yihang answered it affirmatively: As A and B are nonzero rings (0 is not 1), f^{-1}(m_B) is a proper ideal of A. On the other hand, we have an injection A/f^{-1}(m_B) -> B/m_B = F. As a (nonzero) subring of F is a field, f^{-1}(m_B) is a maximal ideal of A so equal to m_A.
[02.11.2014] At the end of today's class I asserted the following: Let rho be a Galois representation Gal(F_S/F) -> GL_n(k), where F_S is the maximal unramified-outside-S extension of F (in \bar{F}), and k is a topological field. (By definition rho is continuous.) I said that the coefficient of the char polynomial of rho(gamma) in each degree is a continuous function from Gal(F_S/F) to k. In fact you can take rho: Gal(F'/F) -> GL_n(k) for any Galois extension F'/F in place of F_S/F for this. Namely if we write
char(rho(gamma))= A_0(gamma) + A_1(gamma)T + ... + A_n(gamma)T^n
then the assertion is that each A_i:Gal(F'/F)->k is a continuous function. In case you don't see this easily, let me outline a proof and leave you to fill in the details.
1. Check that the trace map GL_m(k) -> k is continuous for every m>=1.
2. Let V:=k^n and \wedge^i(V) := i-th exterior power of V. Verify that the natural representation GL(V) -> GL(\wedge^i(V)) is continuous.
3. Check that A_i(gamma) = (-1)^i tr(\wedge^i(\rho(gamma))). Conclude that A_i is continuous.
[02.08.2014] Regarding 02.13 lecture, review the notion of unramified/tamely ramified/wildly ramified extensions of local fields, cf. [Mil-ANT] 7.48-7.59. Especially important is the structure of the tame inertia group, cf. [Ser-LF] IV.2, Exercise 2 or [Fro-LF] section 8, esp. Cor 2 and 3. A fuller discussion of the Weil groups and Weil-Deligne representations in a broader context is found in [Tat79] Sections 1 and 4.1. The original reference for Grothendieck's l-adic monodromy theorem is appendix of [ST68].
[02.08.2014] Regarding 02.11 lecture, here are some further references for basic facts in algebraic number theory. (1) Chebotarev density: [Mil-CFT] Theorem V.3.23; (2) Decomposition group in the global Galois group = local Galois group: [Mil-ANT] Prop 8.10 (via prime ideals) or [Neu] Prop II.9.6 (via valuations)
