Hi Frank,

Here are some comments.  Feel free to reply to this email regarding my
comments, if you want.  If I'm totally wrong about something, I want
to know!

Overall:

It's a great research plan and the length is just right. 

Specific Remarks:

 - Definition 1.  For fun, I challenge you to find a person that knows 
   exactly what an "overconvergent p-adic modular form of tame level N and 
   \kappa is" but does not know exactly what the slope of such a form is.
   I'm not suggesting you change anything; I'm just making a remark.

 - Is the "equivalent reformulation" of Gouvea-Mazur in terms of
   overconvergent forms, which you allude to in 3.1, really equivalent?  
   One might guess that it is stronger, because it says something about
   all overconvergent forms, even the non-classical ones.  

 - I'm unpracticed with overconvergent modular forms.  I take it that
   you don't have to say for k>>0 in Conjecture 1 like you must do with
   classical modular forms. 

 - I don't quite like your statement of Buzzard's conjectures.  We can
   also use modular symbols or trace formulas to "predict all the
   slopes of particular integral weight and level" and the modular
   symbols and trace formula algorithms are also "combinatorial,
   although they are effective."  Could you say how Kevin's are
   distinguished?  They are a simple recursive formula for the slopes,
   that doesn't require computed anything else, such as Hecke
   operators.

 - You say "rhofbar is always the reduction of some crystalline 
   representation", but rhofbar is a representation of the global
   G_{Q,S} instead of the local D_p, and you've only defined "crystalline"
   for local representations.   Do you mean rhofbar|D_p is the reduction
   of a crystallline rho : D_p ---> GL_2(k) for every single p?  Or,
   do you mean there's a rho : G_{Q,S} --> GL_2(k) that is such that
   rho|D_p is crystalline for all p?

 - That quotation looks really weird.  Are you sure you didn't delete too
   much or change something?  Surely, in the phrase "consider, given a residual
   representation $\rb$, the set of distinct eigenforms of level
   $p$, all weights, \emph{and coefficients in} $\Z_p$", there should
   be some requirement that the eigenforms reduce to $\rb$?

 - "[\mathbb T_k \otimes_{\Q} \Qbar:\Q] = \dim(S_k)"
   What??? First, that ":\Q]" should be ":\Qbar]".  Second,
   is S_k classical over overconvergent.  I should first assume
   the latter because you defined M_k to be overconvergent.
   (You never define S_k.)  But then \dim(S_k) is infinite...

- You say: "Let $\Tp$ be the classical Hecke algebra of modular
  forms of weight $k$ on  $\Gamma_1(N)$."   and 
      $$\Tp \otimes \Qbar_p = \prod K_i$$
  I don't believe this, in general, unless you remove the Hecke 
  operators T_n from \Tp where gcd(N,n) =/= 1.  In general, 
  "the classical Hecke algebra" will have nilpotent elements,
  and tensoring with \Qbar_p does nothing to kill them.

- For conjecture 2, you might give Kevin's special case that 
  the bound for p=2 and level 1 is 1 (I think!!! Double check this or
  try the computation yourself..)


Non-mathematical Remarks:

 - On page 1, Dwork sticks out into the margin

 - A secret: Ribet told me that Serre prefers "J-P. Serre" 
   to "J.-P. Serre". 

 - It looks like you mostly follow the Ribet "First time the name
   appears use the initial for the first name" rule.  It looks like 
   you violate it with "Gouvea".

 - I would write "Section 4.2" instead of "section 4.2", as you do
   in the introduction.

 - The title of Section 2.2 is "Almost rational torsion points",
   which suggests that you are using the generally European convention
   of capitalizing only the first word of titles.  However, this is
   not the case, as the title of Section 3.1, "Explicit Slope Conjectures"   
   reveals.

 - "Control theorems of Coleman [7],[8] ..."
   There should be space between the command and "[8".

 - The sentence above Conjecture 1 suggests that "(tau^k, k)" is what you
   mean by an "integral weight".  However, the statement of Conjecture 1
   suggests that by "integral weight" you mean an integer.

 - Why "Theorem 2 (Buzzard,Calegari)" instead of 
       "Theorem 2 (Buzzard, Calegari)"??

 - The sentence "This local situation is in stark contrast to what is "
   sounds clumsy to me.  I might write something more like: 
      "This local situation sits in stark contrast to what is "
                            ^^^^
   or 
      "This local situation is strikingly different from the global one."
  
 - This sentence is awkward; it almost made me fall over when I read it:
    "It seems, however, that classical 
    eigenforms having small fields of definition over $\Q_p$
    is the norm rather than the exception."
    
 - In the biblio: "construction d'un anneau de Barsottii--Tate.},"
   Drop the "."??

    
Best regards,
  William