Dear Lloyd,

Here are some comments about your "Some examples of non-Gorenstein
Hecke algebras..." paper.

1) Change "magma" to either "Magma" or "MAGMA".

2) You might want to change the reference [11] to 
     'Amod Agashe and William Stein: Apendix to Joan-C. Lario \ 
      and \ Ren\'e Schoof: Some computations with Hecke 
      rings and deformation rings.'
   since the latter will be published, whereas [11] won't be.
   The content is similar, except the latter is only one page
   long and is more careful. 

3) >"...certain maximal prime ideal"
   Using both "maximal" and "prime" is redundant.  Just write   
   "...certain maximal ideal"
   
4) >"proving that the localizations of these rings at their maximal
   > ideals m are Gorenstein."
   I would write "at certain of their maximal ideals".

5) >"these forms are can be attached to elliptic curves, using the
   > Shimura-Taniyama-Weil conjecture; in our case, we are dealing with
   > semistable elliptic curves, so we refer to
   > Wiles-Taylor-Wiles~\cite{taylor-wiles,wiles} for the proof. We call
   > the elliptic curve associated to the modular form $E$."

   If you start with the form and then produce the elliptic curve, you
   are **NOT** using the Shimura-Taniyama-Weil conjecture.  (Watch out --
   if you include "Weil" in the name of the conjecture, then Lang might 
   try to beat you up.)  The association f |---> E = E_f is due to 
   Shimura; see, I think, Theorem 7.4 (??) of his book "Introduction to
   the Arithmetic Theory of Automorphic Forms".

6) >"those used in the first draft of the proof"
   
   The "first draft" of Wiles's proof is very famous for being incorrect.
   Some people might think you mean that version.  Clarify.

7) > "then the trace is even. If $r$ splits into two primes, 
   > then the trace is odd."

   The trace is an element of the finite field F2.  It sounds funny
   calling 0 "even" and 1 "odd".

8) > the image of $Frob_r$ is the identity in the 
   Should be {\rm Frob}.

9) > equivalent to the representation being of mod~2 multiplicity one.
   Do you mean "mod~$\ell$?"

10) > "Proposi
tion 2.4 of that paper."
    Hopefully the broken proposition is a result of using email...

11) > "These calculations were made possible with the hecke package,
    > running on the magma operating system. Magma provides an environment
    > for specialised number-theoretic calculations, and also incorporates

    a) Write {\sc Hecke} or Hecke.  
    b) Magma, MAGMA, or maybe even {\sc Magma}, but not "magma". 
    c) Magma is *not* an operating system, it is a computer 
       algebra system.   Linux is an operating system.

12) > "In Ribet-Stein~\cite{ribet-stein} the existence of
    > non-Gorenstein Hecke algebras is discussed in the context of the
    > level-lowering procedure associated with Serre's conjecture (see
    > section 3.7.1)."

    Ribet is trying to change the name from "level-lowering" to 
    "level optimization".  I think we use the latter term in our paper.

13) > "so do all give the Galois representation."

    Huh? 

14) > "We now use the hecke.m package, on the Magma system"
   
    Huh?  The "hecke.m package" sounds a bit too informal.

15) > "Magma provides an environment for specialised number-theoretic
      calculations"

Change to "Magma includes an environment...", since Magma does a lot
more than just number theory.

16) > "using the $\magma$ command ``AbelianIntersection''"

I explain the theory behind this command, along with a proof in Agashe
and Stein, "Visibility of Shafarevich-Tate groups of abelian
varieties: Evidence for the Birch and Swinnerton-Dyer conjecture".
(That paper is still changing a little, so I can't provide a section
number.)

17) > "the Gorenstein property fail to hold here."
                               ^^^^

18) > \begin{question} Are there infintely many prime and squarefree $q
    > \in \zed$ such that $\hecke{q}$ is not Gorenstein, and hence where
    > mod~2 multiplicity one fails?  \end{question}

    a) You mistyped "infinitely". 
    b) "square-free" instead of "squarefree"?  
       but this is stupid styly issue, and who cares.  
       Knuth says use "squarefree".
    c) It is a very difficult open problem to prove there are infinitely
       elliptic curves of prime conductor.   (i just noticed that 
       you mention this later)
  
19) 
   > of the factorisation of the characteristic polynomial of the Hecke
   > operator $T_3$ at level 431, and the characteristic polynomial of
   > $T_11$ at level 503, identifying factors of the polynomials with
   > conjugacy classes. These are the first characteristic polynomials
   > where all the forms are distinct.

   a) You should mention this section earlier in the paper, if you didn't   
      already.  
   b) Replace $T_11$ by $T_{11}$.

20) Bibliography: 
   I prefer "W.\thinspace{}A. Stein" to "W. Stein".