Dear Helena,

I read your "Proof of equality of two L-series".  It looks fine, but
here are some comments which might make it more readable. 

1. In the 4th line of the statement of Theorem 4.3, delete "by K_s".

2. In condition 2 of Theorem 4.3, {\em i.} and {\em ii.} should both
   be indented.  [Yes, Livne's doesn't indent them, but he should have.]

3. In condition 2.i., shouldn't you say something about what "non cubic" 
   means?

4. In the second paragraph after the theorem, where it says 
   "2-adic cohomology", there is way too much space between 2, the dash,
   and the word "adic". 

5. In the next paragraph, line 3, where it says "rho : Gal(Qbar/Q)--->"
   there is too much space between rho and the colon and Gal(Qbar/Q).

  *General remark: You've typeset "Gal" and "GL" in math italic font.  
   I think it is better to typeset these in roman font, (e.g., as Serre 
   and Ribet in their Inventiones papers).  In AMSLaTeX the standard way 
   to do this is with the "DeclareMathOperator" command: 
        \usepackage{amsmath}
       	\DeclareMathOperator{\GL}{GL}
        \DeclareMathOperator{\Gal}{Gal}
        \DeclareMathOperator{\SL}{SL}

6. In the same paragraph (with the "rho:Gal(Qbar"), line 4, you 
   say "such that for all primes p, Tr(rho(Frob_p))=b_p...".  This
   does not make sense because Frob_p is only *defined* for p
   not dividing ell and N.  Put the "for all p not dividing ell*N"
   at the beginning of the sentence.

7. Last line of page 1: I would delete the "trust PARI, or" and just
   leave "(look at the first few terms of ...)."  People who know what
   PARI is will automatically think to use it.  

8. pg. 2, First line of proof of Lemma 1.2, you start using chi but,
   as far as I can tell, you've never even mentioned that you mean
   chi to be the p-adic cyclotomic character.

9. Line 4 of the proof: replace "resepcts" by "respects".
 
10. Line 7 of the proof: Technically it doesn't make sense to 
   write "det Frob_p".  Instead you should probably write 
   "det (rho_1(Frob_p))".   

  * General Remark:  
   I just realized that you are implicitely using something of the 
   nature: "if det (rho_1(Frob_p)) = det(rho_2(Frob_p)) for all p
   for which this makes sense, then det(rho_1(x))=det(rho_2(x)) for
   ALL x in Gal(Qbar/Q)."  
   This is nontrivial, and relies on the Cebetarov Density theorem.   
   Perhaps it's worth mentioning this.   

11. Proof of lemma 1.2, lines 6-7:  Replace "eigen values" by "eigenvalues".
    You say "alp_1, alp_2, then by duality, they are also 
    p^3alp_1, p^3alp_2".   Replace "p^3alp_1, p^3alp_2" by 
    "p^3/alp_1, p^3/alp_2". 
   
12. Line -4 from end of proof of Theorem 1.3: "eg" should be "e.g.,"
    I think.  Am I right about this? 
    
13. Next line: capitilize "theorem". 



  XXX,    
   William