Much of my recent work has involved making explicit number theoretical
                                       ^^^^^^
                                change to "doing" or "carrying out"?

   Verification 
   can lead to better understanding, and refinement of the conjectures.

Passive construction.

   The algorithms I have recently been working on are an algorithm for
                                              ^^^^^^^^^^^^^^
                                          sounds unnatural...

   about supersingular primes of elliptic curves. with Noam Elikies, or 
                                              ^^^^^^^^^^^
                                               
   At Copenhagen University,
   where with Ian Kiming we are building on my computational work, which has
   motivated the production of more general theorems and methods.

This sentence just sounds a little choppy.  How about,

"At Copenhagen University Ian Kiming and I are building on our past 
 computational work; this has resulted in our joint discover of a 
 theorem about [ ] and a method to [ ]."

1) {\bf L-series of varieties:} I have explicitly computed the L-series of 
^^^^^
It looks better if you have space between enumerated items.

   not a simple way to construct a completion of $X$ in each case.
                                             ^^^^^^^^^
                                             What is X?

   This is because the cohomology of a 
   Calabi-Yau threefold is very simple, and although the number of points
   on a Calabi-Yau threefold 
   over ${\mathbb F}_p$ does not give the coefficient $a_p$ of 
   the L-series of $H^3(X,{\mathbb Q}_\ell)$ in such as simple way as one
   obtains coefficients of the L-series of $H^1(E,{\mathbb Q}_\ell)$ for
   an elliptic curve $E$, the difference is typically given by a polynomial in
   $p$ of degree $3$, with coefficients involving certain Kronecker symbols,
   and possibly also coefficients of modular forms
   of lower weight.  

This is a big sentence.


   I have been looking at the possibility of extending 
   Livne's result in various ways, such as, if
   one knows that the L-series of two varieties differ by a polynomial in
   $p$ for enough $p$, can one conclude they are equal?  

Consider the following slight modification:   

   I am attempting to extend Livne's result; my investigation 
   is driven by the following question: if we know that the 
   $L$-series of two varieties differs by a polynomial in~$p$,
   for enough~$p$, are they are equal?  
   I have verified this in [ number ] of cases, and am
   aware of no counterexamples.
   [Answer the questions of why YOU are the person capable of
   answering this question and why it is an important question.]

By the way, this statement of yours strikes me as absurd.  If the
$L$-series are really the same, as you hope to conclude, then the
polynomial must be 0, as you've verified for the first n terms that
the $L$-series are the same.  But then Livne's original result
applies.  Do you really mean that the $L$-series differ by a
polynomial?  (I suppose I know the answer because I know your work,
but most readers won't.)

  non rigid Calabi-Yau fibered threefolds.
    ^^^
I prefer "non-rigid"


   ``transportable modular symbols'', and  give an algorithm to
   compute the period lattice of any modular form of weight $k\ge 2$,
   generalizing work of Cremona [C], who computed period lattices of

It sounds wrong; who is generalizing?

   I have computed the images
   of several mod-$p$ modular representations, starting from 
             ^^^^^^^^^
      I asked Ken his opinion between "mod-p" and "mod p$ and he
      said he prefers the later because it is most comment, though 
      the first makes more sense gramatically.  It's always best 
      to do what Ken does.


   \rightarrow\GL_2(\overline{{\mathbb F}_p})$ be the 
                   ^^^^^^^^^^^^^^^^^^^^^^^^^^

I recommend "\overline{\mathbb F}_p"

   a cuspidal eigen form $f(\tau)$.
                ^^^^^^^
              eigenform  (just copy what everyone else does here)
          or "cuspform"
         
   The Lang-Trotter conjecture  ([LT])
   gives a conjectures  for the distribution of
      ^^^^^^^^^^^^^^


-- William