I will give you some simplistic comments, which could make 
your proposal sound a little nicer.

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

   modularity conjecture for rigid Calabi-Yau threefolds, 
   formulated by Noriko Yui [Y].

I see Noriko is using this to her advantage :)   

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

Passive construction.

   By computation of examples I have also found new  phenomena,
                             ^^
                        insert comma

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

   William Stein, who is also applying to participate in this program,
   ^^^^^^

I regret not mentioning you in my MSRI-oriented paragraph.

   be useful to have the opportunity to talk about the arithmetic of
   K3 surfaces with Hiro Goto, who is applying
   for general membership, or 
    ^^^^^^^^^^^^^^^^^^^^
Do you know this only because a member of the hiring committee told you?

   about supersingular primes of elliptic curves. with Noam Elikies, or 
                                              ^^^^^^^^^^^
                                                huh?


related to number theory and arithmetic geometry, as described below,
                                                ^^^^^^^^^^^^^^^^^^^^
                                                   delete
   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 wrong.  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 [ ]."

Does this sound any crisper and easier to understand?

1) {\bf L-series of varieties:} I have explicitly computed the L-series of 
^^^^^
It looks better if you have space between enumerated items, and you don't.

   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 great big sentence that gives me the impression of slipping
along a multi-dimensional slide. 


   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 question of why YOU are the person capable of
   answering this question, and also 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?

Also I am interested in computing the L-series of other varieties.
One approach towards computing L-series, in the case of fibered varieties,
is to use the relation with the Picard-Fuchs equation of the
fibration.  
In [V1], [V2], and [V3],  
a fibration of a Calabi-Yau threefold is given,
$X\rightarrow{\mathbb P}^1$, with unifomizing parameter
$t$ of ${\mathbb P}^1$ a weight-$0$ modular function $t(\tau)$.
Then in certain cases, for $q=\exp(2\pi i\tau)$, one has
$$\frac{q}{t(\tau)}\frac{df}{dq} = g(\tau) + Ag(B(\tau),$$
for some integers $A,B$.  Here $f(\tau)$, 
a solution of the Picard-Fuchs differential equation,
is a weight-$2$ modular form.


I intend to pursue the study of computation of L-series
further, in particular with regard to 
fibrations over other spaces than~${\mathbb P}^1$, and other varieties, 
such certain
non rigid Calabi-Yau fibered threefolds.
For example, for what I call the $A_1\times A_2$ case in the notation of [V0], 
it is difficult to obtain either the L-series or the solutions of
the Picard-Fuchs equation, but perhaps, combining partial results
about the L-series and the Picard-Fuchs equation, one
might be able to compute both simultaneously.


2) {\bf Periods of modular forms:}  I have worked with modular symbols to
compute periods of modular forms.  In [SV] we introduce the idea of
``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
rational weight-$2$ modular forms.
Our algorithms  have applications to
verification of parts of the Bloch-Kato conjectures, and
also can be used to give a numerical approximation to the intermediate
Jacobian of rigid Calabi-Yau threefolds as is partly explained in [V2]
and [SV].

3) {\bf Images of modular representations:}  
I have computed the images
of several mod-$p$ modular representations, starting from 
the data given by the coefficients of a cuspidal eigenform.  
Currently, with Ian Kiming, I am working on related matters,
such as computation of more examples, and 
generalizing the approach taken in the examples to give an
algorithm, which, given an arbitrary  cuspidal eigenform $f(\tau)$, and
any prime $p$, will compute the image of the corresponding 
mod-$p$ Galois representation.  
Some of our results are in [KV1] and [KV2].
These results are obtained by applying
a generalization of results obtained by Kiming [K].

Currently my main interest is in understanding modular mod-$p$
Galois representations and their images.
Given a cuspidal eigenform
$f(\tau)$, with associated Galois representation $\rho: 
\Gal(\overline{\mathbb Q}/{\mathbb Q})\rightarrow\GL_2({\mathbb Q}_l)$, 
let $\rho_p: \Gal(\overline{\mathbb Q}/{\mathbb Q})
\rightarrow\GL_2(\overline{{\mathbb F}_p})$ be the 
semisimplification of a mod-$p$ reduction of $\rho$.
By work of Ribet [R], there are only finitely many $p$ for
which the  image of $\rho_p$ is not as large as possible,
but there is still no algorithm to give a list of all the $p$ for
which this happens.  I am hoping to obtain such an algorithm, and understand
the exceptional cases.   
The work I am currently undertaking with Ian
Kiming is a first step in this direction.

I am also interested in understanding the supersingular primes of
a cuspidal eigen form $f(\tau)$.
A prime $p$ is supersingular for a rational cuspidal eigenform
$f(\tau)=\sum a_n\exp(2\pi i\tau)$ if $p$ divides $a_p$.  
The Lang-Trotter conjecture  ([LT])
gives a conjectures  for the distribution of
supersingular primes, and Elkies [E] proved that there are
infinitely many supersingular primes for weight-$2$ modular forms with 
rational coefficients.  However, what happens in higher weight is not
yet known.   I hope my computations 
will give a better understanding of this problem, 
and will help me in working towards concrete results.


I believe that the environment of the MSRI Algorithmic number theory 
program will be excellent for work on the above projects.
I fully expect to
contribute to the program, with further advances in connections
with the above circle of ideas, in particular 
contributing  in the area of arithmetic geometry computations.

\begin{thebibliography}{SB}


\bibitem[C]{C} Cremona, J.:  
{\em Algorithms for modular elliptic curves, 2nd edition},
Cambridge University Press, (1997).


\bibitem[E]{E} Elkies, N.:  {\em The existence of infinitely many
    supersingular primes for every elliptic curve over ${Q}$.} Invent. Math. 89 (1987), no. 3, 561--567. 

\bibitem[K]{K} Kiming, I.:  
{\em On the liftings of $2$-dimensional projective Galois representations over
  $\bold Q$.} J. Number Theory 56 (1996), no. 1, 12--35. 

\bibitem[KV1]{KV1} Kiming, I and Verrill, H.: 
{\em Computing the image of modular Galois representations},   report, in
preparation.

\bibitem[KV2]{KV2} Kiming, I and Verrill, H.: 
{\em Twists of modular mod-$\ell$ representations with exceptional images},   
preprint, in preparation.

\bibitem[LT]{LT} Lang, S. and Trotter, H.:
{\em Frobenius distributions in ${\rm GL}\sb{2}$-extensions. }
Lecture Notes in Mathematics, Vol. 504. 

\bibitem[R]{R} Ribet, K.:  
{\em On $l$-adic representations attached to modular forms.} 
Invent. Math. 28 (1975), 245--275.

\bibitem[SV]{SV} Stein, W., and Verrill, H.:
{\em Computing period lattices of newforms}, in preparation.

\bibitem[V1]{V0} Verrill, H.: Root lattices and pencils of varieties, Kyoto Journal of mathematics, Volume 36. Number 2, July 1996, 421--446.


\bibitem[V1]{V1} Verrill, H.: 
 {\em The L-series of some Calabi-Yau threefolds}, to appear in the Journal 
of number theory.

\bibitem[V2]{V2} Verrill, H.: 
{\em The L-series of rigid Calabi-Yau threefolds from fiber products of
elliptic curves}, in preparation.

\bibitem[V3]{V3} Verrill, H.: 
{\em Some congruences related to modular forms}, Max-Planck-Institut f\"ur
Mathematik preprint 1999 (26).

\bibitem[VY]{VY} Verrill, H. and Yui, N: 
{\em Thompson series, and the mirror maps of pencils
of K3 surfaces}, to appear in the Proceedings of the NATO ASI/CRM Summer School ``The Arithmetic and Geometry of Algebraic Cycles" Banff Alberta Canada 1998.

\bibitem[Y]{Y} Yui, N.: 
{\em The arithmetic of Calabi--Yau varieties and mirror symmetry}
to appear from the Institute for Advanced Study.


\end{thebibliography}

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