\documentclass[12pt]{article} \usepackage{amsmath} \usepackage{amssymb} \usepackage[dvips]{rotating} \newcommand{\email}{{\tt verrill@math.ku.dk}} \newcommand{\myname}{\LARGE Statement of Purpose} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Gal}{Gal} %\textwidth=1.2\textwidth \textwidth=1.2\textwidth %\hoffset=-3em \hoffset=-6em \textheight=1.1\textheight \usepackage{fancyhdr,ifthen} \pagestyle{fancy} \cfoot{} % no footers (in pagestyle fancy) % running left heading \lhead{{\bfseries\LARGE\em \noindent{}\hspace{-.2em}\myname{}} \hfill \today{}\vspace{2ex}\\ \large{Helena A. Verrill}} % running right heading \rhead{ \email{}} % adjust, because the header is now taller than usual. \setlength{\headheight}{8ex} \rfoot{\begin{tabular}{ll} \multicolumn{2}{l}{$\phantom{nothing}$}\\ \multicolumn{2}{l}{ Research Statement: page \thepage/2}\\ See also& CV\\ %&teaching statement\\ &cover letter\\ &abstracts \end{tabular} } % bold sans-serif label for margin labels. \newcommand{\marginlabel}[1]{\textsf{\textbf{#1}}} %\newcommand{\marginlabel}[1]{\textsc{\textsc{#1}}} \newcommand{\entrylabel}[1]{\mbox{\marginlabel{#1}}\hfill} \begin{document} Much of my recent work has involved making explicit number theoretical computations, which has included the application and development of algorithms. These computations arise from and are connected with many different areas of mathematics such as the study of Calabi-Yau threefolds, the Bloch-Kato conjectures and modular Galois representations. I am particularly interested in computations and algorithms related to the theory of modular forms. Despite recent major advances in this area, there are many difficult outstanding problems, such as the Birch--Swinnerton-Dyer conjecture, the related Bloch--Kato conjecture, and the Lang--Trotter conjecture. Computation allows one to verify conjectures in specific examples, which give better understand of the conjectures, and may lead to refinements of the conjectures. Computations can also to find new interesting phenomena. Since I am interested in a wide range of computational problems related to number theory and arithmetic geometry, I am convinced that the MSRI Algorithmic Number Theory program will be an excellent opportunity for me to interact fruitfully and successfully with other participants interested in problems of a similar nature. Some of my computational work is as follows: 1) {\bf L-series of varieties:} I have explicitly computed the L-series of the middle cohomology of several rigid Calabi-Yau threefolds ([V1], [V2]), obtaining weight-$4$ modular forms. I am interested in using the methods to produce an algorithm for giving the L-series of rigid Calabi-Yau threefolds more generally. * I believe an algorithmic approach should be possible, given the defining equations over ${\mathbb Q}$ for an affine piece of the Calabi-Yau variety in ${\mathbb P}^4$ 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 may be given by a polynomial in $p$ of degree $3$, with coefficients involving certain Kronecker symbols, and possibly there may be involved the coefficients of modular forms of lower weight. In any case, one might hope it is possible to get some handle on this. One result one might work towards would be to extend Livn\'e's work, (based on Serre and Faltings work). Livn\'e gives a result telling you how many coefficients of the L-series of two Galois representations one has to compute and show equality of the L-series. This theorem has become an important result in the area of computing L-series. One might hope to extend 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? * Also I am interested in computing the L-series of other varieties. One approach towards computing L-series, in the case of fibred varieties, is to use the relation with the Picard-Fuchs equation of the fibration. In [V1], [V2], and [V4], this approach is refered to. There, a fibration of a Calabi-Yau threefold is given, $X\rightarrow{\mathbb P}^1$, with unifomising 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 $A,B$, where $f(\tau)$, a weight-$2$ modular form, is the solution of the Picard-Fuchs differential equation. I intend to persue this further, in particular with regard to fibrations to other spaces than ${\mathbb P}^1$, and other varieties, such certain non rigid Calabi-Yau fibred 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 [V3] I introduce the idea of ``transportable modular symbols'', and in [SV] we 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 ([V2], [SV]). 3) {\bf Images of modular representations:} Currently my main interest is in understanding modular mod-$p$ Galois representations and their images. Given a cuspidal eigen-form $f(\tau)$, 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 reduced representation of $\rho$. For example, one may take reduced representations coming from mod-$p$ modular forms. In [V5] I computed the images of several mod-$p$ modular representations. Currently, with Ian Kiming, we are working on related matters. In particular, we give a proof that the comuptations given in [V5], which required certain assumptions, do give the stated images, and we generalise the approach 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. These results are obtained by applying a generalisation of results obtained by Kiming [K]. 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 some estimate 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 on this problem. I believe that the environement of the MSRI Algorithmic number theory program will be excellent for work on the above projects I fully expect to be able to contribute to the program, with further advances in connections with above circle of ideas, in particular contributing in the area of arithmetic geometery computations. \begin{thebibliography}{SB} \bibitem[C]{C} Cremona, J.: \bibitem[E]{E} Elkies, N.: \bibitem[K]{K} Kiming, I.: \bibitem[LT]{LT} Lang and Trotter: \bibitem[R]{R} Ribet, K.: \bibitem[SV]{SV} Stein, W., and Verrill, H.: Computing period lattices of newforms, in preparation. \bibitem[V0]{V0} Verrill, H.: Root Lattices and Picard-Fuchs equations. \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 fibre products of elliptic curves}, in preparation. \bibitem[V3]{V3} Verrill, H.: {\em Transportable modular symbols}, preprint. \bibitem[V4]{V4} Verrill, H.: {\em Some congruences related to modular forms}, Max-Planck-Institut f\"ur Mathematik preprint 1999 (26). \bibitem[V5]{V5} Verrill, H.: {\em Computing the image of modular Galois representations: Examples}, preprint. \end{thebibliography} \end{document}