\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, including application and development of algorithms. Some of my computational work is as follows: 1) {\bf L-series of varieties:} I have explicitly computed the L-series 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. Also I am interested in computing the L-series of other varieties. 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, 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:} In [V5] I computed the images of several mod-$p$ modular representations. Currently, with Ian Kiming, we are working on related matters, such as producing an algorithm to compute the image of an arbitrary mod-$p$ modular representation. This will generalize the method used in the examples in [V5]. As indicated, 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 and refine conjectures, and better understand them, and also to find new interesting phenomena. There is an interesting relationship between certain elliptic curves and Calabi-Yau threefolds given in [V2], for which I hope to find a good explanation. In these examples, $X$ is a rigid Calabi-Yau threefold. The Mellin transform of the L-series of $X$, is computed to be a weight-$4$ modular form $g(\tau)$. There is a fibration $X\rightarrow{\mathbb P}^1$, with $4$ singular fibres. Let $E$ be the elliptic curve obtained as a double cover of ${\mathbb P}^1$ branched at these $4$ points, and let $f(\tau)$ be the Mellin transform of the L-series of $E$. In each of the six examples in [V2] we have $$g(2\tau)=(f(\tau))^2.$$ The effort to simplify computations motivates other work. For example, to find the L-series of a fibered variety, one may look at the relationship to the Picard-Fuchs equation. Consider a rigid Calabi-Yau varieties with K3 fibrations to ${\mathbb P}^1$. Suppose that the parameter of the fibration is given as a weight-$0$ modular function $t(\tau)$, and the solution of the Picard-Fuchs differential equation is given as a weight-$2$ modular form $f(\tau)$, then in certain cases $$\frac{q}{t(\tau)}\frac{df}{dq} = g(\tau) + Ag(B(\tau),$$ for some integers $A,B$. This is referred to in [V1], [V2] and [V4], and is something I am interested in pursuing further. For example, I would like to elaborate on congruences results of Stienstra relating the L-series to solution of the Picard-Fuchs differential equation. I also intend to work with fibrations to other spaces than ${\mathbb P}^1$. Given the L-series of a fibred variety, one should be able to find the Picard-Fuchs equation, and conversely, the Picard-Fuchs equation should give the L-series. I would like to develop an algorithm for computation of the L-series and Picard-Fuchs equation for a fibred variety in tandem. I.e., rather than making a complete computation of one, then using this to find the other, make computations towards both, and combine these results to complete the computations simultaneously. 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 $\rho: \Gal(\overline{\mathbb Q}/{\mathbb Q})\rightarrow\GL_2({\mathbb Q}_l)$, one can define the reduced representation $\rho_p: \Gal(\overline{\mathbb Q}/{\mathbb Q}) \rightarrow\GL_2(\overline{{\mathbb F}_p})$. One knows, by work of Ribet, 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. %I working towards this by computation %of many examples, and also working on obtaining more abstract results with %Ian Kiming. I am also interested in understanding the super-singular primes of a cuspidal eigen form $f(\tau)$. A prime $p$ is super-singular for an integral modular form $f(\tau)=\sum a_n\exp(2\pi i\tau)$ if $p$ divides $a_p$. The Lang-Trotter conjecture gives some estimate for the distribution of supersingular primes, and Elkies proved that there are infinitely many super-singular 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, as many participants will be interested in the above topics. I hope to learn from other, to start new projects related to the above, and to contribute particularly in the area of arithmetic geometery computations. \begin{thebibliography}{SB} \bibitem[SV]{SV} Stein, W., and Verrill, H.: Computing period lattices of newforms, in preparation. \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}