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Author: William A. Stein
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%CV.tex -- the ultimate CV LaTeX file, September 1999.
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Berkeley, CA 94709\\
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USA}
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\begin{document}
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\begin{MainList}{104pt}
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\item [EDUCATION]
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\begin{bulletlist}
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\item {\bf University of California at Berkeley}\\
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Ph.D., mathematics, expected May 2000.\\
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{\bf Explicit approaches to modular abelian varieties}\vspace{-2ex}\\
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\item {\bf Northern Arizona University}, Flagstaff.\\
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M.S. student, mathematics, 1994--1995.\\
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B.S., mathematics, 1994.
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\end{bulletlist}
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\item [AWARDS]
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\begin{bulletlist}
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\item Cal@SiliconValley university fellowship, 1999--2000.
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\item Sarah M.\ Hallam department fellowship, Spring 1999.
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\item Vice Chancellor research grant (computing equipment), 1999.
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\item Graduate student researcher, Fall 1998.
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\item Outstanding mathematics senior, 1994.
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\item Applied math modeling contest, meritorious ranking, 1994.
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\end{bulletlist}
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\item [IN PROGRESS]
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% Some of the following items can be obtained from\\
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% \mbox{}
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% \hspace{2em} {\tt http://shimura.math.berkeley.edu/\~{}was/Tables}
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\begin{bulletlist}
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\item {\em Lectures on Serre's conjectures},
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with K.\thinspace{}A.~Ribet, to appear in the
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IAS/Park City Mathematics Institute Lecture Series.
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\item {\em Mod~$5$ approaches to modularity of icosahedral
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Galois representations}, with K.\thinspace{}M.~Buzzard.
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\item {\em Explicit approaches to modular abelian
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varieties}, UC Berkeley Ph.D.\ thesis
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under H.\thinspace{}W.~Lenstra.
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\item {\em Hecke: The modular forms calculator}, computer software.
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\item {\em The modular forms database:}\\
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\mbox{}\hspace{3em}
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{\tt http://shimura.math.berkeley.edu/\~{}was/Tables}
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\item {\em Component groups of optimal quotients
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of Jacobians}.
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\item {\em Visibility of Shafarevich-Tate
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groups of modular abelian varieties}, with A. Agash\'{e}.
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\item {\em Computing analytic invariants of modular eigenforms},
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with H.~Verrill.
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\item {\em Lectures on modular forms and
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Galois representations}, with K.\thinspace{}A. Ribet,
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intended for Springer-Verlag's Universitext series.
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\end{bulletlist}
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\item [PUBLICATIONS]
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\begin{bulletlist}
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\item {\em Empirical evidence for
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the Birch and Swinnerton-Dyer conjectures for modular Jacobians of
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genus 2 curves}, with E.\thinspace{}V.~Flynn, F.~Lepr\'{e}vost,
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E.\thinspace{}F.~Schaefer,
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M.~Stoll, J.\thinspace{}L.~Wetherell, submitted.
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\item {\em Parity structures and generating functions from Boolean rings},
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with D.\thinspace{}P.~Moulton, submitted.
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\item {\em Fallacies, Flaws, and
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Flimflam \#92: An Inductive Fallacy},
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with A.~Riskin, College Math.\ Journal 26:5 (1995), 382.
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\end{bulletlist}
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\newpage
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\item [TEACHING]
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{\bf NSF Sponsored IAS/Park City Mathematics Institute}
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\begin{bulletlist}
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\item {\em Teaching Assistant}, Summer 1999. Led problem sessions
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and prepared notes for Ken Ribet's course on Serre's
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conjectures for advanced number theory graduate students.
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\end{bulletlist}
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{\bf University of California at Berkeley}
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\begin{bulletlist}
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\item {\em Curriculum Development}, Fall 1997--Summer 1998. Developed
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curriculum materials and {\sc Matlab} software for workshop based
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calculus and linear algebra courses at UC Berkeley.
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\item {\em Instructor}, Summer 1997. Taught discrete mathematics to
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a class of 20--30 undergraduates. Duties included preparing and delivering
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lectures, writing and grading exams, and conducting office hours.
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\item {\em Teaching Assistant}, Fall 1995--Spring 1997.
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Led discussion sections for the full range of undergraduate
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linear algebra and calculus courses.
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\item {\bf Evaluation average: 5.8 out of 7 from 300 students.}
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\end{bulletlist}
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{\bf Northern Arizona University}
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\begin{bulletlist}
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\item {\em Teaching}, 1994--1995. Taught two semesters
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of college algebra and other freshman topics to a class
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of 30--40 undergraduates.
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Duties included all aspects of organizing a course, including
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preparing and delivering lectures and grading examinations.
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\end{bulletlist}
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\item [LECTURES]
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\begin{bulletlist}
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\item Delivered over twenty talks and organized graduate student
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seminars on modular forms and abelian varieties
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at UC Berkeley.
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\item {\em Participant in panel discussion on the use of technology
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in the classroom},
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Park City Math.\ Inst., July, 1999.
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\item {\em Demonstrations of number theory software},
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Park City Math.\ Inst., July, 1999.
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\item {\em Shafarevich-Tate groups of modular abelian varieties},
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Park City Math.\ Inst., June, 1999.
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\item {\em Shafarevich-Tate groups of modular abelian varieties},
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Advances in Number Theory, Leiden, Netherlands, April, 1999.
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\item {\em Visibility of Shafarevich-Tate groups},
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Arizona Winter School, 1999.
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\end{bulletlist}
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\item [COMPUTING]
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Extensive experience with {\tt C++}, UNIX, {\sc Magma}, and PARI.
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\item [ADDRESS]
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\address{}
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\item [PERSONAL]
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US Citizen, born February, 1974.
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\newpage
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\item [REFERENCES]
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\begin{bulletlist}
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\item {\bf Kevin M. Buzzard}\\
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{\em +44 207 594 8523}\\
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Department of Mathematics\\
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Huxley Building\\
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Imperial College\\
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180 Queen's Gate\\
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London, SW7 2BZ\\
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England\\
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{\tt buzzard@ic.ac.uk}
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\vspace{1ex}
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\item {\bf Professor Robert Coleman}\\
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{\em (510) 642-5101}\\
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Department of Mathematics \#3840\\
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University of California\\
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Berkeley, CA 94720-3840\\
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{\tt coleman@math.berkeley.edu}
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\vspace{1ex}
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\item {\bf Professor Hendrik W. Lenstra}\\
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{\em (510) 643-7857}\\
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Department of Mathematics \#3840\\
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University of California\\
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Berkeley, CA 94720-3840\\
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{\tt hwl@math.berkeley.edu}
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\vspace{1ex}
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\item {\bf Professor Barry Mazur}\\
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{\em (617) 495-2171 ext.~512}\\
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Department of Mathematics\\
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Harvard University\\
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One Oxford Street\\
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Cambridge, MA 02138\\
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{\tt mazur@abel.math.harvard.edu}
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\vspace{1ex}
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\item {\bf Professor Lo\"\i{}c Merel}\\
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{\tt merel@math.jussieu.fr}
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\vspace{1ex}
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\item {\bf Professor Kenneth A. Ribet}\\
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{\em (510) 642-0648}\\
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Department of Mathematics \#3840\\
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University of California\\
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Berkeley, CA 94720-3840\\
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{\tt ribet@math.berkeley.edu}
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\end{bulletlist}
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\newpage
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\item [ABSTRACTS]
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\begin{bulletlist}
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\item {\em Lectures on Serre's conjectures (77 pages):}
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This is an expository paper based on Ken Ribet's lectures at
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the 1999 Park City Mathematics Institute; it will be
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published in the conference proceedings.
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\item \vspace{1ex}{\em Mod~$5$ approaches to modularity of icosahedral
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Galois \nobreak{representations} (16 pages):}
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Consider a continuous odd irreducible representation
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$\rho:\mbox{\rm Gal}(\overline{\mathbf{Q}}/\mathbf{Q})
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\rightarrow\mbox{\rm GL}_2(\mathbf{C}).$
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A special case of a general conjecture of Artin
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is that the $L$-function $L(\rho,s)$ associated to~$\rho$
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is entire.
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Buzzard and I give new examples of representations~$\rho$ that satisfies
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this conjecture. These were obtained by applying a recent
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theorem of Buzzard and Taylor to a mod~$5$ reduction of~$\rho$,
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combined with a computational verification of modularity of a
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related mod~$5$ representation.
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\item \vspace{1ex}{\em Explicit approaches to modular abelian
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varieties (130 pages):}
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I give algorithms for computing on modular
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abelian varieties of any dimension.
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More precisely, I describe how to compute congruences, the modular degree,
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the rational part of the special value of the $L$-function and
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of its twists, the component group at primes of multiplicative
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reduction, the period lattice, and the real and imaginary volumes.
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There are still many invariants that I have not been able to
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compute in all cases, including the exact structure of the torsion
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subgroup and the regulator.
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The second part of my thesis contains
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investigations into several open problems, including
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the Birch and Swinnerton-Dyer conjecture, Artin's conjecture
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on complex Galois representations, and Serre's conjecture.
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\item\vspace{1ex} {\em Hecke: The modular forms calculator:}
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Hecke is a {\tt C++} package for computing with spaces of modular forms
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and modular abelian varieties.
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I have been invited to visit the {\sc Magma} group in
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Sydney in order to make Hecke a part of their computer
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algebra system, and I have already ported my code to {\sc Magma}.
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\item\vspace{1ex} {\em The modular forms database:}
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This is a collection of modular eigenforms,
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special values of $L$-functions, arithmetic invariants of
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modular abelian varieties, and other data.
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These tables, which are freely
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available on the Internet, have already been used
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by many people.
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\item\vspace{1ex} {\em Component groups of optimal quotients of Jacobians (16 pages):}
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Let~$A$ be an optimal quotient of~$J_0(N)$.
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The main theorem of this paper gives a relationship between
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the modular degree of~$A$ and the order of the component group
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of~$A$. From this I deduce a computable formula for the
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component group of any optimal \nobreak{quotient}
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of~$J_0(N)$ at a prime of multiplicative
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reduction. I then compute over one \nobreak{thousand}
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examples leading me
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to conjecture that the torsion and component groups of
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quotients of $J_0(p)$ are as simple as possible.
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\item\vspace{1ex} {\em Visibility of Shafarevich-Tate
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groups of modular abelian varieties (20 pages):}
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We study Mazur's notion of visibility
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of Shafarevich-Tate of modular \nobreak{abelian} varieties,
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and use it to verify the conjecture of Birch and Swinnerton-Dyer
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for several specific abelian varieties.
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\item\vspace{1ex} {\em Lectures on modular forms and
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Galois representations (170 pages):} In 1996, Ken Ribet taught an
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advanced course on modular forms and Galois representations.
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In collaboration with Ribet, I am
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turning my course notes into a book that is intended for
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publication in Springer-Verlag's Universitext series.
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\item\vspace{1ex} {\em Empirical evidence for the Birch and
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Swinnerton-Dyer conjecture for modular Jacobians
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of genus~2 curves (22 pages):}
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We provide systematic numerical evidence for the BSD conjecture
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in the case of dimension two.
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This conjecture relates six quantities associated to
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a Jacobian over the rational numbers. One of these
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quantities is the size~$S$ of the Shafarevich-Tate group.
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Unable to compute~$S$ directly, we compute the five other
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quantities and solve for the conjectural value~$S_?$ of~$S$.
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For all 32~curves considered, the real number~$S_?$
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is very close to either~$1$,~$2$, or~$4$, and
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agrees with the size of the 2-torsion of the
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Shafarevich-Tate group, which we could compute.
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\item\vspace{1ex} {\em Parity structures and generating functions from Boolean rings
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\hspace{.2em}(8 pages):}
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Let~$S$ be a finite set and~$T$ be a subset of the power set of~$S$.
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Call~$T$ a {\em parity structure} for~$S$ if, for
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each subset~$b$ of~$S$ of odd size, the number of subsets of~$b$
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that lie in~$T$ is even. We classify parity structures using generating
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functions from a free boolean ring.
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We also show that if~$T$ is a parity structure, then, for each
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subset~$b$ of~$S$ of even size, the number of subsets of~$b$ of
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odd size that lie in~$T$ is even. We then give several other
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properties of parity structures and discuss a generalization.
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\end{bulletlist}
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\end{MainList}
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\end{document}
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%There are 9 lectures: Introduction to Serre's conjectures;
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%The weak and strong conjectures; The weight in Serre's conjecture;
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%Galois representations from modular forms; Introduction to level lowering;
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%Approaches to level lowering; Mazur's principle; Level lower
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%without multiplicity one; Level lowering with multiplicity one;
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%Other directions.
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