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Author: William A. Stein
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\newcommand{\myname}{William A.\ Stein}
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Berkeley, CA 94709\\
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USA}
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\begin{document}
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\begin{center}
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\LARGE\bf \doc
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\end{center}
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\mysection{Introduction}
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My research program reflects the essential interplay between abstract
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theory and explicit machine computation during
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the latter half of the twentieth century;
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it sits at the intersection of recent work of
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B.~Mazur, K.~Ribet, R.~Taylor, and A.~Wiles
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on Galois representations
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with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
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on explicit computations involving modular abelian varieties.
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My work on the Birch and Swinnerton-Dyer
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conjecture for modular abelian varieties and search for new
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examples of modular icosahedral Galois representations
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has led me to discover and implement algorithms for
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explicitly computing with modular forms.
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\mysection{Objectives}
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The main outstanding problem in my field
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is the conjecture of Birch and Swinnerton-Dyer (\nobreak{BSD} conjecture),
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which ties together the constellation of
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arithmetic invariants of an
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elliptic curve.
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There is still no general class of elliptic curves for which
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the full BSD conjecture is known to hold.
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Approaches to the BSD conjecture that rely on congruences
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between modular forms are likely to
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require a deeper understanding of the analogous
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conjecture for modular abelian varieties, which are higher
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dimensional analogues of elliptic curves.
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As a first step, I have obtained theorems that make possible
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computation of some of the arithmetic invariants of
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modular abelian varieties.
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My objective is to find ways to explicitly compute all of
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the arithmetic invariants.
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Cremona has enumerated these invariants for the first
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few thousand elliptic curves, and I am working to do the same
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for abelian varieties.
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It is hoped that this work will continue to
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yield theoretical results.
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I am also writing modular forms software that I hope will be used
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by many mathematicians and have practical applications in
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the development of elliptic curve cryptosystems.
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My long-range goal is to give a general hypothesis, valid for
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infinitely many abelian varieties,
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under which the full BSD conjecture holds.
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My approach involves combining Euler system techniques of K.~Kato and
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K.~Rubin with visibility and congruence ideas of Mazur and Ribet.
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\mysection{Modular abelian varieties}
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My primary objective is to verify the BSD conjecture for
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specific modular abelian varieties, by using the
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rich theory of their arithmetic.
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The BSD conjecture asserts that if~$A$ is a modular abelian
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variety with $L(A,1)\neq 0$, then
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$$\frac{L(A,1)}{\Omega_A} =
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\frac{\#\Sha(A)\cdot\prod c_p}
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{\# A(\Q)_{\tor}\cdot\#\Adual(\Q)_{\tor}}.$$
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Here $A(\Q)_{\tor}$ is the group of rational torsion points on~$A$;
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the Shafarevich-Tate group $\Sha(A)$ is a measure of the failure
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of the local-to-global principle; the Tamagawa numbers~$c_p$ are the
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orders of certain component groups associated to~$A$;
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the real number~$\Omega_A$ is
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the volume of~$A(\R)$ with respect to a basis of differentials having
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everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.
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\mysubsection{The ratio $L(A,1)/\Omega_A$}
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Extending Manin's work on elliptic curves, A.~Agash\'{e} and I
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found a computable formula for the rational number $L(A,1)/\Omega_A$.
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Using similar techniques, I hope to find computable formulas
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for rational parts of special values of twists, and of $L$-functions
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attached to forms of weight greater than two. I have
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already computed $L(A,1)/\Omega_A$ for several thousand abelian
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varieties, and hope to extend these computations.
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\mysubsection{The Tamagawa numbers $c_p$}
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When~$A$ has semistable reduction at~$p$, I have
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found a way to explicitly compute
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the number~$c_p$, up to a power of~$2$.
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I hope to find a way to compute~$c_p$ in the remaining cases.
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\mysubsection{Bounding $\#\BigSha$}
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V.~Kolyvagin and K.~Kato
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obtained upper bounds on~$\#\Sha(A)$.
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To verify the full BSD conjecture for certain abelian
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varieties, it is necessary is to make these bounds explicit.
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Kolyvagin's bounds involve computations with Heegner points,
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and Kato's involve a study of the Galois representations
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associated to~$A$. I plan to carry out such
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computations in many specific cases.
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One approach to showing that~$\Sha(A)$ is as large as predicted
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by the BSD conjecture is suggested by Mazur's notion of
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the visible part of~$\Sha(A)$.
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Consider an abelian variety~$A$ that sits naturally
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in the Jacobian $J_0(N)$ of the modular curve $X_0(N)$.
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The {\em visible part} of $\Sha(A)$ is the collection of those
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elements of $\Sha(A)$
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that go to~$0$ under the natural map to $\Sha(J_0(N))$.
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Cremona and Mazur observed that if an element of order~$p$
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in~$\Sha(A)$ is visible,
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then it is explained by a jump in the rank of Mordell-Weil,
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in the sense that there is another abelian subvariety $B\subset J_0(N)$
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such that $p \mid \#(A\intersect B)$ and~$B$ has many rational points.
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I am trying to find the precise degree to which this observation
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can be turned around: if there is
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another abelian variety~$B$ with many rational points and
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$p\mid \#( A\intersect B)$, then under what hypotheses
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is there an element of~$\Sha(A)$ of order~$p$?
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\mysection{Icosahedral Galois representations}
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E.~Artin conjectured
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that the $L$-series associated to any continuous irreducible
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representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.
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Recent exciting work of Taylor and others suggests that a complete proof
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of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,
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is on the horizon.
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By combining the main result of
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a recent paper of K.~Buzzard and Taylor
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with a computer computation,
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Buzzard and I recently
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proved that the icosahedral Artin representations of
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conductor~$1376=2^5\cdot 43$ are modular.
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If I can extend a congruence result of J.~Sturm, then
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our method will yield several more examples.
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These ongoing computations are laying a part of the
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foundation necessary for a full proof of the Artin conjecture
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for odd two-dimensional~$\rho$, as well as stimulating
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the development of new algorithms for computing with modular forms
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in characteristic~$\ell$.
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\end{document}
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