Sharedwww / job / SteinProp.texOpen in CoCalc
Author: William A. Stein
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\font\cyr=wncyr10 scaled \magstep 1
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\font\cyrbig=wncyr10 scaled \magstep 2
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\begin{document}
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\normalsize
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\baselineskip=15.9pt
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\begin{center}
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{\Large\bf Research Plan}\\
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William A.\ Stein\\
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\today
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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, J-P.~Serre, R.~Taylor, and A.~Wiles
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on Galois representations attached to modular abelian varieties
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(see \cite{ribet:modreps, serre:conjectures,
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taylor-wiles:fermat, wiles:fermat})
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with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
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on explicit computations involving modular forms
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(see \cite{cremona:algs, elkies:ffield}).
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In 1969 B.~Birch~\cite{birch:bsd} described computations that
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led to the most fundamental open
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conjecture in the theory of elliptic curves:
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\begin{quote}
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I want to describe some computations undertaken by myself and
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Swinnerton-Dyer on EDSAC
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by which we have calculated the zeta-functions of certain elliptic
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curves. As a result of these computations we have found an analogue
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for an elliptic curve of the Tamagawa number of an algebraic group;
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and conjectures (due to ourselves, due to Tate, and
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due to others) have proliferated.
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\end{quote}
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The rich tapestry of arithmetic conjectures and theory
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we enjoy today would not exist without the ground-breaking
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application of computing by Birch and Swinnerton-Dyer.
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Computations in the 1980s by Mestre were key in convincing
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Serre that his conjectures on modularity of odd
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irreducible Galois representations
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were worthy of serious consideration (see~\cite{serre:conjectures}).
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These conjectures have inspired much recent work;
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for example, Ribet's proof of the
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$\epsilon$-conjecture, which played an essential role in
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Wiles's proof of Fermat's Last Theorem.
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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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My research, which involves finding ways to compute
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with modular forms and modular abelian varieties,
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is driven by outstanding conjectures in number theory.
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\mysection{Invariants of modular abelian varieties}
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Now that the Shimura-Taniyama conjecture has been proved,
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the main outstanding problem in the field
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is the Birch and Swinnerton-Dyer conjecture (BSD conjecture),
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which ties together the arithmetic invariants of an elliptic curve.
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There is no general class of elliptic curves for which
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the full BSD conjecture is known.
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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 higher-dimensional abelian varieties.
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As a first step, I have obtained theorems that make possible explicit
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computation of some of the arithmetic invariants of modular
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abelian varieties.
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\mysubsection{The BSD conjecture}
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By~\cite{breuil-conrad-diamond-taylor} we now know
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that every elliptic curve over~$\Q$ is
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a quotient of the curve~$X_0(N)$ whose complex points
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are the isomorphism classes of pairs consisting of a
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(generalized) elliptic curve and a cyclic subgroup of order~$N$.
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Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian
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variety of dimension equal to the genus of~$X_0(N)$ whose points
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correspond to the degree~$0$ divisor classes on~$X_0(N)$.
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An {\em optimal quotient} of $J_0(N)$ is a quotient by an abelian subvariety.
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Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.
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By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A/\Q)$
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are both finite.
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The BSD conjecture asserts that
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$$\frac{L(A,1)}{\Omega_A} =
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\frac{\#\Sha(A/\Q)\cdot\prod_{p\mid N} c_p}
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{\# A(\Q)\cdot\#\Adual(\Q)}.$$
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Here the Shafarevich-Tate group $\Sha(A/\Q)$ 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 the component groups of~$A$; 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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My goal is to verify the full
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conjecture for many specific abelian varieties on a case-by-case basis.
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This is the first step in a program to verify the above
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conjecture for an infinite family of quotients of~$J_0(N)$.
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\mysubsection{The ratio $L(A,1)/\Omega_A$}
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Following Y.~Manin's work on elliptic curves,
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A.~Agash\'e and I proved the following
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theorem in~\cite{stein:vissha}.
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\begin{theorem}\label{thm:ratpart}
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Let~$m$ be the largest square dividing~$N$.
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The ratio $L(A,1)/\Omega_A$ is a rational number that can be
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explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.
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\end{theorem}
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The proof uses modular symbols combined with an extension of the argument
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used by Mazur in~\cite{mazur:rational} to bound the Manin constant.
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The ratio $L(A,1)/\Omega_A$ is expressed as the lattice index of
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two modules over the Hecke algebra.
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I expect the method to give similar results
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for special values of twists, and of
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$L$-functions attached to eigenforms of higher weight.
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I have computed $L(A,1)/\Omega_A$ for all optimal
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quotients of level $N\leq 1500$; this table continues to be
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of value to number theorists.
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\mysubsection{The torsion subgroup}
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I can compute upper and lower bounds on $\#A(\Q)_{\tor}$, but I can not
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determine $\#A(\Q)_{\tor}$ in all cases. Experimentally, the deviation
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between the upper and lower bound is reflected in congruences with
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forms of lower level; I hope to exploit this in a precise way.
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I also obtained the following
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intriguing corollary that suggests cancellation between
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torsion and~$c_p$; it generalizes to higher weight forms, thus
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suggesting a geometric explanation for reducibility
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of Galois representations.
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\begin{corollary}
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Let~$n$ be the order of the image of
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$(0)-(\infty)$ in $A(\Q)$, and
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let~$m$ be the largest square dividing~$N$.
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Then $\displaystyle n\cdot L(A,1)/\Omega_A$ is an integer,
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up to a unit in $\Z[1/(2m)]$.
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\end{corollary}
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\mysubsection{Tamagawa numbers}
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\begin{theorem}\label{thm:tamagawa}
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When $p^2\nmid N$, the number~$c_p$ can be explicitly computed
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(up to a power of~$2$).
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\end{theorem}
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I prove this in~\cite{stein:compgroup}. Several related problems
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remain: when $p^2 \mid N$ it may be possible to compute~$c_p$
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using the Drinfeld-Katz-Mazur interpretation
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of~$X_0(N)$; it should also be possible to use my
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methods to treat optimal quotients of $J_1(N)$.
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I was surprised to find that systematic computations using this
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formula indicate the following conjectural refinement of a result of
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Mazur~\cite{mazur:eisenstein}.
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\begin{conjecture}
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Suppose~$N$ is prime and~$A$
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is an optimal quotient of $J_0(N)$. Then $A(\Q)_{\tor}$ is
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generated by the image of $(0)-(\infty)$
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and $c_p = \#A(\Q)_{\tor}$. Furthermore,
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the product of the~$c_p$ over all optimal factors
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equals the numerator of $(N-1)/12$.
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\end{conjecture}
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I have checked this conjecture for all $N\leq 997$ and,
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up to a power of~$2$, for all $N\leq 2113$.
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The first part is known when~$A$ is an elliptic
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curve (see~\cite{mestre-oesterle:crelle}).
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Upon hearing of this conjecture, Mazur proved it
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when all ``$q$-Eisenstein quotients'' are simple.
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There are three promising approaches to finding
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a complete proof. One involves the explicit
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formula of Theorem~\ref{thm:tamagawa};
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another is based on Ribet's level lowering theorem,
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and a third makes use of a simplicity result of Merel.
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Theorem~\ref{thm:tamagawa}
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also suggests a way to compute Tamagawa numbers of
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motives attached to eigenforms of higher weight.
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These numbers appear in the conjectures of Bloch and Kato,
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which generalize the BSD conjecture to motives (see~\cite{bloch-kato}).
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\mysubsection{Upper bounds on $\#\BigSha$}
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V.~Kolyvagin and K.~Kato~\cite{kolyvagin:structureofsha, scholl: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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\mysubsection{Lower bounds on $\#\BigSha$}
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One approach to showing that~$\Sha$ 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~$\Sha$ (see~\cite{cremona-mazur, mazur:visthree}).
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Let~$\Adual$ be the dual of~$A$.
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The visible part of $\Sha(\Adual/\Q)$ is the
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kernel of $\Sha(\Adual/\Q)\ra \Sha(J_0(N))$.
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Mazur observed that if an element of order~$p$
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in~$\Sha(\Adual/\Q)$ 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 $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.
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I think that this observation can be turned around: if there is
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another abelian
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variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,
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then, under mild hypotheses, there is an element of~$\Sha(\Adual/\Q)$
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of order~$p$. Thus the theory of congruences between modular
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forms can be used to obtain a lower bound on~$\#\Sha(\Adual/\Q)$.
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I am trying to use the cohomological methods of~\cite{mazur:tower}
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and suggestions of B.~Conrad and Mazur to prove the following conjecture.
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\begin{conjecture}
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Let~$\Adual$ and~$B$ be abelian subvarieties of~$J_0(N)$.
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Suppose that $p\mid \#(\Adual\intersect B)$, that~$p\nmid N$,
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and that~$p$ does not divide the
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order of any of the torsion subgroups
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or component groups of~$A$ or~$B$. Then
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$(B(\Q) \oplus \Sha(B/\Q))\tensor\Z/p\Z
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\isom (\Adual(\Q)\oplus \Sha(\Adual/\Q) )\tensor\Z/p\Z$.
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\end{conjecture}
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Unfortunately, $\Sha(\Adual/\Q)$ can fail to be
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visible inside~$J_0(N)$.
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For example, I found that the BSD conjecture
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predicts the existence of invisible elements of odd order in~$\Sha$
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for at least~$15$ of the~$37$ optimal quotients of prime
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level $\leq 2113$.
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For every integer~$M$ (Ribet~\cite{ribet:raising} tells us which~$M$
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to choose), we can consider the images of~$\Adual$ in $J_0(NM)$.
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There is not yet enough evidence to conjecture the existence of
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an integer~$M$ such that all of $\Sha(\Adual/\Q)$ is visible in
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$J_0(NM)$.
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I am gathering data to determine whether or not
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to expect the existence of such~$M$.
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\mysubsection{Motivation for considering abelian varieties}
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If $A$ is an elliptic curve, then explaining~$\Sha(A/\Q)$ using
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only congruences between elliptic curves is bound to fail.
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This is because pairs of nonisogenous elliptic curves
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with isomorphic $p$-torsion are, according to
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E.~Kani's conjecture, extremely rare.
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It is crucial to understand what happens in all dimensions.
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Within the range accessible by computer,
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abelian varieties exhibit more richly textured
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structure than elliptic curves.
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For example, I discovered a visible element of prime order $83341$
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in the Shafarevich-Tate group of an abelian variety of
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prime conductor~$2333$; in contrast, over all optimal elliptic
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curves of conductor up to $5500$, it appears that
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the largest order of an element of a Shafarevich-Tate
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group is~$7$.
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\mysection{Conjectures of Artin, Merel, and Serre}
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\mysubsection{Icosahedral Galois representations}
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E.~Artin conjectured in~\cite{artin:conjecture}
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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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This case of Artin's conjecture is known when the image
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of~$\rho$ in $\PGL_2(\C)$ is solvable
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(see~\cite{tunnell:artin}), and in infinitely
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many cases when the image of~$\rho$ is not solvable (see~\cite{bdsbt}).
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In 1998, K.~Buzzard suggested a way to combine the
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main theorem of~\cite{buzzard-taylor}, along with
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a computer computation, to deduce modularity of certain
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icosahedral Galois representations. Buzzard and I recently obtained
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the following theorem.
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\begin{theorem}
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The icosahedral Artin representations of conductor~$1376=2^5\cdot 43$
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are modular.
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\end{theorem}
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We expect our method to yield several more examples.
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These ongoing computations are laying a small part of the technical
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foundations 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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using modular symbols in characteristic~$\ell$.
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\mysubsection{Cyclotomic points on modular curves}
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If~$E$ is an elliptic curve over~$\Q$
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and~$p$ is an odd prime, then the $p$-torsion
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on~$E$ can not all lie in~$\Q$; because of
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the Weil pairing the $p$-torsion generates a field
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that contains~$\Q(\mu_p)$.
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For which primes~$p$ does
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there exist an elliptic curve~$E$ over $\Q(\mu_p)$
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with all of its $p$-torsion rational over $\Q(\mu_p)$?
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When $p=2,3,5$ the corresponding moduli space has genus zero
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and infinitely many examples exist. Recent work of L.~Merel,
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combined with computations he enlisted me to do,
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suggest that these are the only primes~$p$ for
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which such elliptic curves exist.
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In~\cite{merel:cyclo}, Merel exploits
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cyclotomic analogues of the techniques used
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in his proof of the uniform boundedness conjecture
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to obtain an explicit criterion that can
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be used to answer the above question for many primes~$p$,
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on a case-by-case basis.
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Theoretical work of Merel, combined with my computations of
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twisted $L$-values and character groups of tori,
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give the following result (see~\cite[\S3.2]{merel:cyclo}):
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\begin{theorem}
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Let $p \equiv 3\pmod{4}$ be a prime satisfying
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$7 \leq p < 1000$. There are no elliptic curves
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over $\Q(\mu_p)$ all of whose $p$-torsion is rational
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over $\Q(\mu_p)$.
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\end{theorem}
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The case in which~$p$ is congruent to~$1$ modulo~$4$ presents
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additional difficulties that involve showing that~$Y(p)$ has no
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$\Q(\sqrt{p})$-rational points. We hope to tackle these in
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the near future.
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\mysubsection{Genus one curves}
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The index of an algebraic curve~$C$ over~$\Q$ is the
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order of the cokernel of the degree map $\Div_\Q(C)\ra\Z$;
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rationality of the canonical divisor implies that the index
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divides $2g-2$, where~$g$ is the genus of~$C$.
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When $g=1$ this is no condition at all; Artin conjectured, and
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Lang and Tate~\cite{lang-tate} proved, that for every integer~$m$
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there is a genus one curve of index~$m$ over some number field.
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Their construction yields genus one curves over~$\Q$ only for a few
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values of~$m$, and they ask whether one can find genus one curves
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over~$\Q$ of every index. I have answered
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this question for odd~$m$.
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\begin{theorem}
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Let~$K$ be any number field. There are genus one curves over~$K$
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of every odd index.
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\end{theorem}
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The proof involves showing that enough cohomology classes in
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Kolyvagin's Euler system of Heegner points do not vanish
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combined with explicit Heegner point computations.
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I hope to show that curves of every index occur, and
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to determine the consequences of my
384
nonvanishing result for Selmer groups. This can be viewed
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as a contribution to the problem of understanding $H^1(\Q,E)$.
386
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\mysubsection{Serre's conjecture modulo $pq$}
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Let~$p$ and~$q$ be primes, and consider a continuous
389
representation $\rho:\GQ\ra \GL(2,\Z/pq\Z)$
390
that is irreducible in the sense that its reductions
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modulo~$p$ and modulo~$q$ are both irreducible.
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Call~$\rho$ {\em modular} if there is a modular
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form~$f$ such that a mod~$p$ representation attached to~$f$
394
is the mod~$p$ reduction of~$\rho$, and ditto for~$q$.
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I have carried out specific computations suggested by Mazur
396
in hopes of determining when one should expect
397
that such mod~$pq$ representations are modular; the computation
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suggests that the right conjectures are elusive.
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Ribet's theorem (see~\cite{ribet:raising})
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produces infinitely many levels $pq\ell$ at which there is
401
a form giving rise to $\rho$~mod~$p$ and another
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giving rise to $\rho$~mod~$q$; we hope to determine if for some~$\ell$
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there is a single form giving rise to both reductions.
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\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\newcommand{\closer}{\vspace{-1.5ex}}
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\begin{thebibliography}{10}
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\bibitem{agashe}
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A.~Agash\'{e}, \emph{On invisible elements of the {T}ate-{S}hafarevich group},
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C. R. Acad. Sci. Paris S\'er. I Math. \textbf{328} (1999), no.~5, 369--374.
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\closer
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\bibitem{stein:vissha}
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A.~Agash\'{e} and W.\thinspace{}A. Stein, \emph{Visibility of
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{S}hafarevich-{T}ate groups of modular abelian varieties}, in preparation
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(1999).
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\closer
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\bibitem{artin:conjecture}
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E.~Artin, \emph{{\"U}ber eine neue {A}rt von {L}-{R}eihen}, Abh. Math. Sem. in
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Univ. Hamburg \textbf{3} (1923), 89--108.
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\closer
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\bibitem{birch:bsd}
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B.\thinspace{}J. Birch, \emph{Elliptic curves over \protect{${\mathbf{Q}}$:
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{A}} progress report}, 1969 Number Theory Institute (Proc. Sympos. Pure
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Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math.
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Soc., Providence, R.I., 1971, pp.~396--400.
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\closer
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\bibitem{bloch-kato}
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S.~Bloch and K.~Kato, \emph{\protect{${L}$}-functions and \protect{T}amagawa
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numbers of motives}, The Grothendieck Festschrift, Vol. \protect{I},
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Birkh\"auser Boston, Boston, MA, 1990, pp.~333--400.
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\closer
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\bibitem{breuil-conrad-diamond-taylor}
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C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of
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elliptic curves over~\protect{$\Q$}},
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in preparation.
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\closer
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\bibitem{bdsbt}
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K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor, \emph{On
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icosahedral {A}rtin representations},
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available at {\tt http://www.math.harvard.edu/\~{}rtaylor/}.
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\closer
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\bibitem{buzzard-taylor}
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K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one
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forms}, Annals of Math. (1999).
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\closer\vspace{-3ex}
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\bibitem{cremona:algs}
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J.\thinspace{}E. Cremona, \emph{Algorithms for modular elliptic curves}, second
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ed., Cambridge University Press, Cambridge, 1997.
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\closer
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\bibitem{cremona-mazur}
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J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements in the
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\protect{Shafarevich-Tate} group}, Proceedings of the Arizona Winter School
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(1998).
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\closer
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\bibitem{elkies:ffield}
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N.\thinspace{}D. Elkies, \emph{Elliptic and modular curves over finite fields
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and related computational issues}, Computational perspectives on number
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theory (Chicago, IL, 1995), Amer. Math. Soc., Providence, RI, 1998,
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pp.~21--76.
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\closer
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\bibitem{kolyvagin:structureofsha}
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V.\thinspace{}A. Kolyvagin, \emph{On the structure of {S}hafarevich-{T}ate
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groups}, Algebraic geometry (Chicago, IL, 1989), Springer, Berlin, 1991,
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pp.~94--121.
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\closer
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\bibitem{kolyvagin-logachev:totallyreal}
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V.\thinspace{}A. Kolyvagin and D.\thinspace{}Y. Logachev, \emph{Finiteness of
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\protect{$\Sha$} over totally real fields}, Math. USSR Izvestiya \textbf{39}
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(1992), no.~1, 829--853.
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\closer
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\bibitem{lang-tate}
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\end{document}
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