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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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\begin{document}
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\begin{center}
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{\LARGE\bf
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Publications}\\
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(published, accepted for publication, or submitted)
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\end{center}
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\begin{enumerate}
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\item {\tfont Lectures on modular forms and
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Galois representations} (170 pages), with K.~Ribet: In 1996,
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Ken Ribet taught an advanced
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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 has been accepted for
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publication in Springer-Verlag's Universitext series.
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The chapters are: Overview; Modular representations; Modular forms;
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Embedding Hecke operators in the dual; Rationality and integrality
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questions; Modular curves; Higher weight modular forms; Newforms;
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Some explicit genus computations; The field of moduli;
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Hecke operators as correspondences; Abelian varieties from modular forms;
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The Gorenstein property; Local properties of $\rho_\lambda$; Serre's
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conjecture; Fermat's Last Theorem; Deformations; The Hecke
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algebra $\mathbf{T}_{\Sigma}$.
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Though a version of the notes has been circulating
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for a few years, significant polishing remains to be done.
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\item {\tfont HECKE: The modular forms calculator:}
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I wrote {\sc Hecke}, a popular {\tt C++} package for
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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 {\sc 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 {\tfont Explicit approaches to modular abelian
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varieties} (130 pages), Ph.D. thesis
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under H.\thinspace{}W.~Lenstra:
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In the first part of my thesis, I give algorithms for
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computing with modular abelian varieties.
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These include algorithms for computing congruences between
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modular eigenforms, the modular degree, the rational parts
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of the special values of the $L$-function,
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component groups at primes of multiplicative reduction, period
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lattices, and the real volume.
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In the second part of my thesis, I use these algorithms to
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investigate several open problems.
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The first concerns the Birch and Swinnerton-Dyer conjecture,
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which ties together the constellation of invariants attached
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to an abelian variety; I verify this conjecture for certain specific
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abelian varieties of dimension greater than one.
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The key idea is to use B.~Mazur's notion of visibility, coupled with
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explicit computations, to produce lower bounds on Shafarevich-Tate groups.
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I also describe related computations that are used in an
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upcoming joint paper with L.~Merel on
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rational points of $X_0(N)$ over $\Q(\mu_p)$.
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E.~Artin conjectured in 1924 that the $L$-series associated to
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a representation $\rho:\GQ\rightarrow\GL_2(\C)$ is entire.
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K.~Buzzard and I verify this conjecture in eight new icosahedral cases;
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the argument uses a theorem of K.~Buzzard and R.~Taylor along
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with a computer computation, which is used to fill in a missing
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mod-$5$ modularity result.
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Finally I turn to Serre's conjecture. I consider a family of
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exceptional cases, and show that the conjecture appears to fail
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as badly as possible. I also describe numerical experiments
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towards a mod-$pq$ extension of Serre's conjecture.
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I expect to finish my thesis before May 2000.
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\newpage
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\item {\tfont Lectures on Serre's conjectures} (77 pages), with K.~Ribet:
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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 IAS/Park City Mathematics Institute Lecture Series.
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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 lowering
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without multiplicity one; Level lowering with multiplicity one;
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Other directions. We have finished
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all but the last three sections, and expect to finish
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by early December, 1999.
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\item {\tfont 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), with E.\thinspace{}V.~Flynn,
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F.~Lepr\'{e}vost, E.\thinspace{}F.~Schaefer,
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M.~Stoll, J.\thinspace{}L.~Wetherell:
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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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This paper has been submitted.
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\item {\tfont Parity structures and generating
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functions from Boolean rings \hspace{.2em}}(8 pages), with D.~Moulton:
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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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This paper has been submitted.
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\item {\tfont Fallacies, Flaws, and
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Flimflam \#92: An Inductive Fallacy}, (1 page)
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with A.~Riskin: College Math.\ Journal 26:5 (1995), 382.
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\end{enumerate}
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\vspace{1ex}
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\begin{center}
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\LARGE\bf
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Work in progress
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\end{center}
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\begin{enumerate}
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\item {\tfont The modular forms database:}
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This is an evolving collection of tables 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 are available at\\
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{\tt http://shimura.math.berkeley.edu/\~{}was/Tables/}.
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\item {\tfont Visibility of Shafarevich-Tate
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groups of modular abelian varieties} (20 pages), with
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A.~Agash\'{e}:
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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. We also give evidence
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that visibility is rare. The computations are done, but
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we are still adding to the theoretical content of the paper
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and polishing the presentation.
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\item {\tfont 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 $J_0(p)$, for~$p$ prime, are as simple as possible.
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This paper is almost ready to be submitted.
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\item {\tfont Mod~$5$ approaches to modularity of icosahedral
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Galois \nobreak{representations}} (16 pages), with K.~Buzzard:
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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
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satisfy this conjecture.
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We obtained these examples 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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The computations of the paper are complete, and have been
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mostly written up. We expect to finish the paper in time
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for the Hot Topics workshop at MSRI in December, 1999.
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\item {\tfont Indexes of genus one curves} (10 pages):
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I prove results about the complexity of genus one curves.
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The greatest common divisor of the degrees of the fields in
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which a curve of genus~$g$ over~$\mathbf{Q}$ has a rational
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point divides $2g-2$; when $g=1$ this is no condition at all.
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In the 1950s S.~Lang and J.~Tate asked whether, given a positive integer~$m$,
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there exists a genus one curve so that the smallest number
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field in which it has a rational point is of degree~$m$ over~$\mathbf{Q}$.
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Using Euler systems, I prove this when~$m$ is not
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divisible by~$4$. This paper is almost ready to be
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submitted.
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\end{enumerate}
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\end{document}
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