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Author: William A. Stein
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% vissha.tex
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\chapter{Application: The Birch and Swinnerton-Dyer conjecture}
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\label{chap:bsd}
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In this chapter, we study visibility of Shafarevich-Tate
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group of abelian varieties.
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Consider an abelian variety~$A$ and an injection
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$\iota:A\ra J$ of~$A$ into an abelian variety~$J$.
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Mazur (see~\cite{cremona-mazur})
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introduced the subgroup $H^1(\Q,A)^\circ$\label{defn:visiblepart}
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of elements of $H^1(\Q,A)$ \defn{visible inside of~$J$}.
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This is the kernel of the map $H^1(\Q,A)\ra H^1(\Q,J)$
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induced by~$\iota$.
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Of particular interest is the visible subgroup $\Sha(A/\Q)^\circ$
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consisting of elements of $\Sha(A/\Q)$ that lie in
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$H^1(\Q,A)^\circ$.
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In~\cite{cremona-mazur}, Cremona and Mazur asked the following
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question: if $E\subset J_0(N)$ is a elliptic curve, how much of
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$\Sha(E/\Q)$ is visible in an abelian surface inside~$J_0(N)$?
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In their numerical experiment, visibility appears to occur surprisingly
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often. However, they warn that the conductors considered ($N\leq 5500$)
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are quite small, and the situation is likely to change as the conductors
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get larger.
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In~\cite{mazur:visthree}, Mazur
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proved that if~$E$ is an elliptic curve over a number field~$K$, then
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every element of order~$3$ in $\Sha(E/K)$ is visible in some abelian
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surface.
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In this paper, we consider questions of visibility for abelian
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varieties~$A$ of dimension possibly greater than one.
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This extension is important for several reasons. The numbers become
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large at much smaller conductor, so behavior that is not
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observed when considering elliptic curves becomes apparent.
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Certain moduli surfaces are of general type, so one does not
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expect visibility, in abelian surfaces,
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of large order Shafarevich-Tate group of elliptic curves; it is thus
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essential to consider visibility in higher dimensional abelian
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varieties. J.~de Jong proved in~\cite[\S3]{cremona-mazur}
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that every element of $\Sha(E/\Q)$ becomes visible in some abelian variety.
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%Because the conductors in our computations are small, it
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%is feasible to investigate whether an element of~$\Sha$ that
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%is invisible at level~$N$ becomes visible at level~$\ell N$
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%for~$\ell$ a small prime.
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\section{The Birch and Swinnerton-Dyer conjecture}
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In~\cite{tate:bsd}, Tate stated a generalization of the
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conjecture of Birch and Swinnerton-Dyer to abelian varieties over
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number fields. We recall a special case of the conjecture.
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Consider an abelian variety~$A$ over~$\Q$ that is a quotient
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of~$J_0(N)$ for some~$N$, and let~$\cA/\Z$ be its N\'{e}ron model.
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For each prime~$p$, let $C_{A,p}$ denote the
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component group of the closed fiber $\cA_{\Fp}$.
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A basis $h_1,\ldots,h_d$ for $H^0(\cA,\Omega_{\cA/\Z})$
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defines a measure~$\mu$ on $A(\R)$.
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Under the further assumption that $L(A,1)\neq 0$,
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it is proved in \cite{kolyvagin-logachev:finiteness} that~$A(\Q)$
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and~$\Sha(A/\Q)$ are both finite groups.
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Birch, Swinnerton-Dyer, and Tate gave a conjectural relationship
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between all of these finite groups that expresses the special
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value at~$1$ of $L(A,s)$ in terms of the volumes of all of these
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groups of rational points.
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\begin{conjecture}\label{conj:bsd}
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If~$A$ is a modular abelian variety of analytic rank~$0$, then
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$$L(A,1) = \frac{\prod_{p|N} \#C_{A,p}(\Fp)}
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{\#A(\Q)\cdot\#\Adual(\Q)}
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\cdot \#\Sha(A/\Q) \cdot \mu(A(\R)).
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$$
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\end{conjecture}
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As evidence for this conjecture, Tate proved that both the left and
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right hand sides are invariant under isogeny. Recent work of Kato
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gives bounds on $\#\Sha(A/\Q)$ that are consistent with this conjecture.
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[[Reference the blue book with the article on this by Scholl.]]
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In what follows we will frequently write $c_p=\#C_{A,p}(\Fp)$
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and $\Omega_\cA=\mu(A(\R))$.
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\section{Tables}
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This section contains tables computed using the above algorithms
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as implemented in the author's program {\tt HECKE} (a {\tt C++} program using
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{\tt LiDIA} and {\tt NTL}), David Kohel's {\tt Magma} software,
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and {\tt PARI}.
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Each factor $A_f$ of $J_0(N)$ is denoted as follows:
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\begin{center}
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{\bf N\, isogeny-class\, dimension}
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\end{center}
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The dimension frequently determines the factor, so it
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is included in the notation.
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We consider only the odd part of $\Sha$ so we
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only computed the odd parts of the arithmetic invariants
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of $A_f$. Thus at this point we make the
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\begin{center}
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{\bf WARNING: ONLY ODD PARTS OF INVARIANTS ARE GIVEN!}\\
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\end{center}
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\noindent{\bf Tables 1-3: New Visible $\Sha$}\\
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Let $n_f$ be the largest odd square dividing the numerator
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of $L(A_f,1)/\Omega_f$. Table 1 lists those $A_f$ such that,
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for $p|n_f$ there exists a {\em new} factor $B_g$ of $J_0(N)$,
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of positive analytic rank, and such that
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$(\Adual_f\intersect B_g^{\vee})[p]\neq 0$.
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This is necessary (and usually sufficient) for the $p$-torsion
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in the new visible part of $\Sha$ to be nonzero.
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In many cases it could be seen that there were no other appropriate
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new factors by looking at the signs of the Atkin-Lehner involutions.
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Up to level $1001$ our search was systematic.
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The two examples after level $1001$ were not found by
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systematic search (i.e., there may be a gap).
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In those cases for which $4|N$, we put $c_2=a$, as we don't know
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how to compute $c_2$ exactly when the reduction is additive.
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Table 2 contains further arithmetic information about each
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explanatory factor.
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%Table 3 gives the prime factorizations
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%of the composite levels.
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The explanatory factors of level $\leq 1028$ are {\em exactly} the set of
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rank $2$ elliptic curves of level $\leq 1028$.
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By \cite{brumer:rank}, the explanatory factor at level $1061$ is the
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first surface of rank $4$ (and prime level).
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\noindent{\bf Table 4: Component groups}\\
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Table 4 gives the quantities involved in the formula for Tamagawa numbers,
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for each of the $A_f$ from table 1.
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\noindent{\bf Table 5: Odd square numerator}\\
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In order to find the $A_f$, we first enumerated those
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$A_f$ for which the numerator
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of $L(A_f,1)/\Omega_f$ is divisible by an odd square $n_f$.
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For $N<1000$, these are given in table 5.
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Any odd visible $\Sha$ coprime to
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primes dividing torsion and $c_p$ must show up as a divisor
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of the numerator, and given BSD, it must show up as a square
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divisor because the Mordell-Weil rank of the explanatory factor
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is even. It would be interesting to compute the conjectural
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order of $\Sha$ for each abelian variety in this table, but
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not in table 1, and show (when possible) that the visible $\Sha$
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is old.
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\noindent{\bf Large Sha}\\
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Upon seeing an early draft of this paper,
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Armand Brumer provided us with the first quotients of $J_0(p)$
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having positive rank and dimension $3$ and $4$. Computing further
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I've found the following spectacular table.
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The listed orders of $\Sha$ {\em really} is the numerator
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of the $L$-function as a long computation showed.
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\begin{center}
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$$\begin{array}{|lcl|}\hline
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\mathbf{A_f} & \text{divisior of $|\Sha|$} & \mathbf{B_g}\\\hline
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\mathbf{389E20} &5 &\mathbf{389A1}\\
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\mathbf{1061D46}&151 &\mathbf{1061B2}\\
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\mathbf{1693B72}&1301 &\mathbf{1693A3}\\
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\mathbf{2593B109}&67\cdot 2213 &\mathbf{2593A4}\\\hline
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\end{array}$$
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\end{center}
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For 1693 the odd part of the special value is $1301^2/141$.
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For 2593 it is $67^2\cdot 2213^2/3^3$.
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\begin{table}\label{table:newvis}
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\caption{New visible Shafarevich-Tate}\vspace{-2ex}
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$$\begin{array}{|lccccccl|}\hline
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\mathbf{A_f} &n_f &w_q& c_p &T&TL(1)/\Omega_f &\delta_A & \mathbf{B_g}\\\hline
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%\vspace{-2.5ex} & & &&& & & \\
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\mathbf{389E20}&5^2&-&97&97&5^2&5&\mathbf{389A1}\\
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\mathbf{433D16}&7^2&-&3^2&3^2&7^2&3\cdot 7\cdot 37&\mathbf{433A1}\\
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\mathbf{446F8}&11^2&+- &1,3&3&11^2&11\cdot359353&\mathbf{446B1}\\
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\mathbf{563E31} & 13^2 & - & 281 &281 &13^2 &13 &\mathbf{563A1}\\
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\mathbf{571D2} & 3^2 & - & 1 &1 & 3^2 & 3^2\cdot 127&\mathbf{571B1}\\
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\mathbf{655D13} & 3^4 &+- & 1,1 & 1 & 3^4 & 3^2\cdot 19\cdot 515741&\mathbf{655A1}\\
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\mathbf{664F8}&5^2 &-+ &a,1&1& 5^2 &5 & \mathbf{664A1}\\
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% Sha dim Wq c_p T TL/O delta B
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\mathbf{681B1}& 3^2 &+- & 1,1 & 1 & 3^2 & 3\cdot 5^3 & \mathbf{681C1}\\
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\mathbf{707G15}& 13^2 &+- & 1,1 & 1 & 13^2 & 13\cdot 800077& \mathbf{707A1}\\
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\mathbf{709C30}& 11^2 &- & 59 &59 & 11^2 & 11 & \mathbf{709A1}\\
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\mathbf{718F7}& 7^2 &+- & 1,1 & 1 & 7^2 &7\cdot 151\cdot 35573 & \mathbf{718B1}\\
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\mathbf{794G14}& 11^2 &+- & 3,1 & 3 & 11^2 &3\cdot7\cdot11\cdot47\cdot35447& \mathbf{794A1}\\
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\mathbf{817E15}& 7^2 &+- & 1,5 & 5 & 7^2 & 7\cdot 79 & \mathbf{817A1}\\
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\mathbf{916G9}& 11^2 &-+ & a,1 & 1 & 11^2 & 3^9\cdot 11\cdot 17\cdot 239 & \mathbf{916C1}\\
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\mathbf{944O6}& 7^2 &+- & a,1 & 1 & 7^2 & 7 & \mathbf{944E1}\\
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\mathbf{997H42}& 3^4 &- & 83 & 83 & 3^4 & 3^2 & \mathbf{997B1,C1}\\
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\mathbf{1001L7}& 7^2 &+-+& 1,1,1 & 1 & 7^2 & 7\cdot19\cdot47\cdot2273&\mathbf{1001C1}\\
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\mathbf{1028E14}&3^2\cdot 11^2&-+ & a,1 & 3 & 3^4\cdot 11^2 & 3^{13}\cdot 11 & \mathbf{1028A1}\\
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\mathbf{1061D46} &151^2 &- &5\cdot 53&5\cdot 53&151^2& 61\cdot 151\cdot 179 &\mathbf{1061B2}\\
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\mathbf{2593B109}&67^2\cdot 2213^2 & - & ? & ? & ? & ? & \mathbf{2593A4}\\\hline
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\end{array}$$
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\end{table}
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\begin{table}
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\caption{Explanatory factors}\vspace{-2ex}
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$$\begin{array}{|lcccccc|}\hline
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\mathbf{B_g}&\text{rank}&w_q&c_p&T&\delta_B&\text{Comments}\\\hline
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%\vspace{-2.5ex} & & && & & \\
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\mathbf{389A1}& 2 &-&1 &1&5&\text{first curve of rank $2$}\\
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\mathbf{433A1}&2 &-&1&1&7&\\
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\mathbf{446B1}&2 &+-&1,1& 1 &11&\text{this is $\mathbf{446D}$ in \cite{cremona:algs}}\\
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\mathbf{563A1}&2 &- & 1 & 1 & 13 & \\
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\mathbf{571B1}&2 &- & 1 & 1 & 3 & \\
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\mathbf{655A1}&2 &+-& 1,1 & 1 & 3^2 & \\
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\mathbf{664A1}&2 &-+& 1,1 & 1 & 5 & \\
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% RANK wq g_p Tor delta comments
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\mathbf{681C1} & 2 & +- & 1,1 & 1 & 3 & \\
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\mathbf{707A1} & 2 & +- & 1,1 & 1 & 13 & \\
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\mathbf{709A1} & 2 & - & 1 & 1 & 11 & \\
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\mathbf{718B1} & 2 & +- & 1,1 & 1 & 7 & \\
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\mathbf{794A1} & 2 & +- & 1,1 & 1 & 11 & \\
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\mathbf{817A1} & 2 & +- & 1,1 & 1 & 7 & \\
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\mathbf{916C1} & 2 & -+ & 3,1 & 1 & 3\cdot 11 & \\
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\mathbf{944E1} & 2 & +- & 1,1 & 1 & 7 & \\
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\mathbf{997B1} & 2 & - & 1 & 1 & 3 & \\
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\mathbf{997C1} & 2 & - & 1 & 1 & 3 & \\
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\mathbf{1001C1} & 2 & +-+& 1,3,1& 1 & 3^2\cdot 7 & \\
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\mathbf{1028A1} & 2 & -+ & 3,1 & 1 & 3\cdot 11& \text{intersects $\mathbf{1028E}$ mod $11$}\\
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\mathbf{1061B2}& 4 & - & 1 & 1 & 151 & \text{first prime level surface of pos. even rank \cite{brumer:rank}}\\
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\hline\end{array}$$
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\end{table}
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\begin{table}
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\caption{Factorizations}\vspace{-1ex}
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$$\begin{array}{|llll|}\hline
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\mathbf{446}=2\cdot 223&
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\mathbf{655}=5\cdot 131&
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\mathbf{664}=2^3\cdot 83&
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\mathbf{681}=3\cdot 227\\
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\mathbf{707}=7\cdot 101&
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\mathbf{718}=2\cdot 359&
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\mathbf{794}=2\cdot 397&
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\mathbf{817}=19\cdot 43\\
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\mathbf{916}=2^2\cdot 229&
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\mathbf{944}=2^4\cdot 59&
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\mathbf{1001}=7\cdot 11\cdot 13&
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\mathbf{1028}=2^2\cdot 257\\
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\hline\end{array}$$
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\end{table}
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\begin{table}
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\caption{Component groups}\vspace{-2ex}
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$$\begin{array}{|lccccc|}\hline
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\vspace{-2ex}&&&&&\\
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\mathbf{A_f} & p & w_p &|\coker| &|\disc(X[I_f])| &|\Phi(\Fpbar)| \\
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\vspace{-2ex}& & & & & \\\hline
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\mathbf{389E20}& 389&-& 97 & 5\cdot 97 & 97 \\
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\mathbf{433D16}& 433&-& 3^2& 3^3\cdot 7\cdot 37 & 3^2 \\
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\mathbf{446F8} & 223&-& 3 & 3\cdot 11\cdot 359353 & 3 \\
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& 2 &+ & 3 & 3\cdot 11& 3\cdot 359353\\
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\mathbf{563E31}& 563&-& 281& 13\cdot 281& 281 \\
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\mathbf{571D2} & 571&-& 1 & 3^2\cdot 127 &1 \\
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\mathbf{655D13}& 131&-& 1 & 3^{2}\cdot19\cdot515741 & 1\\
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& 5&+& 1 & 3^2 & 19\cdot 515741 \\
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\mathbf{664F8} & 83&+& 1 & 5 & 1 \\
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\mathbf{681B1} & 227&-& 1 & 3\cdot 5^3 & 1 \\
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& 3&+& 1 & 3\cdot 5^2& 5 \\
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\mathbf{707G15}& 101&-& 1 & 13\cdot800077 & 1 \\
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& 7&+& 1 & 13& 800077 \\
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\mathbf{709C30}& 709&-& 59& 11\cdot 59 & 59 \\
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\mathbf{718F7} & 359&-& 1 & 7\cdot 151\cdot 35573 &1 \\
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& 2 &+& 1 & 7 & 151\cdot 35573 \\
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\mathbf{794G14}& 397&-& 3 & 3^2\cdot7\cdot11\cdot47\cdot35447 & 3 \\
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& 2&+& 3 & 3\cdot11& 3^2\cdot 7\cdot 47\cdot 35447 \\
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\mathbf{817E15}& 43&- & 5 & 5\cdot7\cdot 79 & 5 \\
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& 19&+ &1 & 7 & 79 \\
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\mathbf{916G9} & 229&+ &1 & 3^9\cdot 11\cdot 17\cdot 239 & 1 \\
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\mathbf{944O6} & 59&-& 1 & 7 & 1\\
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\mathbf{997H42}& 997&-& 83& 3^2\cdot 83 & 83 \\
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\mathbf{1001L7}& 13&+& 1& 7\cdot 19\cdot 47\cdot 2273& 1\\
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& 11&-& 1& 7\cdot19\cdot47\cdot2273 & 1 \\
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& 7&+& 1& 7\cdot 19\cdot 47 & 2273 \\
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\mathbf{1028E14}&257&+& 1 & 3^{13}\cdot 11 & 1 \\
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\mathbf{1061D46}&1061&-& 5\cdot53 & 5\cdot 53\cdot 61\cdot 151\cdot 179 & 5\cdot 53 \\
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\hline \end{array}$$
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\end{table}
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\begin{table}
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\caption{Odd square numerator}\vspace{2ex}
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\begin{tabular}{|llllll|}\hline
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$\mathbf{305D7}:3$&
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$\mathbf{309D8}:5$&
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$\mathbf{335E11}:3^2$&
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$\mathbf{389E20}:5$&
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$\mathbf{394A2}:5$&
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$\mathbf{399G5}:3^4$\\
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$\mathbf{433D16}:7$&
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$\mathbf{435G2}:3$&
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$\mathbf{436C4}:3$&
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$\mathbf{446E7}:3$&
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$\mathbf{446F8}:11$&
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$\mathbf{455D4}:3$\\
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$\mathbf{473F9}:3$&
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$\mathbf{500C4}:3$&
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$\mathbf{502E6}:11$&
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$\mathbf{506I4}:5$&
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$\mathbf{524D4}:3$&
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$\mathbf{530G4}:7$\\
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$\mathbf{538E7}:3$&
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$\mathbf{551H18}:3$&
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$\mathbf{553D13}:3$&
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$\mathbf{555E2}:3$&
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$\mathbf{556C7}:3$&
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$\mathbf{563E31}:13$\\
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$\mathbf{564C3}:3$&
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$\mathbf{571D2}:3$&
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$\mathbf{579G13}:3\cdot 5$&
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$\mathbf{597E14}:19$&
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$\mathbf{602G3}:3$&
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$\mathbf{604C6}:3 $\\
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$\mathbf{615F6}:5 $&
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$\mathbf{615G8}:7 $&
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$\mathbf{620D3}:3$&
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$\mathbf{620E4}:3$&
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$\mathbf{626F12}:5$&
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$\mathbf{629G15}:3$\\
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325
$\mathbf{642D2}:3$&
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$\mathbf{644C5}:3$&
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$\mathbf{644D5}:3$&
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$\mathbf{655D13}:3^2$&
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$\mathbf{660F2}:3$&
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$\mathbf{662E10}:43$\\
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$\mathbf{664F8}:5$&
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$\mathbf{668B5}:3$&
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% $\mathbf{675O}:3$& % REMOVED, since new algorithm gives bsd value of 1.
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$\mathbf{678I2}:3$&
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$\mathbf{681B1}:3$&
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$\mathbf{681I10}:3$&
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$\mathbf{682I6}:11$\\
339
340
$\mathbf{707G15}:13$&
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$\mathbf{709C30}:11$&
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$\mathbf{718F7}:7$&
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$\mathbf{721F14}:3^2$&
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$\mathbf{724C8}:3$&
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$\mathbf{756G2}:3$\\
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347
$\mathbf{764A8}:3$&
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$\mathbf{765M4}:3$&
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$\mathbf{766B4}:3$&
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$\mathbf{772C9}:3$&
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$\mathbf{790H6}:3$&
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$\mathbf{794G12}:11$\\
353
354
$\mathbf{794H14}:5^2$&
355
$\mathbf{796C8}:3$&
356
$\mathbf{817E15}:7$&
357
$\mathbf{820C4}:3$&
358
$\mathbf{825E2}:3$&
359
$\mathbf{844C10}:3^2$\\
360
361
$\mathbf{855M4}:3$&
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$\mathbf{860D4}:3$&
363
$\mathbf{868E5}:3$&
364
$\mathbf{876E5}:3$&
365
$\mathbf{878C2}:3$&
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$\mathbf{884D6}:3$\\
367
368
$\mathbf{885L9}:3^2$&
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$\mathbf{894H2}:3$&
370
$\mathbf{902I5}:3$&
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$\mathbf{913G17}:3$&
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$\mathbf{916G9}:11$&
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$\mathbf{918O2}:5$\\
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375
$\mathbf{918P2}:3$&
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$\mathbf{925K7}:3$&
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$\mathbf{932B13}:3^2$&
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$\mathbf{933E14}:19$&
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$\mathbf{934I12}:7$& %-+
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$\mathbf{944O6}:7$\\
381
382
$\mathbf{946K7}:3$&
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$\mathbf{949B2}:3$&
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$\mathbf{951D19}:3$&
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$\mathbf{959D24}:3$&
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$\mathbf{964C12}:3^2$& % -+, same as EC 964A but that has rank=0.
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$\mathbf{966J1}:3$\\
388
389
$\mathbf{970I5}:3$&
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$\mathbf{980F1}:3$&
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$\mathbf{980J2}:3$&
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$\mathbf{986J7}:5$&
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$\mathbf{989E22}:5$&
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$\mathbf{993B3}:3^2$\\
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396
$\mathbf{996E4}:3$&
397
$\mathbf{997H42}:3^2$&
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$\mathbf{998A2}:3$& % ++
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$\mathbf{998H9}:3$&
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$\mathbf{999J10}:3$& \\
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%$\mathbf{1001L7}:7$&
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%$\mathbf{1001M8}:3$\\
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\hline\end{tabular}
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\end{table}
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\section{Prime conductor}
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Using [REFS] we computed the odd part $\Shaan$ of the
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analytic order of the Shafarevich and Tate group of every
410
rank zero optimal quotient of $J_0(p)$ for $p \leq 2593$.
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Those for which $\Shaan > 1$ are laid out in the
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first column of Table~\ref{table:primesha}.
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The fourth column lists the explanatory factor, if there is
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one at the same level. If $\Sha$ is not visible we
415
were sometimes able to find an explanatory factor
416
at higher level; this is discussed in the next section.
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418
\begin{table}\label{table:primesha}
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\caption{Odd parts of Shafarevich-Tate groups at prime level}
420
$$
421
\begin{array}{lccclcc}
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\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan & \mbox{\rm mod deg} & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} \\
423
& & & & & & \vspace{-2ex} \\
424
\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\
425
\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\
426
\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\
427
\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\
428
\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\
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\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\
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\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\
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\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\
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\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\
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\mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\
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\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\
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\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\
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\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\
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\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\
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\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\
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\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\
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\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\
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\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\
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\mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\
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\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\
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\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\
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\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\
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\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\
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\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\
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\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\
465
\end{array}
466
$$
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\end{table}
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Table~\ref{table:primesha} has many interesting properties.\footnote{
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Examples {\bf 1091C} and {\bf 1429B} were
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discoverd in \cite{agashe} and
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{\bf 1913B} in \cite{cremona-mazur}.}
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There are $23$ examples in which $\Sha$ is (almost certainly)
475
visible and $18$ examples in which $\Sha$ is invisible. The largest visible
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$\Sha$ found occurs at level $2333$ and has order at least $83341^2$
477
($83341$ is prime). Believe it or not, the largest invisible $\Sha$
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occurs in a $112$ dimensional abelian variety at level
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$2111$ and has order at least $211^2$. Furthermore,
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\begin{enumerate}
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\item The Shafarevich-Tate group is never Eisenstein in these examples
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[[recheck, as I've added new examples!]]
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\item {\bf 1283C}. This example demonstrates that $\Shaan$ can divide the
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modular degree, yet still be invisible. The other factor
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{\bf 1283A} (of dimension 2) which satisfies a $5$-congruence with
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{\bf 1283C} has analytic rank (and hence algebraic rank) $0$.
488
\end{enumerate}
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\comment{
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\section{Explanatory factors at higher level}
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Consider one of the items in Table~\ref{table:primesha} for which
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$\Sha$ is invisible. It is natural to ask whether these
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elements of $\Sha$ ``become visible somewhere.''
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For example, Barry Mazur \cite{mazur:visthree} proved that if
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$E$ is an elliptic curve and $c\in\Sha(E)$ has order $3$ then
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there is some abelian surface $A$ and an
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injection $\iota: E\hookrightarrow A$ such that
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$\iota_*(c)=0\in H^1(\Q,A)$. T. Klenke has proved
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a partial statement in this direction for elements of
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order $2$ as part of his Harvard Ph.D. thesis.
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Johann de Jong (see \cite[Remark 3]{cremona-mazur})
503
showed that every element of the Shafarevich-Tate
504
group of an elliptic curve is visible in some Jacobian.
505
506
Consider an abelian variety $A$, taken
507
from Table~\ref{table:primesha}, for which
508
$\Sha$ has an invisible element $c$. Thus
509
$A$ sits inside $J_0(p)$ for some prime $p$,
510
and we ask ``is there a prime $q$ such that $\delta(c)=0$
511
for one of the degeneracy maps
512
$\delta : J_0(p)\ra J_0(pq)$?''
513
514
The author has no idea\footnote{Lo\"\i{}c Merel suspects the answer
515
might be yes whereas Richard Taylor is more skeptical.}.
516
To get a feeling for what might happen we consider in
517
detail abelian variety $A=A_f$ at level $p=1091$ in
518
which $\Shaan$ is divisible by $7$.
519
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There is a prime $\lambda$ of the ring
521
$\Z[f] = \Z[\ldots a_n\ldots]$ attached to $A$.
522
The Fourier coefficients of $f$ modulo $\lambda$ are
523
$$\begin{array}{rcccccccccccccccc}
524
p= &2 &3 &5 &7 &11 &13 &17 &19 &23 &29 &31 &37 &41 &43 &47 &53\\
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a_p= &3 &0 &1 &5 &0 &2 &0 &5 &4 &6 &3 &3 &5 &5 &6 &5
526
\end{array}$$
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These were computed by finding an eigenvector in $H_1(X_0(N);\F_7)$.
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[[SAY MORE ABOUT THE TRICK FOR FINDING ALL RIBET $q$'s.]]
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According to Ribet's level raising theorem \cite{ribet:raising}
531
there is a newform $g$ of level $1091\ell$ such that
532
$f\con g$ modulo [[something]] if $a_\ell = \pm (\ell+1)\pmod{\lambda}$.
533
Fortunately this criterion is already satisfied for $\ell=2$.
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Looking closely at level $2\cdot 1091$ (for example,
535
in Cremona's online tables \cite{cremona:onlinetables})
536
we find an elliptic curve $E$ whose corresponding newform
537
$g=\sum b_n q^n$ has Fourier coefficients
538
$$\begin{array}{rcccccccccccccccc}
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p = &2 & 3 & 5 & 7& 11& 13& 17& 19& 23& 29& 31& 37&41&43&47&53\\
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b_p=&1 & 0 & 1 &-2& 0 & -5& 0 & -2& 4& 6& -4& -4&-9&-9&-8&-2\\
541
\end{array}$$
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This is convincing evidence that one of the
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two images of $A$ in $J_0(2\cdot 1091)$ shares some
545
$7$-torsion with the elliptic curve \abvar{2182B}.
546
This can be [[WILL BE!!, EASILY]] established by
547
a direct computation with the period lattices.
548
This is at first disconcerting because the rank of
549
this elliptic curve is {\em not} $2$. However, the
550
rank is still positive; it is $1$ with
551
Mordell-Weil group $\E(\Q)=Z$.
552
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I would not be at all surprised if your
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$7$-torsion in Sha does become visible in $J_0(2\cdot1091)$.
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The curve \abvar{2182B}, which shares 7-torsion with $A$ is
556
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\begin{verbatim}
558
e=ellinit([1,-1,1,-67,67]);
559
The Tamagawa number c_2 is 14 (!!)
560
? elllocalred(e,2)
561
%2 = [1, 18, [1, 0, 0, 0], 14]
562
The Tamagawa number c_1091 is 1.
563
? elllocalred(e,1091)
564
%3 = [1, 5, [1, 0, 0, 0], 1]
565
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I have this feeling that the right statement about congruence
567
and mordell-Weil is really something like
568
congruence ==> "Selmer + Comp group"'s are identified.
569
Anyway, the extra component group of order 7 may perhaps
570
account for the other nontrivial element of Sha. This might
571
just be wild speculation.
572
573
Good luck.
574
575
william
576
Barry,
577
578
Amod asked me to investigate whether his element of order 7
579
in the winding quotient J_e at level 1091 becomes visible at
580
higher level. Luckily, Ribet's level raising theorem predicts the
581
existence of a form at level 2*1091 congruent mod a prime over
582
7 to the form corresponding to J_e. Even more luckily, one of the
583
two rational newforms does the trick. Thus an image of J_e in
584
J_0(2*1091) shares 7-torsion with an elliptic curve E (2182B
585
in Cremona's tables). This elliptic curve has:
586
587
E(Q) = Z
588
Sha(E/Q) = 0
589
c_2 = 14, c_1091 = 1
590
L^(1)(f,1)/r! = 4.27332686791516
591
592
593
So there is reasonable hope that the elements of order 7 in
594
Sha(J_e) are visible at this higher level, even though they
595
are invisible even in J_1(1091).
596
597
Best,
598
William
599
600
601
Dear William,
602
This is terrific. I assume that you will be showing that for J_e the
603
winding
604
(not quite quotient, but more conveniently sub-thing) in J_0(1091),
605
the image of
606
607
Sha(J_e) ---> Sha(J_0(2*1091))
608
609
just dies? Since our working hope, I think, is that for any N there
610
is an
611
M so that
612
Sha(J_0(N)) ---. Sha(J_0(N.M))
613
614
615
dies, this suggests returning to the (mod 3) N=2849 example, where I
616
"know" that there must exist such an M (because all three-torsion in Sha
617
on elliptic curves is visible in some appropriate abelian surface which is
618
isogenous to a prouct of two elliptic curves, and therefore, is abelian
619
surface is probably "modular"). But I don't know a specific M.
620
621
Barry
622
623
\end{verbatim}
624
}
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