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Author: William A. Stein
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\documentclass{birkart}
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\newcommand{\testbnd}{25000}
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% ---- SHA ----
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\DeclareMathOperator{\Tr}{Tr}
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\begin{document}
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\title{Shafarevich--Tate Groups of Nonsquare Order}
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\author[W. A. Stein]{William A. Stein}
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\address{515 Science Center\\ Department of Mathematics\\ Harvard University\\ }
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\email{was@math.harvard.edu}
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\begin{abstract}
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Let~$A$ denote an abelian variety over~$\Q$.
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We give the first known examples in which $\#\Sha(A/\Q)$
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is neither a square nor twice a
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square. For example, let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of
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conductor~$37$. We prove that for every odd prime
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$p< \testbnd$ (with $p\neq 37$), there is a twist~$A$
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of $E\cross\cdots \cross E$ ($p-1$ copies)
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such that $\#\Sha(A/\Q)=p n^2$ for some integer~$n$.
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We prove this by showing under certain hypothesis on~$E$
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and~$p$ that there is an exact sequence
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$$
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0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra
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\Sha(E/\Q)[p^\infty] \ra 0,
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$$
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where $K$ is a certain abelian extension of $\Q$ of degree $p$.
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\end{abstract}
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\maketitle
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\section{Introduction}
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The Shafarevich--Tate group of
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an abelian variety~$A$ over a number field~$F$ is
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$$
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\Sha(A/F) := \Ker\left(H^1(F,A) \ra \bigoplus_{\text{all $v$}} H^1(F_v,A)\right).
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$$
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What are the possibilities for the group structure of $\Sha(A/F)$?
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It is conjectured that $\Sha(A/F)$ is finite and this
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is known in some cases.
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\begin{theorem}[Kato, Kolyvagin, Wiles, et al.]\label{thm:finite}
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Suppose~$A$ is an elliptic curve over~$\Q$.
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(1) If $\ord_{s=1}L(A,s)\leq 1$, then $\Sha(A/\Q)$ is finite.
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(2) If~$\chi$ is a character of the Galois group of an abelian
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extension~$K$ of~$\Q$ and
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$L(A,\chi,1)\neq 0$, then the
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$\chi$-component of $\Sha(A/K)\tensor_\Z\Z[\chi]$
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is finite. (Here $\Z[\chi]$ is generated by
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the image of~$\chi$.)
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\end{theorem}
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The Cassels--Tate pairing $\Sha(A/F)\cross \Sha(A^{\vee}/F)\ra \Q/\Z$
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imposes strong constraints on the structure of $\Sha(A/F)$.
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\begin{theorem}[Tate, Flach]\label{thm:tate}
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Let~$p$ be a prime and suppose that there is a polarization
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$\lambda : A \ra A^{\vee}$ of degree coprime to~$p$.
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If $p=2$ assume also that~$\lambda$ arises from an $F$-rational
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divisor on~$A$ (this hypothesis is automatic if~$A$ is
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an elliptic curve, but can fail in general).
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If $\Sha(A/F)[p^\infty]$ is finite then $\#\Sha(A/F)[p^\infty]$
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is a perfect square.
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\end{theorem}
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\begin{proof}
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If~$\lambda$ is $F$-rational, the
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Cassels--Tate pairing on $\Sha(A/F)[p^\infty]$ (induced by~$\lambda)$
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is nondegenerate and alternating (see \cite{tate:duality}),
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so $\#\Sha(A/F)[p^\infty]$ is a
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perfect square. Even when~$\lambda$ is not $F$-rational, the
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Cassels--Tate pairing is nondegenerate and antisymmetric (see
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\cite{flach:pairing}), which when~$p$ is odd implies that
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$\#\Sha(A/F)[p^\infty]$ is a perfect square.
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\end{proof}
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It is tempting to
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conjecture that $\#\Sha(A/F)$ is always a perfect square. Perhaps
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squareness is a fundamental property of Shafarevich--Tate groups?
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While implementing algorithms based on \cite{poonen-schaefer} for
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computing with Jacobians of hyperelliptic curves, M.~Stoll was shocked to
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discover an example of an abelian variety of dimension two such
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that $\#\Sha(A/F)[2^{\infty}]=2$. This was surprising because,
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for example, one finds in the literature \cite[pg.149]{sd:bsd}
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the following statement:
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``[The group $\Sha(A/F)$] is conjectured to be finite,
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and Tate [26] has shown that if it is finite its order
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is a perfect square.''
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Stoll and B.~Poonen discovered what hid behind
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this and other similar examples in which $\#\Sha(A/F)$ is twice
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a perfect square.
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An algebraic curve~$X$ of genus~$g$ over a local
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field~$k$ is {\em deficient} if~$X$ has no $k$-rational divisor
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of degree $g-1$.
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\begin{theorem}[Poonen-Stoll \cite{poonen-stoll}]\label{thm:ps}
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Suppose~$A$ is the Jacobian of an algebraic curve over~$F$ that
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is deficient at an odd number of places. If $\#\Sha(A/F)$ is
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finite, then $\#\Sha(A/F)$ is twice a square.
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\end{theorem}
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For example, they prove that the
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Jacobian~$J$ of the nonsingular projective curve defined by
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$$
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y^2 = -3(x^2+1)(x^2-6x+1)(x^2+6x+1)
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$$
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has Shafarevich--Tate group of order~$2$
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(to see that $\#\Sha(J)\mid 2$ they observe that~$J$ is isogenous
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to a product of CM elliptic curves and apply a theorem of Rubin;
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see~\cite[Prop.~27]{poonen-stoll} for details).
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Also, Jordan and Livn\'e
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\cite{jordan-livne:sha} give an infinite
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family of Atkin--Lehner quotients of Shimura curves which are deficient
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at an odd number of places.
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Though $\#\Sha(A/F)$ need not be square, one might still be tempted to
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conjecture that $\Sha(A/F)$ must have order
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either a square or twice a square. Let~$p$ be an odd prime.
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In this paper, we construct (under certain hypotheses that are
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satisfied for $p<25000$) abelian varieties~$A$ such
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that $\#\Sha(A/\Q)=pn^2$ for some integer~$n$. For
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example (see Section~\ref{sec:ex}):
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\begin{theorem}
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Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
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For every odd prime $p<\testbnd$ (with $p\neq 37$), there is a twist~$A$ of
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$E^{\cross (p-1)}$ such that $\#\Sha(A/\Q)=pn^2$ for
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some integer~$n$.
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\end{theorem}
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%The abelian variety $A$ of the theorem is the kernel of the
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%trace map $R\ra E$, where $R=\Res_{K/\Q}(E_K)$ is the restriction
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%of scalars of $E$ from an abelian extension~$K$ of~$\Q$.
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%Since $\#\Sha(R)$ is a perfect square,
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%our construction also gives an example of an isogeny $\phi : A \ra R$
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%of abelian varieties such that $\#\Sha(A)$ and $\#\Sha(B)$ do not
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%differ by a perfect square.
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This paper was originally motivated by the problem of relating the
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conjecture of Birch and Swinnerton-Dyer about the ranks of elliptic
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curves~$E$ to the Birch and Swinnerton-Dyer formula for the orders
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$\#\Sha(A)$ for abelian varieties~$A$ of analytic rank~$0$.
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Let~$p$ be a prime. Under suitable hypotheses, we construct an abelian
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variety~$A$ and a natural map $E(\Q)/p E(\Q) \hra
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\Sha(A/\Q)$. Thus if $E(\Q)\isom \Z$ then $\Sha(A/\Q)$ has a natural
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subgroup of order~$p$, and no other natural subgroup of order~$p$
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presents itself. Moreover, when $E$ is defined by $y^2+y=x^3-x$,
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the Birch and Swinnerton-Dyer formula predicts that $\Sha(A/\Q)[3]$ is
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of order~$3$. Further investigation led to the results of this paper.
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\vspace{2ex}
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\par\noindent{}{\bf{}Acknowledgement: } It is a pleasure
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to thank Kevin Buzzard, Frank Calegari,
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Sol Friedberg, Benedict Gross, Emmanuel Kowalski, Barry Mazur, Bjorn
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Poonen, and David Rohrlich for their helpful comments, and in particular
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Michael Stoll for Lemma~\ref{lem:red_mod_n} and Cristian Gonz\'{a}lez
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for carefully reading this paper, making many comments, and
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sending me a proof of Proposition~\ref{prop:h2}.
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\subsection{Notation}
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If~$G$ is an abelian group and $n$ is an integer, then
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$G[n]$ denotes the subgroup of elements of order~$n$ and
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$G[n^\infty]$ is the subgroup of elements of order any power of~$n$.
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We refer to elliptic curves using the notation of \cite{cremona:algs}.
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\section{Construction of Nonsquare Shafarevich--Tate Groups}
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For the rest of this paper we will work with an elliptic curve~$E$
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over~$\Q$. Aside from the significant use of known cases of the Birch
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and Swinnerton-Dyer conjecture below, much of the construction
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should generalize to the situation when~$E$ is replaced by a
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principally polarized abelian variety over a global field.
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For the rest of this section, fix an elliptic curve~$E$ over~$\Q$.
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By \cite{breuil-conrad-diamond-taylor}, $E$ is modular so
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there is a newform $f=\sum_{n=1}^{\infty} a_n q^n$ of level
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equal to the conductor~$N=N_E$ of~$E$ such that $L(E,s)=L(f,s)$.
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For each prime $q\mid N$,
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the Tamagawa number~$c_q$ of~$E$ at~$q$ is the order of the group of
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rational components of the special fiber of the N\'eron model of~$E$
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at~$q$.
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\subsection{Twisting By Characters of Prime Order}
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Let $p$ be a prime number.
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For any prime $\ell \con 1\pmod{p}$, let
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$$
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\chi_{p,\ell} : (\Z/\ell\Z)^* \to \mu_p \subset \C^*
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$$
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be one of the $p-1$ Galois-conjugate Dirichlet characters of
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order~$p$ and conductor~$\ell$.
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\begin{conjecture}\label{conj:nonvanish}
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Suppose~$p$ is a prime such that
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$\rho_{E,p}:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective. Then there exists a
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prime $\ell\nmid N$ such that $L(E,\chi_{p,\ell},1)\neq 0$,
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$\ell\con 1\pmod{p}$ and $a_\ell\not\con \ell+1 \pmod{p}$.
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\end{conjecture}
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\begin{remarks}\mbox{}\vspace{-4ex}\\
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\begin{enumerate}
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\item Formulas involving modular symbols imply that
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$L(E,\chi_{p,\ell},1)\neq 0$ if
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and only if $L(E,\chi_{p,\ell}^\sigma,1)\neq 0$ for any $\Gal(\Qbar/\Q)$-conjugate~$\chi_{p,\ell}^\sigma$
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of~$\chi_{p,\ell}$.
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\item
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J.~Fearnley proved related nonvanishing results when
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$L(E,1)\neq 0$ in \cite{fearnley:phd}.
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\item
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If~$E$ is the elliptic curve $y^2+y=x^3-x$ of conductor~$37$ and rank~$1$,
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then $\ell=41$ is the only $\ell\con 1\pmod{5}$ with $\ell<1000$
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for which $L(E,\chi_{5,\ell},1)=0$.
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\end{enumerate}
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\end{remarks}
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The following proposition gives evidence for
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Conjecture~\ref{conj:nonvanish} for the lowest-conductor elliptic
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curves of ranks $1$, $2$, and $3$.
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\begin{proposition}\label{prop:conjtest}
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Conjecture~\ref{conj:nonvanish} is true for
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the rank~$1$ elliptic curve \nf{37A} for
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every odd $p<\testbnd$ (with $p\neq 37$).
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The conjecture is true for the rank~$2$ curve
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\nf{389A} for every odd $p<1000$ (with $p\neq 389$).
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The conjecture is true for the rank~$3$ curve
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\nf{5077A} for every odd $p<1000$.
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\end{proposition}
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\begin{proof}
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Consider the modular symbol
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$$
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e_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} \chi_{p,\ell}(a)
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\cdot \left\{0,\,\, \frac{a}{\ell}\right\}
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\in H_1(X_0(N),\Q(\zeta_p)).
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$$
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Then $L(E,\chi_{p,\ell},1)\neq 0$ if and only if the
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image of $e_{p,\ell}$ under
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$$
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H_1(X_0(N),\Q(\zeta_p)) \ra H_1(E,\Q(\zeta_p))
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$$
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is nonzero. In any particular case, we can use modular
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symbols to determine whether or not this image is nonzero.
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When~$p$ is large, it is difficult to compute in the
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field $\Q(\zeta_p)$, so instead we compute in the residue class
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field $\Fell=\Z[\zeta_p]/\m\isom Z/\ell\Z$, where~$\m$ is one of
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the maximal ideals of $\Z[\zeta_p]$ that lies over~$\ell$.
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(Note that $\ell$ splits completely in $\Z[\zeta_p]$
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because $\ell\con 1\pmod{p}$.) After reducing modulo~$\m$,
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we compute the image of
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$$
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\overline{e}_{p,\ell} = \sum_{a\in (\Z/\ell\Z)^*} a^{(\ell-1)/p}
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\cdot \left\{0, \,\,\frac{a}{\ell}\right\}
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\in H_1(X_0(N),\Fell)
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$$
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in $H_1(E,\Fell)$.
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If it is nonzero, then the image of $e_{p,\ell}$
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in $H_1(E,\Q(\zeta_p))$ is nonzero.
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A big computation (that takes hundreds of hours using
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{\sc Magma} \cite{magma})
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shows that the image of $\overline{e}_{p,\ell}$
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is nonzero in the cases asserted by the proposition.
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So the reader can carry out similar computations,
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we include the following {\sc Magma} V2.10-6 code,
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which illustrates verification of the proposition
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for {\bf 37A} for $p<100$:
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\begin{verbatim}
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procedure VerifyConjecture(E, p)
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assert Type(E) eq CrvEll;
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assert Type(p) eq RngIntElt and IsPrime(p) and IsOdd(p);
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N := Conductor(E);
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assert N mod p ne 0;
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M := ModularSymbols(E,+1); // takes a long time if N large!
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ell := 3; t := Cputime();
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printf "p=%o: ", p;
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while true do
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while (ell mod p ne 1) or (N mod ell eq 0) or
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TraceOfFrobenius(ChangeRing(E,GF(ell))) mod p eq (ell+1) do
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ell := NextPrime(ell);
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end while;
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k := FiniteField(ell);
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printf "trying ell=%o...",ell;
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psi := DirichletGroup(ell,k).1;
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eps := psi^(Order(psi) div p); // order p character
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M_k := BaseExtend(M,k);
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phi := RationalMapping(M_k);
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e := TwistedWindingElement(M_k,1,eps);
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if phi(e) ne 0 then
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printf " success! (%o seconds)\n", Cputime(t);
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return;
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end if;
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printf "failed. ";
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ell := NextPrime(ell);
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end while;
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end procedure;
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E := EllipticCurve([0,0,1,-1,0]); // 37A
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for p in [q : q in [3..100] | IsPrime(q) and q ne 37] do
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VerifyConjecture(E,p);
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end for;
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\end{verbatim}
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The above input results in the following abbreviated output:
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\begin{verbatim}
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p=3: trying ell=7... success! (0.021 seconds)
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p=5: trying ell=11... success! (0.039 seconds)
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p=7: trying ell=29... success! (0.121 seconds)
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...
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p=89: trying ell=179... success! (0.739 seconds)
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p=97: trying ell=389... success! (1.491 seconds)
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\end{verbatim}
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\end{proof}
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\subsection{A Restriction of Scalars Exact Sequence}
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As above, $E$ is an elliptic curve over~$\Q$. Let~$p$
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be any prime (note that $p=2$ is allowed).
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Suppose $\ell\con 1\pmod{p}$ is another prime and that
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$\ell\nmid N_E$.
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Let $K\subset \Q(\mu_\ell)$ be the abelian
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extension of~$\Q$ that corresponds to~$\chi_{p,\ell}$
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(thus~$K$ is the unique subfield of $\Q(\mu_\ell)$ of
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degree~$p$).
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Let $R = \Res_{K/\Q}(E_K)$ be the restriction of scalars down
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to~$\Q$ of~$E$ viewed as an elliptic curve over~$K$. Thus~$R$
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is an abelian variety over~$\Q$ of dimension $p=[K:\Q]$.
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It is characterized by the fact that it
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represents the following functor on $\Q$-schemes~$S$:
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$$
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S \mapsto E_K(S_K).
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$$
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As a Galois module,
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$$
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R(\Qbar) = E(\Qbar)\tensor_\Z \Z[\Gal(K/\Q)],
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$$
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where
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$\tau\in \Gal(\Qbar/\Q)$ acts on
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$\sum P_{\sigma} \tensor \sigma$ by
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$$\tau\left(\sum P_\sigma\tensor \sigma\right) =
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\sum \tau(P_\sigma)\tensor \tau_{|K}\cdot\sigma,
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$$
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where $\tau_{|K}$ is the image of~$\tau$ in $\Gal(K/\Q)$.
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\begin{proposition}\label{prop:exactabvar}
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The identity map induces a closed immerion $\iota: E\hookrightarrow
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R$, and the trace $\Tr:K\ra \Q$ induces a surjection $\Tr:R\ra E$
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whose kernel is geometrically connected. Thus we have an exact
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sequence of abelian varieties
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\begin{equation}\label{eqn:exactabvar}
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0 \ra A \ra R \xra{\Tr} E \ra 0.
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\end{equation}
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\end{proposition}
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\begin{proof}
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The existence of~$\iota$ and $\Tr$ follows from Yoneda's lemma.
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The map~$\iota$ is induced by the functorial inclusion
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$E(S)\hookrightarrow E_K(S_K)=R(S)$, so~$\iota$ is injective.
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The $\Tr$ map is induced by the functorial trace map on points
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$R(S)=E_K(S_K)\xrightarrow{\Tr} E(S)$.
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To verify that $\Ker(\Tr)$ is geometrically connected, we base
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extend the exact sequence (\ref{eqn:exactabvar}) to~$\Qbar$. First,
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note that there is an isomorphism
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$$
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R_{\Qbar} \isom E_{\Qbar}\cross \cdots \cross E_{\Qbar}.
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$$
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After base extension, we identify
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the trace map with the summation map
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$$
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+: E_{\Qbar} \cross \cdots \cross E_{\Qbar}
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\longrightarrow E_{\Qbar}.
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$$
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Let $n=[K:\Q]$. The map defined by
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$$\left(a_1,\ldots, a_{n-1}\right) \mapsto
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\left(a_1,a_2,\ldots ,a_{n-1},-\sum_{i=1}^{n-1} a_i\right),$$
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is an isomorphism from
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$E_{\Qbar}^{\times (n-1)}$ to $\Ker(+)=\Ker(\Tr_{\Qbar})$.
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Thus $\Ker(\Tr_{\Qbar})$ is isomorphic to
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a product of copies of $E_{\Qbar}$, and hence is connected.
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\end{proof}
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\begin{corollary}\label{cor:intersection}
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$
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\ds\iota(E)\intersect \Ker(\Tr) = \iota(E)[p].
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$
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\end{corollary}
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\begin{proof}
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The composition
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$\Q\hookrightarrow K\xrightarrow{\Tr} \Q$
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is multiplication by~$p$, so
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the composition
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$E \xra{\,\,\iota\,\,} R \xra{\Tr} E$
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is also multiplication by~$p$.
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Since
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$\iota(E)\intersect \Ker(\Tr)$ is the kernel of~$\Tr\circ\iota = [p]$,
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it equals $E[p]$.
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\end{proof}
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\begin{lemma}\label{lem:powers}
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The abelian varieties
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$A_K$, $R_K$, and $(R/\iota(E))_K$ are all isomorphic to
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a product of copies of $E_K$.
486
\end{lemma}
487
\comment{
488
\begin{proof}
489
Base extending (\ref{eqn:eraprime}) to~$K$
490
we have an exact sequence
491
$$
492
0 \ra E_K \xra{\Delta} E_K^{\times p} \ra A'_K \ra 0.
493
$$
494
Embed $E_K^{\times (p-1)}$ in $E_K^{\times p}$ by
495
$(a_1,\ldots, a_{p-1}) \mapsto (a_1,\ldots, a_{p-1},0)$.
496
Then $E_K^{\times (p-1)}$ maps injectively to $A'_K$, since
497
$E_K^{\times (p-1)}$ has $0$ intersection with the diagonal,
498
so $A'_K \ncisom E_K^{\times (p-1)}$.
499
\end{proof}
500
}
501
502
503
504
\begin{proposition}\label{prop:exactneron}
505
The exact sequence $0\ra A \ra R \ra E\ra 0$ of
506
Proposition~\ref{prop:exactabvar} extends to
507
an exact sequence
508
$ 0 \ra \cA \ra \cR \ra \cE \ra 0$
509
of N\'eron models over~$\Z$.
510
\end{proposition}
511
\begin{proof}
512
We use results of \cite[Ch.~7]{neronmodels} and the
513
fact that formation of N\'eron models commutes with unramified base
514
change (see \cite[\S1.2, Prop.~2]{neronmodels})
515
to prove that for every prime~$q$, the complex
516
\begin{equation}\label{eqn:neron}
517
0 \ra \cA_{\Z_q} \ra \cR_{\Z_q} \ra \cE_{\Z_q}\ra 0
518
\end{equation}
519
is exact.
520
521
First suppose that $q\neq \ell$, and let~$\q$ be a prime
522
of~$K$ lying over~$q$. We use the fact that formation of N\'eron models
523
commutes with unramified base extension and check exactness
524
of (\ref{eqn:neron}) after base extension to the
525
unramified extension $\O_{K,\q}$ of $\Z_q$.
526
By Lemma~\ref{lem:powers}, the generic fiber of the base
527
extension of (\ref{eqn:neron}) to $\O_{K,\q}$ is
528
$$
529
0\ra E_{K,\q}^{\oplus (n-1)} \ra
530
E_{K,\q}^{\oplus n}\xra{\Sigma} E_{K,\q}\ra 0.
531
$$
532
Thus the corresponding complex of N\'eron models over $\O_{K,\q}$ is
533
$$
534
0\ra \cE_{\O_{K,\q}}^{\oplus (n-1)} \ra \cE_{\O_{K,\q}}^{\oplus n}
535
\xra{\Sigma} \cE_{\O_{K,\q}}\ra 0,
536
$$
537
which is exact, since it is exact on $S$-points for {\em any}
538
ring~$S$.
539
540
Suppose that $q=\ell$. Since $p\neq \ell$,
541
\cite[Prop.~7.5.3 (a)]{neronmodels} asserts
542
that the sequence
543
$0 \ra \cA_{\Z_q} \ra \cR_{\Z_q} \ra \cE_{\Z_q}$
544
is exact.
545
Since $p\neq q$,
546
the map $[p]: \cE_{\Z_q} \to \cE_{\Z_q}$ is
547
an \'etale morphism of smooth schemes.
548
Since~$E$ has good reduction at~$q$, we also
549
know that the fibers of $\cE_{\Z_q}$ are geometrically connected,
550
so $[p]$ is surjective (for more details, see the proof
551
of~\cite[Lem.~3.2]{agashe-stein:visibility}).
552
It follows that $\cR_{\Z_q} \ra \cE_{\Z_q}$ is surjective.
553
554
\end{proof}
555
556
557
558
\subsection{The Cokernel of Trace}
559
Let~$\ell$ be a prime as in Conjecture~\ref{conj:nonvanish}.
560
This section is devoted to computing the cokernel
561
of the trace map $R(\Q) \ra E(\Q)$. Note that $R(\Q)=E(K)$, so
562
this cokernel is also $E(\Q)/\Tr_{K/\Q}(E(K))$.
563
564
\begin{lemma}\label{lem:kellsize}
565
Let $K_\ell$ denote the completion of $K$ at the totally ramified
566
prime of~$K$ lying over~$\ell$. Then $E(K)[p] = E(K_{\ell})[p]=0$.
567
\end{lemma}
568
\begin{proof}
569
The characteristic polynomial of
570
$\Frob_{\ell}\in\Gal(\Q_\ell^{\ur}/\Q_\ell)$
571
on $E[p] = E(\Q_\ell^{\ur})[p]$ is $x^2 - a_\ell x + \ell \in \F_p[x]$.
572
By hypothesis $a_\ell \not\con \ell+1\pmod{p}$, so
573
$+1$ is not a root of $x^2 - a_\ell x + \ell$ hence
574
$$
575
E(\Q_\ell)[p] = E(\Q_\ell^{\ur})[p]^{\Frob_{\ell}-1} = 0.
576
$$
577
Since~$K$ is totally ramified at~$\ell$ and~$E$ has
578
good reduction at~$\ell$, $E(K_\ell)[p]=0$ as well,
579
so $E(K)[p]=0$, as required.
580
\end{proof}
581
582
\begin{proposition}\label{prop:coker}
583
$\ds
584
\Coker(R(\Q)\ra E(\Q)) \isom E(\Q)/p E(\Q).
585
$
586
\end{proposition}
587
\begin{proof}
588
By Corollary~\ref{cor:intersection} the
589
the image of $\iota(E(\Q))\subset R(\Q)$ in $E(\Q)$
590
is $p E(\Q)$, so the
591
cokernel of $R(\Q)\ra E(\Q)$
592
is a quotient of $E(\Q)/p E(\Q)$.
593
Thus it suffices to prove that $R(\Q)/\iota(E(\Q))$
594
is {\em finite} of order coprime to~$p$.
595
596
We have an exact sequence
597
$0 \ra E \ra R \ra A' \ra 0$,
598
with $A'$ an abelian variety that is isogenous to~$A$
599
(in fact, $A'$ is the abelian variety dual of~$A$ since~$R$
600
is self dual, but we will not use this fact.)
601
The $L$-series of $A'$ is
602
$\prod_{i=1}^{p-1} L(E,\chi_{p,\ell}^i,s)$,
603
so by hypothesis $L(A',1)\neq 0$ and
604
it follows from Kato's theorem (see \cite[\S8.1]{rubin:kato})
605
that $A'(\Q)$ is finite.
606
Thus $R(\Q)/\iota(E(\Q))$ is finite since $R(\Q)/\iota(E(\Q))\subset A'(\Q)$.
607
By Lemma~\ref{lem:powers}, $A'_K \ncisom E_K^{\times (p-1)}$
608
and by Lemma~\ref{lem:kellsize} $E(K)[p]=0$, so
609
$A'(\Q)[p]=0$, which proves the proposition.
610
\end{proof}
611
612
\subsection{\'Etale Cohomology and Shafarevich--Tate Groups}\label{sec:etale}
613
Fix an elliptic curve~$E$ over~$\Q$ and a prime $p\nmid \prod c_{E,q}$.
614
615
In this section, we use results mostly due to Mazur to relate the
616
Shafarevich--Tate groups of~$A$,~$R$, and~$E$ to certain \'etale
617
cohomology groups.
618
We maintain the notation and assumptions
619
of the previous sections, except
620
that we abuse notation slightly and let $\cA$, $\cR$, and $\cE$ also
621
denote the \'etale sheaves on $\Spec(\Z)$ defined by
622
the N\'eron models $\cA$, $\cR$, and $\cE$.
623
Let $\cB$ be either
624
$\cA$, $\cR$, or $\cE$ and let $B=\cB_\Q$
625
be the corresponding abelian variety.
626
Let $H^q(\Z,\cB)$ be the $q$th \'etale cohomology group of~$\cB$.
627
628
\begin{lemma}\label{lem:red_mod_n}
629
There is an isomorphism
630
$B(\Q_\ell)[p] \isom \cB(\Fell)[p].$
631
\end{lemma}
632
\begin{proof}
633
This follows from
634
\cite[Lem.~2, pg.~495]{serre-tate},
635
but we sketch a proof for the convenience of the reader.
636
Let $B^{1}(\Q_\ell)$ denote the kernel of the natural reduction
637
map $r:B(\Q_\ell)\ra \cB(\Fell)$. Using formal groups and
638
that $p\neq \ell$,
639
one sees that $[p]:B^{1}(\Q_\ell)\ra{}B^{1}(\Q_\ell)$ is an isomorphism.
640
Since $\cB$ is smooth over~$\Q_\ell$,
641
Hensel's lemma (see \cite[\S2.3~Prop.~5]{neronmodels})
642
implies that the reduction map
643
is surjective, so we obtain an exact sequence
644
$$
645
0\ra B^1(\Q_\ell) \ra B(\Q_\ell) \ra \cB(\Fell) \ra 0.
646
$$
647
The snake lemma applied to the multiplication-by-$p$ diagram
648
attached to this exact sequence yields the
649
exact sequence
650
$$0\ra B(\Q_\ell)[p]\ra \cB(\Fell)[p] \ra 0 \ra B(\Q_\ell)/p B(\Q_\ell)
651
\ra \cB(\Fell)/p\cB(\Fell)\ra0,$$
652
which proves the lemma.
653
\end{proof}
654
655
The {\em Tamagawa number} of~$B$ at a prime~$q$
656
is $c_{B,q}=\#\Phi_{B,q}(\F_q)$, where $\Phi_{B,q}$
657
is the component group of the closed fiber
658
of the N\'eron model of~$B$ at~$q$.
659
%The {\em geometric Tamagawa number} is $\cbar_{B,q} = \#\Phi_{B,q}(\Fbar_q)$.
660
661
\begin{lemma}\label{lem:boundcq}
662
$p\nmid c_{B,q}$.
663
\end{lemma}
664
\begin{proof}
665
First suppose $q=\ell$.
666
The cokernel of
667
$\cB(\F_\ell) \ra \Phi_{B,\ell}(\F_\ell)$
668
is contained in $H^1(\F_\ell,\cB^0)$, which
669
is~$0$ by Lang's theorem (see \cite{lang:finitefields} or
670
\cite[\S{}VI.4]{serre:alggroups}),
671
so if $\Phi_{B,\ell}(\F_\ell)[p]\neq 0$ then $\cB(\F_\ell)[p]\neq 0$. But by
672
Lemmas~\ref{lem:powers}, \ref{lem:kellsize}, and~\ref{lem:red_mod_n},
673
$$
674
\cB(\F_\ell)[p] \isom \cB(\Q_\ell)[p] \subset \cB(K_\ell)[p] \isom
675
E(K_\ell)[p]\cross \cdots \cross E(K_\ell)[p] = 0.
676
$$
677
678
Next suppose that $q\neq \ell$. Since formation of N\'eron models
679
commutes with unramified base extension, we have
680
$$
681
\Phi_{B,q}(\Fbar_q)[p] \isom
682
\Phi_{E,q}(\Fbar_q)[p] \cross \cdots \cross \Phi_{E,q}(\Fbar_q)[p] = 0,
683
$$
684
by our hypotheses on~$p$.
685
\end{proof}
686
687
Following the appendix to \cite{mazur:tower}, let
688
$$
689
\Sigma(B/\Q) = \ker\left(H^1(\Q,B)\ra \bigoplus_{\text{all finite $q$}}
690
H^1(\Q_q, B)\right),
691
$$
692
where the product is over all finite primes~$q$ of~$\Q$.
693
If~$p$ is an odd prime, then
694
$\Sigma(B/\Q)[p^\infty] = \Sha(B/\Q)[p^\infty]$;
695
also one can see easily Tate cohomology for the cyclic
696
group $\Gal(\C/\R)$ that
697
$$
698
\Sigma(B/\Q)[2]/\Sha(B/\Q)[2]\subset
699
H^1(\R,B(\C)) \isom B(\R)/B(\R)^0,
700
$$
701
where $B(\R)/B(\R)^0$ has order $2^e$ for some $e\leq \dim B$.
702
\begin{proposition}[Mazur]\label{prop:shah1}
703
Suppose that $a_\ell\not\con \ell+1\pmod{p}$.
704
If $p$ is odd, then
705
$$\ds
706
H^1(\Z,\cB)[p^{\infty}] \isom \Sha(B/\Q)[p^\infty].
707
$$
708
Also,
709
$\#H^1(\Z,\cB)[2^{\infty}] / \Sha(B/\Q)[2^\infty]$
710
divides $\#(B(\R)/B(\R)^0)$.
711
\end{proposition}
712
\begin{proof}
713
It follows from the appendix to \cite{mazur:tower} that there is an
714
exact sequence
715
\begin{equation}
716
0 \ra \Sigma(B)[p^\infty] \ra H^1(\Z,\cB)[p^\infty] \ra
717
\bigoplus_{\text{all finite $q$}} H^1\left(\F_q, \Phi_{B,q}(\Fbar_q)\right)[p^\infty],
718
\end{equation}
719
where $\Phi_{B,q}$ is the component group of the fiber of $\cB$
720
at~$q$.
721
By \cite[VIII.4.8]{serre:localfields},
722
$$
723
\#H^1(\F_q,\Phi_{B,q}(\Fbar_q)) = \#\Phi_{B,q}(\F_q) = c_{B,q},
724
$$
725
so the proposition follows from Lemma~\ref{lem:boundcq}.
726
\end{proof}
727
728
\begin{proposition}\label{prop:h2}
729
$H^2(\Z,\cA)[p] = 0$.
730
\end{proposition}
731
\begin{proof}
732
We apply the lemmas in \cite[\S{}III.6]{schneider:iwasawa}.
733
Note that~$A$ has good reduction at~$p$ by \cite[Prop.~1]{milne:bsdres},
734
and $H^1(\Z,\cA)[p^\infty]$ is finite by Kato's theorem
735
(see \cite[\S8.1]{rubin:kato}) and Proposition~\ref{prop:shah1}.
736
In the proof of Proposition~\ref{prop:coker}, we showed that
737
$A'(\Q)$ is finite of order coprime to~$p$, where $A'$
738
is the abelian variety dual of~$A$. We now use\footnote{Note that
739
the proof of Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}
740
relies on a theorem of
741
Artin and Mazur whose proof they never published;
742
generalizations of this theorem have been published by
743
McCallum \cite[\S5]{mccallum:duality} and
744
Milne \cite[\S{}III.3.4]{milne:duality}, and Mazur assures the author
745
that he and Milne both know the proof of Artin-Mazur duality well.}
746
Lemma~7 of \cite[\S{}III.6]{schneider:iwasawa}, which because
747
$A'(\Q)[p]=0$
748
implies that $H^2(\Z,\cA[p^\infty]) = 0$
749
(Schneider uses $H^q_{\text{fpqf}}$, but this is not a problem
750
since \'etale and \text{fpqf} cohomology agree on the smooth
751
scheme $\cA$.)
752
It is easy to see (see, e.g., the proof of Lemma~6
753
of \cite[\S{}III.6]{schneider:iwasawa}) that
754
the natural map $H^q(\Z,\cA[p^\infty]) \ra H^q(\Z,\cA)[p^\infty]$
755
is surjective for any $q>0$, in particular, for $q=2$,
756
so $H^2(\Z,\cA)[p^\infty]=0$ which proves the proposition.
757
\comment{
758
As discussed in [], there is an exact sequence
759
$$0\ra \cB^0 \ra \cB \ra \bigoplus_{\text{primes }q} \Phi_{B,q} \ra 0,$$
760
which leads to the exact sequence
761
$$ H^2(\Z,\cB^0)[p^\infty] \ra H^2(\Z,\cB)[p^\infty]
762
\ra \bigoplus_{\text{primes $q$}} H^2\left(\F_q, \Phi_{B,q}(\Fbar_q)\right)[p^\infty].$$
763
By \cite[\S{}VIII.4]{serre:localfields} and Lemma~\ref{lem:boundcq},
764
$$
765
\#H^2(\F_q,\Phi_{B,q})[p] = \#H^0(\F_q,\Phi_{B,q})[p] = 1,
766
$$
767
so it suffices to show that $H^2(\Z,\cB^0)[p]=0$.
768
}
769
\end{proof}
770
771
772
773
\subsection{The Main Theorem}
774
Fix an elliptic curve~$E$ over~$\Q$ and a prime
775
$p\nmid \prod c_{E,q}$
776
such that $\rho_{E,p}:G_\Q\to \Aut(E[p])$ is
777
surjective.
778
If $p=2$ assume also that $E(\R)$ is connected.
779
Assume that~$\ell$ is one of the primes whose existence
780
is predicted by Conjecture~\ref{conj:nonvanish}.
781
782
\begin{theorem}\label{thm:nonsquare}
783
There is an exact sequence
784
$$
785
0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(E/K)[p^\infty]\ra
786
\Sha(E/\Q)[p^\infty] \ra 0.
787
$$
788
In particular, if~$E$ has odd rank and $\Sha(E/\Q)[p^\infty]$ is
789
finite, then $\#\Sha(A/\Q)[p^\infty]$
790
is not a perfect square.
791
\end{theorem}
792
\begin{proof}
793
By Proposition~\ref{prop:exactneron} we have an
794
exact sequence of \'etale sheaves
795
$$
796
0 \ra \cA \ra \cR \ra \cE \ra 0,
797
$$
798
which gives rise to an exact
799
sequence of \'etale cohomology groups
800
$$
801
H^0(\Z,\cR) \ra H^0(\Z,\cE)
802
\ra H^1(\Z,\cA) \ra H^1(\Z,\cR) \ra H^1(\Z,\cE) \ra H^2(\Z,\cA).
803
$$
804
We have
805
$$
806
H^0(\Z, \cR) = \cR(\Z) = R(\Q)$$
807
and likewise for $\cE$, so by Propositions~\ref{prop:coker},~\ref{prop:shah1},
808
and \ref{prop:h2} we obtain an exact sequence
809
$$
810
0 \ra E(\Q)/p E(\Q) \ra \Sha(A/\Q)[p^\infty] \ra \Sha(R/\Q)[p^\infty] \ra
811
\Sha(E/\Q)[p^\infty] \ra 0.
812
$$
813
By Shapiro's lemma, there is an isomorphism $\Sha(R/\Q)\isom \Sha(E/K)$
814
(see \cite[\S1.3]{agashe-stein:visibility}), which yields the
815
claimed exact sequence.
816
817
Kato's theorem (\cite[\S8.1]{rubin:kato} and \cite[Cor.~14.3]{kato:secret})
818
implies that $\Sha(E/K)[p^\infty]$ is
819
finite (for the trivial character use our hypothesis
820
that $\Sha(E/\Q)[p^\infty]$
821
is finite, and for the nontrivial characters use our hypothesis
822
that $L(E,\chi_{p,\ell},1)\neq 0$).
823
Theorem~\ref{thm:tate} then implies that $\#\Sha(E/K)[p^\infty]$ is a
824
perfect square. If $E(\Q)$ has odd rank then $\#(E(\Q)/p E(\Q))$
825
is an odd power of~$p$
826
(since $E[p]$ is irreducible), so
827
$\#\Sha(A/\Q)[p^\infty]$ cannot be a perfect square.
828
\end{proof}
829
830
\begin{remark}
831
In the language of visibility of Shafarevich-Tate
832
groups (see~\cite{cremona-mazur}),
833
Theorem~\ref{thm:nonsquare}
834
asserts that the
835
visible subgroup of $\Sha(A)$ with respect to the
836
embedding $A\hra R$ is canonically isomorphic to
837
the Mordell-Weil quotient $E(\Q)/p E(\Q)$.
838
\end{remark}
839
840
\begin{proposition}\label{prop:away_from_p}
841
If $q\neq p$ is a prime, then $\Sha(A/\Q)[q^\infty]$ has order
842
a perfect square.
843
\end{proposition}
844
\begin{proof}
845
Let $\phi$ be the canonical prinicipal polarization on~$R$
846
induced by the canonical polarization on the elliptic curve~$E$.
847
Let~$\lambda$ be the polarization on~$A$ obtained by pulling back~$\phi$.
848
By Corollary~\ref{cor:intersection}, the polarization
849
$\lambda$ has degree~$p^2$, and
850
since~$\phi$ arises from a rational divisor, so does $\lambda$.
851
Now apply Theorem~\ref{thm:tate}.
852
\end{proof}
853
854
855
856
\section{An Example}\label{sec:ex}
857
Combining Proposition~\ref{prop:conjtest}, Theorem~\ref{thm:nonsquare},
858
and Proposition~\ref{prop:away_from_p} yields the following theorem.
859
\begin{theorem}\label{thm:37}
860
Let $E$ be the elliptic curve $y^2 + y = x^3 - x$ of conductor~$37$.
861
For every odd prime $p<\testbnd$ (with $p\neq 37$), there is a twist~$A$ of
862
$E^{\cross (p-1)}$ such that $\#\Sha(A/\Q)=pn^2$
863
for some integer~$n$.
864
\end{theorem}
865
\begin{remark}
866
Using the elliptic curve of conductor $43$ in place of~$E$
867
one can construct an abelian variety $A$ with $\Sha(A/\Q)=37n^2$
868
for some integer~$n$.
869
\end{remark}
870
871
Though unnecessary for Theorem~\ref{thm:37}, we
872
prove below that $\Sha(E/\Q)=0$, which removes our
873
dependence on Proposition~\ref{prop:h2}.
874
We show that $\Sha(E/\Q)[p^\infty]=0$ for all odd~$p$
875
using \cite[Thm.~A]{kolyvagin:euler_systems}, and
876
we use a $2$-descent (with \cite{mwrank}) to see that $\Sha(E/\Q)[2]=0$.
877
\begin{theorem}[Kolyvagin]\label{thm:kolybound}
878
Let~$E$ be an elliptic curve
879
and let $L=\Q(\sqrt{-D})$ be an imaginary quadratic
880
field of odd discriminant $-D$, where all primes dividing the conductor
881
of~$E$ split, and assume
882
that $D\neq 3,4$. If the Heegner point $y_L\in E(L)$ has infinite order (equivalently,
883
by \cite{gross-zagier}, $L'(E/L,1)\neq 0$),
884
then $\#\Sha(E/L)\mid t\cdot [E(L): \Z y_L]^2$,
885
where the only primes that
886
divide~$t$ are~$2$ or primes where $\rho_{E,p}$ is not surjective.
887
\end{theorem}
888
889
By \cite{cremona:algs},~$E$ is isolated in its isogeny class,
890
so $\rho:\Gal(\Qbar/\Q)\ra \Aut(E[p])$ is surjective for all primes~$p$
891
(see \cite[\S1.4]{ribet-stein:serre})
892
hence~$t$ is a power of~$2$.
893
Let $L = \Q(\sqrt{-7})$.
894
To compute $[E(L):\Z{}y_L]$ up to a power of~$2$
895
we use the Gross-Zagier formula and the fact that $[E(L) : E(\Q)+ E^D(\Q)]$
896
is a power of~$2$. By \cite[Thm.~6.3]{gross-zagier},
897
$$
898
h(y_L) = \frac{u^2|D|^{\frac{1}{2}}}{\|\omega_f\|} L'(E,1)L(E^D,1),
899
$$
900
where $D=-7$, $u=1$, and $\|\omega_f\|$
901
is the Peterson norm of the newform~$f$ corresponding to~$E$.
902
Generators for the period lattice of~$E$ are
903
$\omega_1 \sim 2.993459$ and
904
$\omega_2 \sim 2.451389i$, so $\|\omega_f\|\sim 7.338133$.
905
The quadratic twist $E^D$ is the curve \nf{1813B1} in \cite{cremona:onlinetables},
906
and $E^D(\Q)=0$. From \cite{cremona:onlinetables} we find that
907
$L'(E,1) \sim 0.306000$
908
and $L(E^D,1) \sim 1.853076$,
909
so $h(y_L)\sim 0.204446$.
910
The height of a generator of $E(\Q)$ is
911
$\sim 0.051111\sim h(y_L)/4$, so
912
$[E(L):\Z{}y_L]$ is a power of~$2$.
913
(As a double check, and to avoid dependence on
914
the Gross-Zagier formula, we wrote a program using
915
\cite{magma} to compute Heegner points and
916
found that $y_L=(0,0)$, which is a generator for $E(\Q)$.)
917
Thus $\#\Sha(E/L)$ is a power of~$2$.
918
919
To connect $\Sha(E/L)$ with $\Sha(E/\Q)$,
920
use the inflation-restriction exact sequence
921
$$
922
0 \ra H^1(L/\Q, E(L)) \ra H^1(\Q,E(\Qbar)) \ra H^1(L,E(\Qbar)).
923
$$
924
Let~$p$ be an odd prime.
925
Since $H^1(L/\Q,E(L))$ is a $2$-group, the above sequence leads
926
to an injective map
927
$$
928
H^1(\Q,E(\Qbar))[p] \hra H^1(L, E(\Qbar))[p],
929
$$
930
which induces an inclusion
931
$$
932
\Sha(E/\Q)[p] \hra \Sha(E/L)[p] = 0.
933
$$
934
935
936
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
937
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
938
% \MRhref is called by the amsart/book/proc definition of \MR.
939
\providecommand{\MRhref}[2]{%
940
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
941
}
942
\providecommand{\href}[2]{#2}
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\end{document}
1085