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Author: William A. Stein
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\begin{document}
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\title{Shafarevich-Tate Groups of Modular Motives}
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\author{Neil Dummigan}
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\author{William A. Stein}
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%\author{??Mark Watkins (hopefully)??}
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\date{August 2001}
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\subjclass{11F33, 11F67, 11G40.}
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\keywords{modular form,$L$-function, Bloch-Kato conjecture,
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Shafarevich-Tate group.}
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\address{University of Sheffield\\ Department of Pure
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Mathematics\\
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Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
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U.K.}
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\address{Harvard University\\ Mathematics Department\\
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1 Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
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%\address{Penn State Mathematics Department\\
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%University Park, State College, PA 16802\\ U.S.A.}
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\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
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%\email{[email protected]}
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\begin{abstract}
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\end{abstract}
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\maketitle
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\section{Introduction}
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\section{Motives and Galois representations}
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Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
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$\Gamma_0(N)$, with coefficients in an algebraic number field $E$.
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A theorem of Deligne \cite{De1} implies the existence, for each
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(finite) prime $\lambda$ of $E$, of a continuous representation
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$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda})$$
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($V_{\lambda}$ is a two-dimensional vector space over
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$E_{\lambda}$), such that
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\begin{enumerate}
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\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
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$lN$ (where $\lambda \mid l$);
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\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
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characteristic polynomial of $\Frob_p^{-1}$ acting on
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$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
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\end{enumerate}
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Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
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the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
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There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
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both $2$-dimensional $E$-vector spaces. For details of the
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construction see \cite{Sc}. The de Rham realisation has a Hodge
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filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset
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F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
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cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
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cohomology. There are natural isomorphisms $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. Using a basis for singular
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cohomology with $\ZZ$-coefficients, we get
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-modules $T_{\lambda}$
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inside each $V_{\lambda}$. Define
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$A_{\lambda}=V_{\lambda}/T_{\lambda}$. There are two kinds of
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twist we shall have to consider. There is the Tate twist
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$V_{\lambda}(j)$ (for an integer $j$), which amounts to
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multiplying the action of $\Frob_p$ by $p^j$. For $D$ the
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discriminant of a quadratic field, there is the quadratic twist
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$V_{\lambda}(\chi_D)$, which is the tensor product of
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$V_{\lambda}$ with a one-dimensional space on which
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$\Gal(\Qbar/\QQ)$ acts via the quadratic character $\chi_D$.
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Following \cite{BK} (Section 3), for $p\neq l$ (including
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$p=\infty$) let
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$$H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))=\ker (H^1(D_p,V_{\lambda}(\chi_D,j))\rightarrow
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H^1(I_p,V_{\lambda}(\chi_D,j))).$$ The subscript $f$ stands for
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``finite part''. $D_p$ is a decomposition subgroup at a prime
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above $p$, $I_p$ is the inertia subgroup, and the cohomology is
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for continuous cocycles and coboundaries. For $p=l$ let
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$$H^1_f(\QQ_l,V_{\lambda}(\chi_D,j))=\ker (H^1(D_l,V_{\lambda}(\chi_D,j))\rightarrow
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H^1(D_l,V_{\lambda}(\chi_D,j)\otimes B_{\cris}))$$ (see Section 1
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of \cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
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$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(\chi_D,j))$ be the subspace
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of elements of $H^1(\QQ,V_{\lambda}(\chi_D,j))$ whose local
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restrictions lie in $H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$ for all
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primes $p$.
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There is a natural exact sequence
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$$\begin{CD}0@>>>T_{\lambda}(\chi_D,j)@>>>V_{\lambda}(\chi_D,j)@>\pi>>A_{\lambda}(\chi_D,j)@>>>0\end{CD}.$$
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Let
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$H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(\chi_D,j))$.
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Define the $\lambda$-Selmer group \newline
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$H^1_f(\QQ,A_{\lambda}(\chi_D,j))$ to be the subgroup of elements
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of $H^1(\QQ,A_{\lambda}(\chi_D,j))$ whose local restrictions lie
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in $H^1_f(\QQ_p,A_{\lambda}(\chi_D,j))$ for all primes $p$. Note
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that the condition at $p=\infty$ is superfluous unless $l=2$.
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Define the Shafarevich-Tate group
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$$\Sha(\chi_D,j)=\oplus_{\lambda}H^1_f(\QQ,A_l(\chi_D,j))/\pi_*H^1_f(\QQ,V_l(\chi_D,j)).$$
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The length of its $\lambda$-component may be taken for the
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exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Sha(\chi_D,j)$. We shall only concern ourselves with the case
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$j=k/2$, and write $\Sha(\chi_D)$ for $\Sha(\chi_D,j)$.
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Define the set of global points
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$$\Gamma_{\QQ}(\chi_D)=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(\chi_D,k/2)).$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. The length of its $\lambda$-component may be taken
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for the exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Gamma_{\QQ}(\chi_D)$.
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\section{The Bloch-Kato conjecture}
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The Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts that
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$$\frac{L(f,k/2)}{\vol_{\infty}}=\frac{\left(\prod_pc_p(k/2)\right)\cdot\#\Sha}{(\#\Gamma_{\QQ})^2}.$$
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The local factors $\vol_{\infty}$ and $c_p(k/2)$ depend on the
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choice of a basis for $V_{\dR}$, but their product does not. For
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$l\neq p$, $\ord_{\lambda}(c_p(\chi_D,j))$ is defined to be
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$$\length H^1_f(\QQ_p,T_{\lambda}(\chi_D,j))_{\tors.}-\ord_{\lambda}((1-\chi_D(p)a_pp^{-j}+p^{k-1-2j})).$$
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We omit the definition of $\ord_{\lambda}(c_p(\chi_D,j))$ for
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$\lambda\mid p$, which requires one to assume Fontaine's de Rham
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conjecture (\cite{Fo}, Appendix A6). The above formula is to be
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interpreted as an equality of fractional ideals of $E$. (Strictly
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speaking, the conjecture in \cite{BK} is only given for $E=\QQ$.)
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Similarly, the Bloch-Kato conjecture for $M_f(\chi_D,k/2)$
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predicts that
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$$\frac{L(f,\chi_D,k/2)}{\vol_{\infty}(\chi_D)}=\frac{\left(\prod_pc_p(k/2,\chi_D)\right)\#\Sha(\chi_D)}{(\#\Gamma_{\QQ}(\chi_D))^2}.$$
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A careful treatment of real periods
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such as $\vol_{\infty}(\chi_D)$ is given in \cite{De2}. The
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calculation of the periods of Artin motives in terms of
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generalized Gauss sums, in Section 6 of \cite{De2}, yields the
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following lemma.
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\begin{lem}
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If $\chi_D(-1)=1$ (i.e. if $D>0$) then
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$\vol_{\infty}(\chi_D)=\vol_{\infty}/\sqrt{D}.$
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\end{lem}
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\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer
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and $D$ a quadratic discriminant. Then the fractional ideal
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$c_p(\chi_D,j)$ is supported at most on divisors of $2$ and
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divisors of $p$, while $c_p(j)$ is supported at most on divisors
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of $p$.
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\end{lem}
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\begin{proof}
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As on p. 30 of \cite{Fl1}, for odd $l\neq p$,
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$\ord_{\lambda}(c_p(\chi_D,j))$ is the length of the finite
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$O_{\lambda}$-module
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$H^0(\QQ_p,H^1(I_p,T_{\lambda}(\chi_D,j))_{\tors}),$ where $I_p$
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is an inertia group at $p$. If $p$ does not divide $D$ then
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$T_{\lambda}(\chi_D,j)$ is a trivial $I_p$-module, so
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$H^1(I_p,T_{\lambda}(\chi_D,j))$ is torsion-free. In general,
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since $l\neq p$,$$H^1(I_p,T_{\lambda}(\chi_{D},j))\simeq
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T_{\lambda}(\chi_{D},j)/(1-\tau)T_{\lambda}(\chi_{D},j),$$ where
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$\tau$ is a topological generator of the tame quotient of $I_p$.
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If $p\mid D$ then thanks to the quadratic twist ramified at $p$,
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$\,\tau$ acts as multiplication by $-1$ on
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$T_{\lambda}(\chi_{D},j)$, hence
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$H^1(I_p,T_{\lambda}(\chi_{D},j))$ is trivial (remember $l\neq
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2$).
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The assertion about $c_p(j)$ may be proved similarly to the case
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$p\nmid D$ above.
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\end{proof}
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\begin{lem} Let $D$ be a quadratic discriminant, and $p$ a prime such that
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$\chi_D(p)=1$. Then $c_p(\chi_D,j)=c_p(j)$.
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\end{lem}
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\begin{proof}
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This is simply a consequence of the fact that the
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definition of $\ord_{\lambda}(c_p(j))$ depends only on
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$T_{\lambda}$ as a $\Gal(\Qbar_p/\QQ_p)$-module.
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\end{proof}
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\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(\chi_D,j))=0$ unless
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the coefficients of $f=\sum a_nq^n$ satisfy the congruence
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$a_p\equiv \chi_D(p)(p^j+p^{k-1-j})\pmod{\lambda}$ for all primes
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$p\nmid lN$.
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\end{lem}
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This follows from the interpretation of $a_p$ as a trace of
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Frobenius. See \cite{SwD}.
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Putting together the above lemmas we arrive at the following:
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\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime. Let $D>0$ be a
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quadratic discriminant such that $\chi_D(q)=1$ and $\chi_D(p)=1$
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for all primes $p\mid N$. Let $\qq\mid q$ be a prime of $E$ such
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that neither
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$$a_p\equiv \chi_D(p)(p^{k/2}+p^{(k/2)-1})\pmod{\qq}$$
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nor
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$$a_p\equiv p^{k/2}+p^{(k/2)-1}\pmod{\qq}$$
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holds for all $p\nmid qN$. If
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$$\ord_{\qq}(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))>0$$
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then the Bloch-Kato conjecture predicts that
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$\ord_{\qq}(\#\Sha)>0$.
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\end{prop}
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\begin{prop}\label{triv}
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Suppose that $p\nmid DN$, $p>k$ and that $f$ is not
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congruent to another cusp form for $\Gamma_0(N)$ modulo any
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divisor of $p$. Then, for $(k/2)\leq j\leq k-1$, $c_p(j)$ is
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trivial and $c_p(\chi_D,j)$ is at worst supported on divisors of
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$2$.
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\end{prop}
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This may be proved in a similar manner to Theorem 7.6 of
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\cite{Du1}. The third condition in the statement of the
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proposition ensures that the denominator of the projector used to
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cut out the motive $M$ is coprime to $p$. Bearing in mind Flach's
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generalisation of the Cassels-Tate pairing \cite{Fl2}, we see the
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following corollary.
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\begin{cor} The Bloch-Kato conjecture predicts that, for $D$ as in
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Proposition \ref{sha}, the fractional ideal
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$(L(f,k/2)/\sqrt{D}L(f,\chi_D,k/2))$ is a square, up to divisors
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of $2$ and of primes not satisfying the conditions of Proposition
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\ref{triv}.
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\end{cor}
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\section{Constructing elements of the Shafarevich-Tate group}
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Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
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weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
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large enough to contain all the coefficients $a_n$ and $b_n$.
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Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
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g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. For $f$
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we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$.
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Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the
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corresponding objects for $g$. Let $A[\lambda]$ denote the
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$\lambda$-torsion in $A_{\lambda}$. Since $a_p$ is the trace of
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$\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev
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Density Theorem that $A[\qq]$ and $A'[\qq]$ are isomorphic as
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$\Gal(\Qbar/\QQ)$-modules.
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Suppose that $L(g,k/2)=0$. If the sign in the functional equation
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is positive, this implies that the order of vanishing at $s=k/2$
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is at least $2$. According to the Beilinson-Bloch conjecture
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\cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the
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rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational,
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null-homologous algebraic cycles on the motive $M_g$, modulo
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rational equivalence.
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Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
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to $H^1(\QQ,V'_{\qq}(k/2))$, and Nekov\'ar shows in \cite{Ne2}
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that its image is contained in the subspace
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$H^1_f(\QQ,V'_{\qq}(k/2))$. See also 0.13 of \cite{Ne1}. If, as
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expected, the $\qq$-adic Abel-Jacobi map is injective, we get
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(assuming also the Beilinson-Bloch conjecture) a subspace of
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$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
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vanishing of $L(g,s)$ at $s=k/2$.
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If $L(f,k/2)\neq 0$ then it is expected that
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$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that
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$\Sha=H^1_f(\QQ,A_{\qq}(k/2))$.
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\begin{thm}\label{local}
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Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
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$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
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$A[\qq]$ and $A'[\qq]$ are irreducible representations of
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$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm
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1\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
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modulo $\qq$ to any newform of trivial character and level
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dividing $N/p$, with $p$ a prime exactly dividing $N$. Then the
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$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
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$\FF_{\qq}$-rank at least $r-1$.
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\end{thm}\edit{There's something wrong with the statement
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of this theorem, as none of the objects depend on~$g$.
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I think $H^1_f(\QQ,A_{\qq}(k/2))$
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should be replaced by $H^1_g(\QQ,A_{\qq}(k/2))$.}
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\begin{proof}Take a non-zero class $d\in
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H^1_f(\QQ,V'_{\qq}(k/2))$, by continuity and rescaling we may
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assume that it lies in $H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
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H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
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class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$.
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Since $a_p\equiv p^{k/2}+p^{(k/2)-1}\pmod{\qq}$ (for all $p\nmid
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qN$) does not hold (by irreducibility), $H^0(\QQ,A[\qq](k/2))$ is
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trivial, so $H^1(\QQ,A[\qq](k/2))$ injects into
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$H^1(\QQ,A_{\qq}(k/2))$, and we get a non-zero class $\gamma\in
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H^1(\QQ,A_{\qq}(k/2))$.
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First we will show that $\res_p(\gamma)\in
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H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p\nmid qN$.
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Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
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A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
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$H^1(I_p,A'[\qq](k/2))$ injects into $H^1(I_p,A'_{\qq}(k/2))$, and
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it follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
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$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
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of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
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contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
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to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
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$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
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Next we will show that $\res_p(\gamma)\in
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H^1_f(\QQ_p,A_{\qq}(k/2))$ for $p\mid N$. Our first task is to
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show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible. It
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suffices to show that $$\dim H^0(I_p,A'[\qq](k/2))=\dim
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H^0(I_p,V'_{\qq}(k/2)).$$ If the dimensions differ then, given
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that $g$ is not congruent modulo $\qq$ to a newform of level
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strictly dividing $N$, and since $p\neq \pm 1\pmod{q}$,
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Propositions 3.1 and 2.3 of \cite{L} tell us that $A'[\qq](k/2)$
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is unramified at $p$ and that $\ord_p(N)=1$. (Here we are using
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Carayol's result that $N$ is the prime-to-$q$ part of the
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conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of
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\cite{JL} implies the existence of a newform of trivial character
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and level dividing $N/p$, congruent to $g$ modulo $\qq$. This
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contradicts our hypotheses, and we have established that $H^0(I_p,
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A'_{\qq}(k/2))$ is $\qq$-divisible.
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It follows that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
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$H^1(I_p,A[\qq](k/2))$ is zero. By inflation-restriction,
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$\res_p(c)$ then comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$.
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The order of this group is the same as the order of the group
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$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
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of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
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ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
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above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
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H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
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this common dimension is $1$. The (motivic) Euler factor at $p$
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for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
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as multiplication by $\alpha$ on the one-dimensional space
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$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
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\cite{Ca1} that this is the same as the Euler factor at $p$ of
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$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
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follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
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$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
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$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
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$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm
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1\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
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$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
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$H^1_f(\QQ_p,A_{\qq}(k/2))$.
393
394
It remains to deal with the local condition at $p=q$. Since
395
$q\nmid N$ is a prime of good reduction for the motive $M_f$,
396
$\,V_{\qq}$ is a crystalline representation of
397
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V_{\qq})$ and $V_{\qq}$
398
have the same dimension, where
399
$D_{\cris}(V_{\qq}):=H^0(\QQ_q,V_{\qq}\otimes_{\QQ_q} B_{\cris})$.
400
(This is a consequence of Theorem 5.6 of \cite{Fa1}.) It follows
401
from Theorem 4.1(ii) of \cite{BK} that
402
$$D_{\dR}(V_{\qq})/F^{k/2}D_{\dR}(V_{\qq})\simeq H^1_f(\QQ_q,V_{\qq}(k/2)),$$
403
via their exponential map.
404
405
Since $p$ is a prime of good reduction, the de Rham conjecture is
406
a consequence of the crystalline conjecture, which follows from
407
Theorem 5.6 of \cite{Fa1}. Hence
408
$$V_{\dR}\otimes_E E_{\qq}/F^{k/2}V_{\dR}\otimes_E E_{\qq}\simeq H^1_f(\QQ_q,V_{\qq}(k/2)).$$
409
Observe that the dimension of the left-hand-side is $1$.
410
Meanwhile, by 3.8 of \cite{BK}, $H^1_f(\QQ_q,V_{\qq}(k/2))$ is its
411
own annihilator in $H^1(\QQ_q,V_{\qq}(k/2))$ with respect to the
412
Tate duality pairing. Hence $H^1_f(\QQ_q,V_{\qq}(k/2))$ has
413
codimension one in $H^1(\QQ_q,V_{\qq}(k/2))$. It follows that our
414
$r$-dimensional $\FF_{\qq}$-subspace of the $\qq$-torsion in
415
$H^1(\QQ,A_{\qq}(k/2))$ has at least an $(r-1)$-dimensional
416
subspace landing in $H^1_f((\QQ_{\qq},A_{\qq}(k/2))$.
417
418
To justify this argument carefully, we should check that the
419
natural map from $H^1(\QQ_q,V_{\qq}(k/2))$ to
420
$H^1((\QQ_{\qq},A_{\qq}(k/2))$ is surjective. Its cokernel injects
421
into \newline $H^2((\QQ_{\qq},T_{\qq}(k/2))$, which is dual to
422
$H^0((\QQ_{\qq},A_{\qq}(k/2))$ (via local Tate duality). It
423
suffices to check that $H^0((\QQ_{\qq},A[\qq](k/2))$ is trivial.
424
This follows from the description of $A[\qq](k/2)$ as a
425
$\Gal(\Qbar_q/\QQ_q)$-module provided by theorems of Deligne and
426
Fontaine (Theorems 2.5 and 2.6 of \cite{Ed}).
427
\end{proof}
428
429
Theorem 2.7 of \cite{AS} is concerned with verifying local
430
conditions in the case $k=2$, where $f$ and $g$ are associated
431
with abelian varieties $A$ and $B$. (Their theorem also applies to
432
abelian varieties over number fields.) Our restriction outlawing
433
congruences modulo $\qq$ with cusp forms of lower level is
434
analogous to theirs forbidding $q$ from dividing Tamagawa factors
435
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
436
curve with $\ord_l(j(A))<0$, consideration of a Tate
437
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
438
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
439
at $l$.)
440
441
\section{Some Data}
442
\subsection{Prime Level $127$}
443
Using modular symbols, we find that there is a newform
444
$$f = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
445
\in S_4(\Gamma_0(127))$$
446
with $L(f,2)=0$.
447
(In fact, $L'(f,2)=0$, as well.)
448
%The sign of the Atkin-Lehner involution $W_{127}$ on~$f$
449
%is $+1$, so $L'(f,2)=0$ as well (look up reference).
450
There is also a newform $g \in S_4(\Gamma_0(127))$ whose
451
Fourier coefficients generate a number field~$K$ of degree~$17$,
452
and $L(g,2)\neq 0$.
453
The newforms~$f$ and~$g$ are congruent
454
modulo a prime $\qq$ of~$K$ of residue characteristic~$43$.
455
The mod~$\qq$ reductions of~$f$ and~$g$ are both equal to
456
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots.$$
457
The residue characteristics of congruence primes
458
between the cuspidal and Eisenstein subspaces are~$2$,~$3$,~$7$,
459
and $1613$, so (reference) $\rho_{f,\qq}\approx\rho_{g,\qq}$
460
is irreducible.
461
Since $S_4(\SL_2(\ZZ))=0$, we see that~$\fbar$ does not arise from a
462
level~$1$ form of weight~$4$. Thus, if we assume the Beilinson-Bloch
463
conjecture, Theorem~\ref{local} implies that $\Sha(M_g)$ has
464
order divisible by~$43$.
465
466
\begin{verbatim}
467
FROM Mark Watkins:
468
469
Here's a list of stuff; the left-hand (sinister) column
470
contains forms with a double zero at the central point,
471
whilst the right-hand (dexter) column contains forms which
472
have a large square factor in LRatio at 2. The middle column
473
is the large prime factor in ModularDegree of the LHS,
474
and/or the lpf in the LRatio at 2 of the RHS.
475
476
It seems that 567k4L has invisible Sha possibly.
477
The ModularDegree for 639k4B has no large factors.
478
479
127k4A 43 127k4C
480
159k4A 23 159k4E
481
365k4A 29 365k4E
482
369k4A 13 369k4I
483
453k4A 17 453k4E
484
453k4A 23
485
465k4A 11 465k4H
486
477k4A 73 477k4L
487
567k4A 23 567k4G
488
13 567k4L
489
581k4A 19 581k4E
490
639k4B --
491
492
493
Forms with spurious zeros to do:
494
495
657k4A
496
681k4A
497
684k4B
498
95k6A 31,59
499
116k6A --
500
122k6A 73
501
260k6A
502
503
If we allow 5 and 7 to be small primes, then we get more
504
visibility info.
505
506
159k4A 5 159k4E
507
369k4A 5 369k4I
508
453k4A 5 453k4E
509
639k4B 7 639k4H
510
511
Maybe the general theorem does not apply to primes which are so small,
512
but we might be able to show that they are OK in these specific cases.
513
514
I checked the first three weight 6 examples with AbelianIntersection,
515
but actually computing the LRatio bogs down too much. Will need
516
new version.
517
518
95k6A 31 95k6D
519
95k6A 59 95k6D
520
116k6A 5 116k6D
521
122k6A 73 122k6D
522
523
Incidentally, the LRatio for 581k4E has a power of 19^4.
524
Perhaps not surprising, as the ModularDegree of 581k4A
525
has a factor of 19^2. But maybe it means a bigger Sha.
526
\end{verbatim}
527
528
529
\begin{thebibliography}{AS}
530
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
531
Shafarevich-Tate groups of abelian varieties: evidence for the
532
Birch and Swinnerton-Dyer conjecture, in preparation.
533
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
534
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
535
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
536
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
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\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
538
of motives, The Grothendieck Festschrift Volume I, 333--400,
539
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
540
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $l$-adiques
541
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
542
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
543
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
544
Galoisiennes modulo $l$ attach\'ees aux formes modulaires, {\em
545
Duke Math. J. }{\bf 59 }(1989), 785--801.
546
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
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Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
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13--28.
549
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
550
$l$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math. {\bf
551
179, } 139--172, Springer, 1969.
552
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
553
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
554
part 2, 313--346.
555
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
556
Math. Ann. }{\bf 318 }(2000), 621--636.
557
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
558
Selmer groups, {\em Math. Res. Lett., } to appear.
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\bibitem[Ed]{Ed} S. J. Edixhoven, The weight in Serre's
560
conjectures on modular forms, {\em Invent. Math. }{\bf 109
561
}(1992), 563--594.
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\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
563
Galois representations, {\em in }Algebraic analysis, geometry and
564
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
565
Press, Baltimore, 1989.
566
\bibitem[Fl1]{Fl1} M. Flach, On the degree of modular parametrisations,
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S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
568
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
569
Boston Berlin, 1993.
570
\bibitem[Fl2]{Fl2} M. Flach, A generalisation of the Cassels-Tate
571
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
572
\bibitem[Fo]{Fo} J.-M. Fontaine, Sur certains types de
573
repr\'esentations $p$-adiques du groupe de Galois d'un corps
574
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
575
}{\bf 115 }(1982), 529--577.
576
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
577
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
578
51--56.
579
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $l$ Galois
580
representations coming from modular forms, {\em J. Number Theory
581
}{\bf 31 }(1989), 133--141.
582
\bibitem[Ne1]{Ne1} J. Nekov\'ar, On the $p$-adic heights of Heegner
583
cycles, {\em Math. Ann. }{\bf 302 }(1995), 609--686.
584
\bibitem[Ne2]{Ne2} J. Nekov\'ar, Kolyvagin's method for Chow groups of
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Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
586
99--125.
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\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
588
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
589
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $l$-adic representations and
590
congruences for coefficients of modular forms, {\em Modular functions
591
of one variable} III, Lect. Notes Math. {\bf 350, } Springer, 1973.
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\end{document}
596