CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v2.texOpen in CoCalc with one click!
Author: William A. Stein
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\begin{document}
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\title{Constructing elements in
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Shafarevich-Tate groups of modular motives\\
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{\sc (NOT FOR DISTRIBUTION!)}}
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\author{Neil Dummigan}
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\author{William Stein}
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\author{Mark Watkins}
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\date{\today}
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\subjclass{11F33, 11F67, 11G40.}
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\keywords{modular form, $L$-function, visibility, 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\\Department of Mathematics\\
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One 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{watkins@math.psu.edu}
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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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Mention \cite{CM}.
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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 two-dimensional vector space
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$V_{\lambda}$ over $E_{\lambda}$, and a continuous representation
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$$
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\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),
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$$
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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
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$$
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V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset F^k=\{0\}.
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$$
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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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$$\Sha(\chi_D,j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(\chi_D,j))/\pi_*H^1_f(\QQ,V_{\lambda}(\chi_D,j)).$$
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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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Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture for the
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motive $M_f(k/2)$ predicts that
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$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
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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, if $L(f,\chi_D,k/2)\neq 0$ then the Bloch-Kato
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conjecture for $M_f(\chi_D,k/2)$ predicts that
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$${L(f,\chi_D,k/2)\over \vol_{\infty}(\chi_D)}={\left(\prod_pc_p(k/2,\chi_D)\right)\#\Sha(\chi_D)\over (\#\Gamma_{\QQ}(\chi_D))^2}.$$
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Note that if $L(f,k/2)\neq 0$ then it is necessary that
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$\chi_D(-N)=1$ if the sign in the functional equation is to allow
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$L(f,\chi_D,k/2)\neq 0$. A careful treatment of real periods such
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as $\vol_{\infty}(\chi_D)$ is given in \cite{De2}. The calculation
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of the periods of Artin motives in terms of generalized Gauss
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sums, in Section 6 of \cite{De2}, yields the 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}\edit{I switched to using proof environments
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instead of doing it by hand. -- was}
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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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\end{proof}
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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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\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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(with numerator and denominator both non-zero) then the Bloch-Kato
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conjecture predicts that $\ord_{\qq}(\#\Sha)>0$.
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\end{prop}
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\begin{prop}\label{triv}
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Suppose that $p\nmid DN\phi(N)$, $p>k$ 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 conditions in the statement of the proposition
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ensure, among other things, that the denominator of the projector
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used to cut out the motive $M_f$ is coprime to $p$. This is
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essential if the proofs of the lemmas in Section 7 of \cite{Du1}
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are to carry across.
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Bearing in mind Flach's generalisation of the Cassels-Tate pairing
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\cite{Fl2}, we see the 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 $A'[\qq]$, if irreducible, are
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isomorphic as $\Gal(\Qbar/\QQ)$-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, codimension $k/2$ algebraic cycles on the motive
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$M_g$, modulo 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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$\Sha=H^1_f(\QQ,A_{\qq}(k/2))$.\edit{Does one really get {\em all} of
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$\Sha$ as you claim, or just the $\qq$-torsion or $\qq$-part??}
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(For $j\geq k/2$, the rank of
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$H^1_f(\QQ,V_{\qq}(j))$ is expected to be the order of vanishing
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of $L(f,s)$ at $s=k-j$.)
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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\edit{I added this weight~$k$
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assertion. -- was} of weight~$k$, trivial character and level
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dividing $N/p$, with~$p$ any\edit{Change ``a'' to ``any''. -- was}
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prime that exactly divides~$N$. Then
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the $\qq$-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}
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\begin{proof}\edit{This proof is really long. Could it benefit by
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being broken up into more maneagable chunks (lemmas, etc.)?
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If nothing else, a summary paragraph at the beginning would be
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useful. -- was}
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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 $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
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irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
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$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
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we get a non-zero class $\gamma\in H^1(\QQ,A_{\qq}(k/2))$.
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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} (which uses the condition $q>k$) implies the existence
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of a newform of weight $k$, trivial character and level dividing
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$N/p$, congruent to $g$ modulo $\qq$. This contradicts our
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hypotheses, and we have established that $H^0(I_p, A'_{\qq}(k/2))$
400
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
407
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
408
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
409
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
410
H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
411
this common dimension is $1$. The (motivic) Euler factor at $p$
412
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
413
as multiplication by $\alpha$ on the one-dimensional space
414
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
415
\cite{Ca1} that this is the same as the Euler factor at $p$ of
416
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
417
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
418
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
419
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
420
$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm
421
1\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
422
$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
423
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
424
425
It remains to deal with the local condition at $p=q$. Since
426
$q\nmid N$ is a prime of good reduction for the motive $M_f$,
427
$\,V_{\qq}$ is a crystalline representation of
428
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V_{\qq})$ and $V_{\qq}$
429
have the same dimension, where
430
$D_{\cris}(V_{\qq}):=H^0(\QQ_q,V_{\qq}\otimes_{\QQ_q} B_{\cris})$.
431
(This is a consequence of Theorem 5.6 of \cite{Fa1}.) It follows
432
from Theorem 4.1(ii) of \cite{BK} that
433
$$D_{\dR}(V_{\qq})/(F^{k/2}D_{\dR}(V_{\qq}))\simeq H^1_f(\QQ_q,V_{\qq}(k/2)),$$
434
via their exponential map.
435
436
Since $p$ is a prime of good reduction, the de Rham conjecture is
437
a consequence of the crystalline conjecture, which follows from
438
Theorem 5.6 of \cite{Fa1}. Hence
439
$$(V_{\dR}\otimes_E E_{\qq})/(F^{k/2}V_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V_{\qq}(k/2)).$$
440
Observe that the dimension of the left-hand-side is $1$.
441
Meanwhile, by 3.8 of \cite{BK}, $H^1_f(\QQ_q,V_{\qq}(k/2))$ is its
442
own annihilator in $H^1(\QQ_q,V_{\qq}(k/2))$ with respect to the
443
Tate duality pairing. Hence $H^1_f(\QQ_q,V_{\qq}(k/2))$ has
444
codimension one in $H^1(\QQ_q,V_{\qq}(k/2))$. It follows that our
445
$r$-dimensional $\FF_{\qq}$-subspace of the $\qq$-torsion in
446
$H^1(\QQ,A_{\qq}(k/2))$ has at least an $(r-1)$-dimensional
447
subspace landing in $H^1_f(\QQ_{\qq},A_{\qq}(k/2))$.
448
449
To justify this argument carefully, we should check that the
450
natural map from $H^1(\QQ_q,V_{\qq}(k/2))$ to
451
$H^1(\QQ_{\qq},A_{\qq}(k/2))$ is surjective. Its cokernel injects
452
into \newline $H^2(\QQ_{\qq},T_{\qq}(k/2))$, which is dual to
453
$H^0(\QQ_{\qq},A_{\qq}(k/2))$ (via local Tate duality). It
454
suffices to check that $H^0(\QQ_{\qq},A[\qq](k/2))$ is trivial.
455
This follows from the description of $A[\qq](k/2)$ as a
456
$\Gal(\Qbar_q/\QQ_q)$-module provided by theorems of Deligne and
457
Fontaine (Theorems 2.5 and 2.6 of \cite{Ed}).
458
\end{proof}
459
460
Theorem 2.7 of \cite{AS} is concerned with verifying local
461
conditions in the case $k=2$, where $f$ and $g$ are associated
462
with abelian varieties $A$ and $B$. (Their theorem also applies to
463
abelian varieties over number fields.) Our restriction outlawing
464
congruences modulo $\qq$ with cusp forms of lower level is
465
analogous to theirs forbidding $q$ from dividing Tamagawa factors
466
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
467
curve with $\ord_l(j(A))<0$, consideration of a Tate
468
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
469
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
470
at $l$.)
471
472
\section{Eleven examples of Theorem~\ref{local}}
473
\newcommand{\nf}[1]{\mbox{\bf #1}}
474
\begin{figure}
475
\caption{\label{fig:newforms}Newforms Satisfying Theorem~\ref{local}}
476
$$
477
\begin{array}{|ccccc|}\hline
478
f & \deg(f) & g & \deg(g) & q's \\\hline
479
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43\\
480
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23\\
481
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29\\
482
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13\\
483
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17\\
484
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11\\
485
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73\\
486
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23\\
487
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19\\
488
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59\\
489
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73\\
490
\hline
491
\end{array}
492
$$
493
\end{figure}
494
495
496
Table~\ref{fig:newforms} on page~\pageref{fig:newforms}
497
lists eleven pairs of newforms~$f$ and~$g$
498
along with at least one prime~$q$ such that there is a
499
prime $\qq\mid q$ such that by Theorem~\ref{local}
500
the Beilinson-Bloch conjecture
501
implies that
502
$$\dim_{\FF_\qq} \#H^1_f(\QQ,A_{\qq}(k/2))[\qq] > 0.$$
503
Recall that $H^1_f(\QQ,A_{\qq}(k/2))[\qq]$ is a piece of the
504
finite cohomology associated to the motive attached to~$g$.
505
As discussed just before Theorem~\ref{local}, since $L(g,\frac{k}{2})\neq 0$,
506
we expect that $H^1_f(\QQ,A_{\qq}(k/2))[\qq]$ equals the $\qq$-torsion
507
of the Shafarevich-Tate group of the motive attached to~$g$.
508
(Note also that the subscript of $f$ means ``finite'' and has nothing
509
to do with the newform that we denote by ``$f$''.)
510
511
\subsection{Notation}
512
Table~\ref{fig:newforms} is laid out as follows.
513
The first column contains a label whose structure is
514
\begin{center}
515
{\bf [Level]k[Weight][GaloisOrbit]}
516
\end{center}
517
This label determines a newform $f=\sum a_n q^n$, up to Galois conjugacy.
518
For example, \nf{127k4C} denotes a
519
newform in the third Galois orbit of
520
newforms in $S_4(\Gamma_0(127))$.
521
The Galois orbits are ordered
522
first by the degree of $\QQ(\ldots, a_n, \ldots)$,
523
then by $|\mbox{\rm Tr}(a_p(f))|$ for~$p$ not dividing the
524
level, with positive trace being first in the event that the two absolute
525
values are equal, and the first Galois orbit is denoted {\bf A}, the
526
second {\bf B}, and so on.
527
For the purposes of the present paper, the ordering is not important;
528
if you repeat the experiment, you will quickly find which form
529
must be~$f$ and which must be~$g$.
530
The second column contains the degree
531
of the field $\QQ(\ldots, a_n, \ldots)$. The third and forth
532
columns contain~$g$ and its degree, respectively.
533
The fifth columns contains at least one prime~$q$
534
such that there is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$,
535
and so that the hypothesis of Theorem~\ref{local} are satisfied
536
for~$f$,~$g$, and~$\qq$.
537
538
\subsection{The first example in detail}
539
\newcommand{\fbar}{\overline{f}}
540
We describe the first line of Table~\ref{fig:newforms}
541
in more detail. See the next section for further details
542
on how the computations were performed.
543
544
Using modular symbols, we find that there is a newform
545
$$f = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
546
\in S_4(\Gamma_0(127))$$
547
with $L(f,2)=0$.
548
Because $W_{127}(f)=f$, the functional equation has sign~$+1$,
549
so $L'(f,2)=0$ as well.
550
We also find a newform $g \in S_4(\Gamma_0(127))$ whose
551
Fourier coefficients generate a number field~$K$ of degree~$17$,
552
and by computing the image of the modular symbol
553
$XY\{0,\infty\}$ under the period mapping, we find
554
that $L(g,2)\neq 0$. The newforms~$f$ and~$g$ are congruent
555
modulo a prime $\qq$ of~$K$ of residue characteristic~$43$.
556
The mod~$\qq$ reductions of~$f$ and~$g$ are both equal to
557
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
558
There is no form in the Eisenstein subspaces of $M_4(\Gamma_0(127))$
559
whose Fourier coefficients of index~$n$, with $(n,127)=1$, are
560
congruent modulo $43$ to those of $\fbar$, so
561
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible.
562
Since $127$ is prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$
563
does not arise from a
564
level~$1$ form of weight~$4$. Thus, if we assume the Beilinson-Bloch
565
conjecture, Theorem~\ref{local} implies that the Shafarevich-Tate
566
group attached to~$g$ has order divisible by~$43$.
567
568
\subsection{Some remarks on how the computation was performed}
569
We give a brief summary of how the computation was performed. The
570
algorithms that we used were implemented by the second author, and
571
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
572
573
Let~$f$,~$g$, and~$q$ be some data from a line of
574
Table~\ref{fig:newforms} and let~$N$ denote the level of~$f$.
575
We verified the existence of a congruence modulo~$q$, that
576
$L(f,\frac{k}{2})=L'(f,\frac{k}{2})=0$ and $L(g,\frac{k}{2})\neq 0$,
577
and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does not arise from
578
any $S_k(\Gamma_0(N/p))$, as follows:
579
580
To prove there is a congruence, we showed that the corresponding {\em
581
integral} spaces of modular symbols satisfy an appropriate congruence,
582
which forces the existence of a congruence on the level of Fourier
583
expansions. We showed that $\rho_{f,\qq}$ is irreducible by computing
584
a set that contains all possible residue characteristics of congruences
585
between~$f$ and any Eisenstein series of level dividing~$N$, where
586
by congruence, we mean a congruence for all Fourier coefficients of index~$n$
587
with $(n,N)=1$.
588
Similarly, we checked that~$f$ is not congruent to any form~$h$ of
589
level $N/p$ for any~$p$ that exactly divides~$N$ by listing a basis of
590
such~$h$ and finding the possibly congruences, where again we
591
disregard the Fourier coefficients of index not coprime to~$N$.
592
593
To verify that $L(f,\frac{k}{2})=0$, we computed the image of the
594
modular symbol ${\mathbf
595
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a map
596
with the same kernel as the period mapping, and found that the image
597
was~$0$. The period mapping sends the modular symbol~${\mathbf e}$
598
to a nonzero multiple of $L(f,\frac{k}{2})$, so that ${\mathbf e}$
599
maps to~$0$ implies that $L(f,\frac{k}{2})=0$. In a similar way,
600
we verified that $L(g,\frac{k}{2})\neq 0$. Next, we checked
601
that $W_N(f) = f$ which, because of the functional equation, implies
602
that $L'(f,\frac{k}{2})=0$.
603
604
%In the course of our search, we found~$17$ plausible examples and had
605
%to discard~$6$ of them because either the representation $\rho_{f,\qq}$ arose
606
%from lower level, was irreducible, or $p\equiv \pm 1\pmod{q}$ for
607
%some $p\mid N$. Table~\ref{fig:newforms}
608
%is of independent interest because it
609
%includes examples of modular forms of prime level and weight
610
%$>2$ with a zero at $\frac{k}{2}$ that is not forced
611
%by the functional equation. We found no such examples
612
%of weights $\geq 8$.
613
614
\section{Some Remarks on Visibility}
615
It would be nice to add a section on visibility. I.e.,
616
systematically list examples where the rational part of the central
617
value has a big square factor, which {\em probably} should be
618
nontrivial $\Sha$, and say whether or not the theory of this paper
619
predicts that it is visible. Answer the question:
620
``Does it look like all Sha be visible for motives?'' probably
621
``no''.
622
623
624
625
626
627
628
629
\begin{thebibliography}{AS}
630
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
631
Shafarevich-Tate groups of abelian varieties: evidence for the
632
Birch and Swinnerton-Dyer conjecture, in preparation.
633
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
634
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
635
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
636
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
637
638
\bibitem[BCP]{magma}
639
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
640
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
641
235--265, Computational algebra and number theory (London, 1993).
642
643
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
644
of motives, The Grothendieck Festschrift Volume I, 333--400,
645
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
646
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
647
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
648
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
649
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
650
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
651
Duke Math. J. }{\bf 59 }(1989), 785--801.
652
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
653
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
654
13--28.
655
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
656
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
657
{\bf 179, } 139--172, Springer, 1969.
658
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
659
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
660
part 2, 313--346.
661
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
662
Math. Ann. }{\bf 318 }(2000), 621--636.
663
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
664
Selmer groups, {\em Math. Res. Lett., } to appear.
665
\bibitem[Ed]{Ed} S. J. Edixhoven, The weight in Serre's
666
conjectures on modular forms, {\em Invent. Math. }{\bf 109
667
}(1992), 563--594.
668
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
669
Galois representations, {\em in }Algebraic analysis, geometry and
670
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
671
Press, Baltimore, 1989.
672
\bibitem[Fl2]{Fl2} M. Flach, A generalisation of the Cassels-Tate
673
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
674
\bibitem[Fl1]{Fl1} M. Flach, On the degree of modular parametrisations,
675
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
676
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
677
Boston Berlin, 1993.
678
\bibitem[Fo]{Fo} J.-M. Fontaine, Sur certains types de
679
repr\'esentations $p$-adiques du groupe de Galois d'un corps
680
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
681
}{\bf 115 }(1982), 529--577.
682
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
683
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
684
51--56.
685
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
686
representations coming from modular forms, {\em J. Number Theory
687
}{\bf 31 }(1989), 133--141.
688
689
\bibitem[Ne1]{Ne2} J. Nekov\'ar, Kolyvagin's method for Chow groups of
690
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
691
99--125.
692
\bibitem[Ne2]{Ne1} J. Nekov\'ar, On the $p$-adic heights of Heegner
693
cycles, {\em Math. Ann. }{\bf 302 }(1995), 609--686.
694
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
695
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
696
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
697
congruences for coefficients of modular forms,{\em Modular functions
698
of one variable} III, Lect. Notes Math. {\bf 350, } Springer, 1973.
699
\end{thebibliography}
700
701
702
\end{document}
703
\subsection{Plan for finishing the computation}
704
705
What I (William) have to do with the below stuff:\\
706
\begin{enumerate}
707
\item Fill in the missing blanks at these levels:
708
$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
709
\item Make a MAGMA program that has a table of forms to try.
710
\item First, unfortunately, for some reason the
711
labels are sometimes wrong, e.g., \nf{159k4A} should
712
have been \nf{159k4B}, so I have to check all the
713
rational forms to see which has $L(1)=0$.
714
\item Try each level using the MAGMA program (this will take a long time to run).
715
\item Finish the ``Table of examples'' above.
716
\end{enumerate}
717
718
719
\begin{verbatim}
720
721
722
(This stuff below is being integrated into the above, as I do
723
the required (rather time consuming) computations.)
724
**************************************************
725
FROM Mark Watkins:
726
727
Here's a list of stuff; the left-hand (sinister) column
728
contains forms with a double zero at the central point,
729
whilst the right-hand (dexter) column contains forms which
730
have a large square factor in LRatio at 2. The middle column
731
is the large prime factor in ModularDegree of the LHS,
732
and/or the lpf in the LRatio at 2 of the RHS.
733
734
It seems that 567k4L has invisible Sha possibly.
735
The ModularDegree for 639k4B has no large factors.
736
737
127k4A 43 127k4C
738
159k4A 23 159k4E
739
365k4A 29 365k4E
740
369k4A 13 369k4I
741
453k4A 17 453k4E
742
453k4A 23
743
465k4A 11 465k4H
744
477k4A 73 477k4L
745
567k4A 23 567k4G
746
13 567k4L
747
581k4A 19 581k4E
748
639k4B --
749
750
751
Forms with spurious zeros to do:
752
753
657k4A
754
681k4A
755
684k4B
756
95k6A 31,59
757
116k6A --
758
122k6A 73
759
260k6A
760
761
If we allow 5 and 7 to be small primes, then we get more
762
visibility info.
763
764
159k4A 5 159k4E
765
369k4A 5 369k4I
766
453k4A 5 453k4E
767
639k4B 7 639k4H
768
769
Maybe the general theorem does not apply to primes which are so small,
770
but we might be able to show that they are OK in these specific cases.
771
772
I checked the first three weight 6 examples with AbelianIntersection,
773
but actually computing the LRatio bogs down too much. Will need
774
new version.
775
776
95k6A 31 95k6D
777
95k6A 59 95k6D
778
116k6A 5 116k6D
779
122k6A 73 122k6D
780
781
Incidentally, the LRatio for 581k4E has a power of 19^4.
782
Perhaps not surprising, as the ModularDegree of 581k4A
783
has a factor of 19^2. But maybe it means a bigger Sha.
784
\end{verbatim}
785