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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]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
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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{August 29th, 2002}
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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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We study Shafarevich-Tate groups of motives attached to modular
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forms on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a
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criterion for the existence of nontrivial elements of these
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Shafarevich-Tate groups, and give $16$ examples in which a strong
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form of the Beilinson-Bloch conjecture implies the existence of
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such elements. We also use modular symbols and observations about
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Tamagawa numbers to compute nontrivial conjectural lower bounds on
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the orders of the Shafarevich-Tate groups of modular motives of
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low level and weight at most $12$. Our methods build upon the
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idea of visibility due to Cremona and Mazur, but in the context of
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motives instead of abelian varieties.
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\end{abstract}
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\maketitle
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\section{Introduction}
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Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
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be the associated $L$-function. The conjecture of Birch and
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Swinnerton-Dyer \cite{BSD} predicts that the order of vanishing of $L(E,s)$
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at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
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also gives an interpretation of the leading term in the Taylor
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expansion in terms of various quantities, including the order of
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the Shafarevich-Tate group of~$E$.
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Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
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curves over $\QQ$ of conductor $N\leq 5500$, at those with
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nontrivial Shafarevich-Tate group (according to the Birch and
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Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
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group has predicted elements of prime order~$p$. In most cases
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they find another elliptic curve, often of the same conductor,
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whose $p$-torsion is Galois-isomorphic to that of the first one,
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and which has positive rank. The rational points on the second elliptic
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curve produce classes in the common $H^1(\QQ,E[p])$. They show
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\cite{CM2} that these lie in the Shafarevich-Tate group of the
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first curve, so rational points on one curve explain elements of
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the Shafarevich-Tate group of the other curve.
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The Bloch-Kato conjecture \cite{BK} is the generalisation to
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arbitrary motives of the leading term part of the Birch and
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Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
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\cite{B} generalises the part about the order of vanishing at the
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central point, identifying it with the rank of a certain Chow
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group.
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This paper is a partial generalisation of \cite{CM} and \cite{AS}
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from abelian varieties over $\QQ$ associated to modular forms of
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weight~$2$ to the motives attached to modular forms of higher weight.
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It also does for congruences between modular forms of equal weight
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what \cite{Du2} did for congruences between modular forms of different
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weights.
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We consider the situation where two newforms~$f$ and~$g$, both of
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even weight $k>2$ and level~$N$, are congruent modulo a maximal
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ideal $\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
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$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
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to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},
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we give sixteen such examples (all with $k=4$ and $k=6$), and in
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each example, we find that $\qq$ divides the numerator of the
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algebraic number $L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$
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is a certain canonical period.
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In fact, we show how this divisibility may be deduced from the
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vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
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point is, the congruence between$f$ and~$g$ leads to a congruence
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between suitable ``algebraic parts'' of the special values
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$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
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one result of Faltings and Jordan shows that the congruence of
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Fourier expansions leads to a congruence of certain associated
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cohomology classes. These are then identified with the modular
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symbols which give rise to the algebraic parts of special values.
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If $L(g,k/2)$ vanishes then the congruence implies that
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$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
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The Bloch-Kato conjecture sometimes then implies that the
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Shafarevich-Tate group $\Sha$ attached to~$f$ has nonzero
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$\qq$-torsion. Under certain hypotheses and assumptions, the most
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substantial of which is the Beilinson-Bloch conjecture relating
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the vanishing of $L(g,k/2)$ to the existence of algebraic cycles,
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we are able to construct some of the predicted elements of~$\Sha$
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using the Galois-theoretic interpretation of the congruences to
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transfer elements from a Selmer group for~$g$ to a Selmer group
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for~$f$. One might say that algebraic cycles for one motive
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explain elements of~$\Sha$ for the other, or that we use
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congruences to link the Beilinson-Bloch conjecture for one motive
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with the Bloch-Kato conjecture for the other.
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%In proving the local
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%conditions at primes dividing the level, and also in examining the
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%local Tamagawa factors at these primes, we make use of a higher weight
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%level-lowering result due to Jordan and Livn\'e \cite{JL}.
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We also compute data which, assuming the Bloch-Kato conjecture,
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provides lower bounds for the orders of numerous Shafarevich-Tate
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groups (see Section~\ref{sec:invis}).
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%Our data is consistent
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%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
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%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
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%is finite).
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\section{Motives and Galois representations}
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This section and the next provide definitions of some of the
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quantities appearing later in the Bloch-Kato conjecture. Let
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$f=\sum a_nq^n$ 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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which is necessarily totally real. Let~$\lambda$ be any finite
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prime of~$E$, and let~$\ell$ denote its residue characteristic. A
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theorem of Deligne \cite{De1} implies the existence of a
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two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and
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a continuous representation
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$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
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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~$p$
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not dividing~$lN$, and
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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 $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
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F^k=\{0\}$. 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.
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For each prime $\lambda$, there is a natural isomorphism
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$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
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Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
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There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
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which amounts to multiplying the action of $\Frob_p$ by $p^j$.
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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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$$
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H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).
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$$
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The subscript~$f$ stands for ``finite
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part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
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$I_p$ is the inertia subgroup, and the cohomology is for
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continuous cocycles and coboundaries. For $p=l$ let
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$$
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H^1_f(\QQ_l,V_{\lambda}(j))=\ker
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(H^1(D_l,V_{\lambda}(j))\rightarrow
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H^1(D_l,V_{\lambda}(j)\otimes_{\QQ_l} B_{\cris}))
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$$
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(see Section 1 of
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\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
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$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
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elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
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in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
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There is a natural exact sequence
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$$
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\begin{CD}0@>>>T_{\lambda}(j)@>>>
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V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.
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$$
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Let
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$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
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Define the $\lambda$-Selmer group
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$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
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$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
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$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
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condition at $p=\infty$ is superfluous unless $l=2$. Define the
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Shafarevich-Tate group
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$$
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\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
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\pi_*H^1_f(\QQ,V_{\lambda}(j)).
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$$
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Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
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prime ideal~$\lambda$ is the length of the $\lambda$-component of
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$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
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and write~$\Sha$ for~$\Sha(k/2)$. It depends on the choice of
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. But if $A[\lambda]$ is irreducible then
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$T_{\lambda}$ is unique up to scaling and the $\lambda$-part of
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$\Sha$ is independent of choices.
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In the case $k=2$ the motive comes from an (self-dual)
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isogeny class of abelian varieties over $\QQ$, with endomorphism
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algebra containing $E$. If one chooses an abelian variety $B$ in
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the isogeny class and takes all the $T_{\lambda}(1)$ to be
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$\lambda$-adic Tate modules, then what we have defined above
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coincides with the usual Shafarevich-Tate group of $B$. To see
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this one uses 3.11 of \cite{BK}, for $l=p$. For $l\neq p$,
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$H^1(\QQ_p,V_l)=0$. Considering the formal group, every class in
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$B(\QQ_p)/lB(\QQ_p)$ is represented by an $l$-power torsion point
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in $B(\QQ_p)$, so maps to zero in $H^1(\QQ_p,A_l)$.
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Define the group of global torsion points
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$$
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\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
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$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
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which the exponent of any prime ideal~$\lambda$ is the length of
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the $\lambda$-component of $\Gamma_{\QQ}$.
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\section{Canonical periods}
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We assume from now on for convenience that $N\geq 3$. We need to
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choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
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and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
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in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
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agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
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and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
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\cite{DFG} using cohomology, with non-constant coefficients, of
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modular curves. (In \cite{DFG}, see especially Sections 2.2 and
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5.4, and the paragraph preceding Lemma 2.3.)
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For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
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module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
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$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$-module
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$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
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Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
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generalised elliptic curves with full level-$N$ structure. Let
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$\mathfrak{E}$ be the universal generalised elliptic curve over
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$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
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of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
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using a projector on the cohomology of a desingularisation of
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$\mathfrak{E}^{k-2}$). Realising $M(N)(\CC)$ as the disjoint union
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of $\phi(N)$ copies of the quotient
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$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
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completed upper half plane), and letting $\tau$ be a variable on
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$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
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the elliptic curve with period lattice generated by $1$ and
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$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
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the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
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Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
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is a well-defined differential form on (a desingularisation of)
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$\mathfrak{E}^{k-2}$ and naturally represents a generating element
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of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
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primes dividing $Nk!$ so that this is the case.) We shall call
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this element $e(f)$.
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Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
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$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
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a natural action of complex conjugation on $V_B$, breaking it up
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into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
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Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
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to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
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$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
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are rank one $O_E$-modules, but not necessarily free, since the
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class number of $O_E$ may be greater than one. Choose nonzero
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elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
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the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
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$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
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\section{The Bloch-Kato conjecture}\label{sec:bkconj}
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In this section we extract from the Bloch-Kato conjecture for
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$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate
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group, by analysing the other terms in the formula.
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Let $L(f,s)$ be the $L$-function attached to~$f$. For
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$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
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Euler product
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$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
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there is an analytic continuation given by an integral, as
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described in the next section. Suppose that $L(f,k/2)\neq 0$. The
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Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
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following equality of fractional ideals of~$E$:
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$$
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\frac{L(f,k/2)}{\vol_{\infty}}=
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\left(\prod_pc_p(k/2)\right)
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\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
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$$
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Here, {\bf and from this point onwards, }$\pm$ represents the
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parity of $(k/2)-1$. The quantity
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$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$
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multiplied by the determinant of the isomorphism
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$V_B^{\pm}\otimes\CC\simeq (V_{\dR}/F^{k/2})\otimes\CC$,
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calculated with respect to the lattices $O_E\delta_f^{\pm}$ and
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the image of $T_{\dR}$. For $l\neq p$, $\ord_{\lambda}(c_p(j))$ is
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defined to be
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\begin{align*}
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\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
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\ord_{\lambda}(P_p(p^{-j}))\\
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=&\length\>\> \left(H^0(\QQ_p,A_{\lambda}(j))/H^0\left(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
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\end{align*}
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We omit the definition of $\ord_{\lambda}(c_p(j))$ 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), and depends on the choices of
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$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
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concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
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is a prime of good reduction. For such primes, the de Rham
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conjecture follows from Theorem 5.6 of \cite{Fa1}.)
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Strictly speaking, the conjecture in \cite{BK} is only given for
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$E=\QQ$. We have taken here the obvious generalisation of a slight
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rearrangement of (5.15.1) of \cite{BK}. The Bloch-Kato conjecture
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has been reformulated and generalised by Fontaine and Perrin-Riou,
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who work with general $E$, though that is not really the point of
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their work. In Section 11 of \cite{Fo2} it is sketched how to
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deduce the original conjecture from theirs, though only in the
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case $E=\QQ$.
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\begin{lem}\label{vol}
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$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
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$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
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\end{lem}
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\begin{proof}
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We note that $\Omega_{\pm}$ is equal to the determinant of the
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period map from $F^{k/2}V_{\dR}\otimes\CC$ to
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$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
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used above in the definition of $\vol_{\infty}$ (c.f. the last
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paragraph of 1.7 of \cite{De2}). We are using here natural
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pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
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$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
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by noting that, locally away from primes dividing $Nk!$, the index
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of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
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dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}.
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\end{proof}
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\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
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Then the fractional ideal $c_p(j)$ is supported at most on
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divisors 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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$\ord_{\lambda}(c_p(j))$ is the length of the finite
447
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
448
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
449
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
450
torsion free.
451
\end{proof}
452
453
\begin{lem}\label{local1}
454
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
455
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
456
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
457
that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not
458
congruent modulo $\qq$ to any newform of weight~$k$, trivial
459
character, and level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$
460
for all integers~$j$.
461
\end{lem}
462
\begin{proof}
463
There is a natural injective map from
464
$V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A[\qq](j))$.
465
Consideration of $\qq$-torsion shows that
466
$$
467
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}}
468
H^0(I_p,V_{\qq}(j)).
469
$$ To prove the lemma it suffices to show that
470
$$
471
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
472
$$
473
since this ensures that $H^0(I_p,A_{\qq}(j))=
474
V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that
475
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
476
If the dimensions differ then, given that $f$ is not congruent
477
modulo $\qq$ to a newform of level dividing $N/p$, Proposition 2.2
478
of \cite{L} shows that we are in the situation covered by one of
479
the three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv
480
-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
481
unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
482
result that $N$ is the prime-to-$q$ part of the conductor of
483
$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
484
the condition $q>k$) implies the existence of a newform of weight
485
$k$, trivial character and level dividing $N/p$, congruent to~$g$
486
modulo $\qq$. This contradicts our hypotheses.
487
\end{proof}
488
489
\begin{remar}
490
For an example of what can be done when~$f$ is congruent to
491
a form of lower level, see the first example in Section~\ref{sec:other_ex}
492
below.
493
\end{remar}
494
495
\begin{lem}\label{at q}
496
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
497
$\ord_{\qq}(c_q)=0$.
498
\end{lem}
499
\begin{proof}
500
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
501
application, at the end of Section~2.2, of the results of
502
\cite{Fa1}) that $T_{\qq}$ is the
503
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
504
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
505
$\mathbb{V}$. (This property is part of the definition of an
506
$S$-integral premotivic structure given in Section~1.2 of
507
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
508
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
509
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
510
2(h) of \cite{Fa1}.)
511
\end{proof}
512
513
\begin{lem}
514
If $A[\lambda]$ is an
515
irreducible representation of $\Gal(\Qbar/\QQ)$,
516
then
517
$$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$
518
\end{lem}
519
\begin{proof}
520
This follows trivially from the definition.
521
\end{proof}
522
523
Putting together the above lemmas we arrive at the following:
524
\begin{prop}\label{sha}
525
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that
526
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$,
527
where $\qq\mid q$. Assume the same hypotheses as in Lemma
528
\ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which
529
locally at $\qq$ are as in the previous section. If
530
$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato
531
conjecture predicts that
532
$$
533
\ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}).
534
$$
535
\end{prop}
536
537
\section{Congruences of special values}
538
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
539
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
540
large enough to contain all the coefficients $a_n$ and $b_n$.
541
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
542
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume
543
that $A[\qq]$ is an irreducible representation of
544
$\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose
545
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
546
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
547
$T_B^{\pm}$ locally at $\qq$. Make two further assumptions:
548
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
549
550
\begin{prop} \label{div}
551
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
552
\end{prop}
553
\begin{proof} This is based on some of the ideas used in Section 1 of
554
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
555
which presumably should refer to ``Condition 2''. Since
556
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
557
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
558
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
559
$$
560
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
561
$$
562
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
563
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
564
where the integral is taken along the positive imaginary axis,
565
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
566
Thus we are reduced
567
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
568
569
Let $\mathcal{D}_0$ be the group of divisors of degree zero
570
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
571
integer $r\geq 0$, let $P_r(R)$ be the additive group of
572
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
573
groups have a natural action of $\Gamma_1(N)$. Let
574
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
575
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
576
577
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
578
the argument in 1.7 of \cite{V}, the cohomology class
579
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
580
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
581
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
582
now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
583
which is why we insist that $q\nmid \phi(N)$. It follows from the
584
last line of Section~4.2 of \cite{St} that, up to some small
585
factorials which do not matter locally at $\qq$,
586
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
587
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
588
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
589
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
590
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
591
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
592
show is divisible by $\qq$.
593
Similarly
594
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
595
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
596
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
597
Therefore it would suffice to show that, for some $\mu\in O_E$,
598
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
599
$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
600
for some $\mu\in O_E$, the element
601
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
602
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
603
non-constant coefficients. This would be the case if
604
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
605
one-dimensional subspace upon reduction modulo~$\qq$. But this is
606
a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need
607
the irreducibility of $A[\qq]$).
608
\end{proof}
609
\begin{remar}\label{sign}
610
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
611
equal. They are determined by the eigenvalue of the
612
Atkin-Lehner involution~$W_N$,
613
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
614
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
615
characteristic coprime to $2N$. The common sign in the functional
616
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
617
$W_N$ acting on~$f$ and~$g$.
618
\end{remar}
619
620
This is analogous to the remark at the end of Section~3 of \cite{CM},
621
which shows that if~$\qq$ has odd residue characteristic and
622
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
623
at least two at $s=k/2$. Note that Maeda's conjecture
624
implies that there are no examples of~$g$ of
625
level one with positive sign in their functional equation such that
626
$L(g,k/2)=0$ (see \cite{CF}).
627
628
\section{Constructing elements of the Shafarevich-Tate group}
629
Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the
630
previous section. In the previous section we showed how the
631
congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$
632
to the divisibility by $\qq$ of an ``algebraic part'' of
633
$L(f,k/2)$. Conjecturally the former is associated with the
634
existence of certain algebraic cycles (for $M_g$) while the latter
635
is associated with the existence of certain elements of the
636
Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
637
section we show how the congruence, interpreted in terms of Galois
638
representations, provides a direct link between algebraic cycles
639
and the Shafarevich-Tate group.
640
641
For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
642
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
643
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
644
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
645
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
646
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
647
648
Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in
649
the functional equation for $L(g,s)$ is positive (this follows
650
from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of
651
vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the
652
Beilinson-Bloch conjecture \cite{B}, the order of vanishing of
653
$L(g,s)$ at $s=k/2$ is the rank of the group
654
$\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence
655
classes of null-homologous, algebraic cycles of codimension $k/2$
656
on the motive $M_g$. (This generalises the part of the
657
Birch--Swinnerton-Dyer conjecture which says that for an elliptic
658
curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is
659
equal to the rank of the Mordell-Weil group $E(\QQ)$.)
660
661
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
662
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
663
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
664
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
665
get (assuming also the Beilinson-Bloch conjecture) a subspace of
666
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
667
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
668
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
669
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
670
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
671
Sections~1 and~6.5 of \cite{Fo2}. We shall call it the ``strong''
672
Beilinson-Bloch conjecture.
673
674
Similarly, if $L(f,k/2)\neq 0$ then we expect that
675
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
676
coincides with the $\qq$-part of $\Sha$.
677
\begin{thm}\label{local}
678
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
679
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
680
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
681
that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
682
newform of weight~$k$, trivial character and level dividing $N/p$.
683
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
684
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
685
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
686
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
687
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
688
\end{thm}
689
690
\begin{proof}
691
It follows easily from our hypothesis that the rank of the free
692
part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is $r$. The natural map from
693
$H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to
694
$H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in
695
the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in
696
H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the
697
$\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence
698
$$
699
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
700
$$
701
where~$\pi$ is multiplication by a uniformising element of
702
$O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial.
703
Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
704
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
705
we get a nonzero, $\qq$-torsion class $\gamma\in
706
H^1(\QQ,A_{\qq}(k/2))$.
707
708
Our aim is to show that $\res_p(\gamma)\in
709
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
710
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
711
712
\begin{enumerate}
713
\item {\bf $p\nmid qN$. }
714
715
Consider the $I_p$-cohomology of the short exact sequence above.
716
717
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
718
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
719
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
720
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
721
follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that
722
the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is
723
zero, hence that the restriction of~$c$ to
724
$H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence
725
the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also
726
zero. By line~3 of p.~125 of \cite{Fl2},
727
$H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in)
728
the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to
729
$H^1(I_p,A_{\qq}(k/2))$, so we have shown that $\res_p(\gamma)\in
730
H^1_f(\QQ_p,A_{\qq}(k/2))$.
731
732
\item {\bf $p\mid N$. }
733
734
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
735
It suffices to show that
736
$$\hspace{3.5em}
737
\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),
738
$$
739
since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to
740
$H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in
741
the proof of Lemma \ref{local1}. It follows as above that the
742
image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is
743
zero. Then $\res_p(c)$ comes from
744
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
745
order of this group is the same as the order of the group
746
$H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we
747
claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$
748
is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that
749
$V_{\qq}(k/2)$ is ramified at $p$, hence $\dim
750
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
751
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
752
consider the case where this common dimension is $1$. The
753
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
754
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication
755
by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It
756
follows from Theor\'eme A of \cite{Ca1} that this is the same as
757
the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of
758
\cite{AL}, it then follows that $p^2\nmid N$ and
759
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
760
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
761
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as
762
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
763
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
764
$\res_p(\gamma)=0$ and certainly lies in
765
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
766
767
\item {\bf $p=q$. }
768
769
Since $q\nmid N$ is a prime of good reduction for the motive
770
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
771
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
772
$V'_{\qq}$ have the same dimension, where
773
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
774
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
775
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
776
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
777
filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may
778
now prove, in the same manner as Proposition 9.2 of \cite{Du3},
779
that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the
780
convenience of the reader, we give some details.
781
782
In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq
783
0}$ is constructed on the Fontaine-Lafaille category of filtered
784
Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and
785
all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where
786
$1_{FD}$ is the ``unit'' filtered Dieudonn\'e module.
787
788
Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T_{\dR}\otimes
789
O_{\qq}$. By Lemma 4.5 (c) of \cite{BK},
790
$$
791
\hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}),
792
$$
793
where
794
$$
795
\hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow
796
H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq}))
797
$$
798
and
799
$$
800
\hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow
801
H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})).
802
$$ Likewise
803
$h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of
804
\cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules,
805
forgetting the $O_{\qq}$-structure.
806
807
For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration
808
shifted by $j$. Then
809
$$\hspace{3.5em}
810
h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j))
811
$$
812
(as long as $k-p+1<j<p-1$, so that $D(j)$ satisfies the hypotheses
813
of Lemma 4.5 of \cite{BK}). By Corollary 3.8.4 of \cite{BK},
814
$$
815
\hspace{3.5em}
816
H^1_f(\QQ_q,V_{\qq}(j))/H^1_e(\QQ_q,V_{\qq}(j))\simeq
817
(D(j)\otimes_{\ZZ_q}\QQ_q)/(1-f)(D(j)\otimes_{\ZZ_q}\QQ_q),
818
$$
819
where $f$ is the Frobenius operator on crystalline cohomology. By
820
1.2.4(ii) of \cite{Sc}, and the Weil conjectures,
821
$H^1_e(\QQ_q,V_{\qq}(j))=H^1_f(\QQ_q,V_{\qq}(j))$, since $j\neq
822
(k-1)/2$. Similarly
823
$H^1_e(\QQ_q,V'_{\qq}(j))=H^1_f(\QQ_q,V'_{\qq}(j))$.
824
825
We have
826
$$\hspace{3.5em}h^1(D(k/2))\simeq H^1_f(\QQ_q,T_{\qq}(k/2))\quad\text{and}\quad
827
h^1(D'(k/2))\simeq H^1_f(\QQ_q,T'_{\qq}(k/2)).$$
828
The exact sequence in the middle of page 366 of \cite{BK} gives us a
829
commutative diagram.
830
$$\hspace{3.5em}\begin{CD}
831
h^1(D'(k/2))@>\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\
832
@VVV@VVV@VVV\\
833
H^1(\QQ_q,T'_{\qq}(k/2))@>\pi
834
>>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)).
835
\end{CD}$$
836
The vertical arrows are all inclusions and we know that the image
837
of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly
838
$H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is
839
surjective since $h^2(D'(k/2))=0$.
840
841
The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image
842
of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is
843
in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and
844
exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem
845
4.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to
846
$D(k/2)/\qq D(k/2)$.
847
848
It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is
849
in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in
850
the exact sequence analogous to the above. Since the map from
851
$h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective,
852
$\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From
853
this it follows that $\res_q(\gamma)\in
854
H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired.
855
\end{enumerate}
856
\end{proof}
857
858
Theorem~2.7 of \cite{AS} is concerned with verifying local
859
conditions in the case $k=2$, where~$f$ and~$g$ are associated
860
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
861
abelian varieties over number fields.) Our restriction outlawing
862
congruences modulo $\qq$ with cusp forms of lower level is
863
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
864
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
865
curve with $\ord_l(j(A))<0$, consideration of a Tate
866
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
867
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
868
at~$l$.)
869
870
In this paper we have encountered two technical problems which we
871
dealt with in quite similar ways:
872
\begin{enumerate}
873
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
874
\item proving local conditions at primes $p\mid N$, for an element
875
of $\qq$-torsion.
876
\end{enumerate}
877
If our only interest was in testing the Bloch-Kato conjecture at
878
$\qq$, we could have made these problems cancel out, as in Lemma
879
8.11 of \cite{DFG}, by weakening the local conditions. However, we
880
have chosen not to do so, since we are also interested in the
881
Shafarevich-Tate group, and since the hypotheses we had to assume
882
are not particularly strong. Note that, since $A[\qq]$ is
883
irreducible, the $\qq$-part of $\Sha$ does not depend on the
884
choice of $T_{\qq}$.
885
886
\section{Examples and Experiments}
887
\label{sec:examples} This section contains tables and numerical
888
examples that illustrate the main themes of this paper. In
889
Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
890
which contains~$16$ examples of pairs $f,g$ such that the strong
891
Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
892
the existence of nontrivial elements of the Shafarevich-Tate group
893
of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
894
the higher-weight modular symbol computations that were used in
895
making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
896
Table~\ref{tab:invisforms}, which summarizes the results of an
897
extensive computation of conjectural orders of Shafarevich-Tate
898
groups for modular motives of low level and weight.
899
Section~\ref{sec:other_ex} gives specific examples in which
900
various hypotheses fail. Note that in \S 7 ``modular symbol'' has
901
a different meaning from in \S 5, being related to homology rather
902
than cohomology. For precise definitions see \cite{SV}.
903
904
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
905
\begin{table}
906
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
907
908
$$
909
\begin{array}{|c|c|c|c|c|}\hline
910
g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline
911
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
912
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
913
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
914
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
915
\vspace{-2ex} & & & & \\
916
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
917
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
918
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
919
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
920
\vspace{-2ex} & & & & \\
921
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
922
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
923
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
924
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
925
\vspace{-2ex} & & & & \\
926
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
927
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
928
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
929
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
930
\hline
931
\end{array}
932
$$
933
\end{table}
934
935
936
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
937
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
938
along with at least one prime~$q$ such that there is a prime
939
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
940
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
941
The notation is as follows.
942
The first column contains a label whose structure is
943
\begin{center}
944
{\bf [Level]k[Weight][GaloisOrbit]}
945
\end{center}
946
This label determines a newform $g=\sum a_n q^n$, up to Galois
947
conjugacy. For example, \nf{127k4C} denotes a newform in the third
948
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
949
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
950
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
951
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
952
being first in the event that the two absolute values are equal,
953
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
954
and so on. The second column contains the degree of the field
955
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
956
contain~$f$ and its degree, respectively. The fifth column
957
contains at least one prime~$q$ such that there is a prime
958
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
959
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
960
satisfied for~$f$,~$g$, and~$\qq$.
961
962
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
963
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
964
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
965
at the end of 7.3 below.
966
967
968
We describe the first line of Table~\ref{tab:newforms}
969
in more detail. See the next section for further details
970
on how the computations were performed.
971
972
Using modular symbols, we find that there is a newform
973
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
974
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
975
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
976
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
977
coefficients generate a number field~$K$ of degree~$17$, and by
978
computing the image of the modular symbol $XY\{0,\infty\}$ under
979
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
980
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
981
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
982
both equal to
983
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
984
+ \cdots\in \FF_{43}[[q]].$$
985
986
There is no form in the Eisenstein subspaces of
987
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
988
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
989
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
990
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
991
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
992
of Theorem~\ref{local}, so if $r$ is the dimension of
993
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
994
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
995
996
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
997
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
998
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
999
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
1000
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
1001
Bloch-Kato conjecture.
1002
1003
For particular examples of elliptic curves one can often find and
1004
write down rational points predicted by the Birch and
1005
Swinnerton-Dyer conjecture. It would be nice if likewise one could
1006
explicitly produce algebraic cycles predicted by the
1007
Beilinson-Bloch conjecture in the above examples. Since
1008
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
1009
0.3.2 of \cite{Z}), so ought to be trivial in
1010
$\CH_0^{k/2}(M_g)\otimes\QQ$.
1011
1012
\subsection{How the computation was performed}\label{sec:howdone}
1013
We give a brief summary of how the computation was performed. The
1014
algorithms that we used were implemented by the second author, and
1015
most are a standard part of MAGMA (see \cite{magma}).
1016
1017
Let~$g$,~$f$, and~$q$ be some data from a line of
1018
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
1019
verified the existence of a congruence modulo~$q$, that
1020
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
1021
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
1022
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
1023
1024
To prove there is a congruence, we showed that the corresponding
1025
{\em integral} spaces of modular symbols satisfy an appropriate
1026
congruence, which forces the existence of a congruence on the
1027
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
1028
irreducible by computing a set that contains all possible residue
1029
characteristics of congruences between~$g$ and any Eisenstein
1030
series of level dividing~$N$, where by congruence, we mean a
1031
congruence for all Fourier coefficients of index~$n$ with
1032
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
1033
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
1034
listing a basis of such~$h$ and finding the possible congruences,
1035
where again we disregard the Fourier coefficients of index not
1036
coprime to~$N$.
1037
1038
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
1039
modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$
1040
under a map with the same kernel as the period mapping, and found that the
1041
image was~$0$. The period mapping sends the modular
1042
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
1043
so that ${\mathbf e}$ maps to~$0$ implies that
1044
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
1045
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
1046
=(-1)^{k/2} g$ which, because of the functional equation, implies
1047
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
1048
independent interest because it includes examples of modular forms
1049
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
1050
the functional equation. We found no such examples of weights
1051
$\geq 8$.
1052
1053
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
1054
In this section we apply some of the results of
1055
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
1056
of Shafarevich-Tate groups of many modular motives. The results of
1057
this section suggest that~$\Sha$ of a modular motive is usually not
1058
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
1059
which agrees with the observations of \cite{CM} and \cite{AS}. For
1060
example, when $k>6$ we find many examples of conjecturally
1061
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
1062
1063
For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}
1064
L(f^{(i)},s)$ where $f^{(i)}$ runs over the
1065
$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
1066
$\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
1067
is defined by integrating integral cuspidal modular symbols (for
1068
$\Gamma_0(N)$) against the conjugates of~$f$. Let
1069
$\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$
1070
eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$
1071
for complex conjugation on~$T$.
1072
1073
1074
{\begin{table}
1075
\vspace{-2ex}
1076
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
1077
\vspace{-4ex}
1078
1079
$$
1080
\begin{array}{|c|c|c|c|}\hline
1081
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1082
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
1083
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
1084
\nf{263k4B} & 39 & 41^{2} & 263 \\
1085
\nf{269k4C} & 39 & 23^{2} & 269 \\
1086
\nf{271k4B} & 39 & 29^{2} & 271 \\
1087
\nf{281k4B} & 40 & 29^{2} & 281 \\
1088
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
1089
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
1090
%\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
1091
% 319k4C removed since Lemma not satisfied.
1092
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
1093
\hline
1094
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
1095
\nf{101k6B} & 24 & 17^{2} & 101 \\
1096
\nf{103k6B} & 24 & 23^{2} & 103 \\
1097
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
1098
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
1099
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
1100
\nf{157k6B} & 34 & 251^{2} & 157 \\
1101
\nf{167k6B} & 40 & 41^{2} & 167 \\
1102
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
1103
\nf{173k6B} & 39 & 71^{2} & 173 \\
1104
\nf{181k6B} & 40 & 107^{2} & 181 \\
1105
\nf{191k6B} & 46 & 85091^{2} & 191 \\
1106
\nf{193k6B} & 41 & 31^{2} & 193 \\
1107
\nf{199k6B} & 46 & 200329^2 & 199 \\
1108
\hline
1109
\nf{47k8B} & 16 & 19^{2} & 47 \\
1110
\nf{59k8B} & 20 & 29^{2} & 59 \\
1111
\nf{67k8B} & 20 & 29^{2} & 67 \\
1112
\nf{71k8B} & 24 & 379^{2} & 71 \\
1113
\nf{73k8B} & 22 & 197^{2} & 73 \\
1114
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
1115
\nf{79k8B} & 25 & 307^{2} & 79 \\
1116
\nf{83k8B} & 27 & 1019^{2} & 83 \\
1117
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
1118
\nf{89k8B} & 29 & 44491^{2} & 89 \\
1119
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
1120
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
1121
\nf{103k8B} & 32 & 75367^{2} & 103 \\
1122
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
1123
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
1124
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
1125
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
1126
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
1127
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
1128
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
1129
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
1130
\hline
1131
\end{array}
1132
$$
1133
\end{table}
1134
\begin{table}
1135
$$
1136
\begin{array}{|c|c|c|c|}\hline
1137
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1138
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
1139
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
1140
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
1141
\nf{125k8D} & 16 & 179^{2} & 5 \\
1142
\nf{127k8B} & 39 & 59^{2} & 127 \\
1143
\nf{128k8F} & 4 & 11^{2} & 1 \\
1144
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
1145
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
1146
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
1147
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
1148
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
1149
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
1150
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,
1151
639259, 17440535
1152
97287 \\
1153
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,
1154
769, 36583
1155
\\
1156
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,
1157
518737, 9837145
1158
699 \\
1159
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
1160
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
1161
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
1162
1163
\hline
1164
1165
\nf{43k10B} & 17 & 449^{2} & 43 \\
1166
\nf{47k10B} & 20 & 2213^{2} & 47 \\
1167
\nf{53k10B} & 21 & 673^{2} & 53 \\
1168
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
1169
\nf{59k10B} & 25 & 37^{2} & 59 \\
1170
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
1171
\nf{64k10K} & 2 & 19^{2} & 3 \\
1172
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
1173
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
1174
\nf{71k10B} & 30 & 1103^{2} & 71 \\
1175
1176
\hline
1177
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
1178
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
1179
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
1180
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
1181
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
1182
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
1183
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
1184
\nf{49k12H} & 12 & 271^{2} & 7 \\
1185
\hline
1186
\end{array}
1187
$$
1188
\end{table}
1189
1190
\begin{lem}\label{lem:lrat}
1191
If $p\nmid Nk!$ is such that $f$ is not congruent to any of its
1192
Galois conjugates modulo a prime dividing $p$ then the $p$-parts
1193
of
1194
$$
1195
\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad
1196
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)
1197
$$
1198
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1199
\end{lem}
1200
1201
\begin{proof} (Sketch.) Let $H$ be the $\ZZ$-module of all
1202
integral cuspidal modular symbols for $\Gamma_0(N)$. Let $I$ be
1203
the image of $H$ under the projection into $H\otimes\QQ$
1204
corresponding to $f$ and its Galois conjugates. $I$ is not
1205
necessarily contained in $H$ since we will have inverted the
1206
residue characteristics of any primes of congruence between $f$
1207
and cuspforms for $\Gamma_0(N)$ which are not Galois conjugate to
1208
$f$.
1209
1210
Now $\mathcal{L}$ is (up to divisors of $Nk!$) the lattice
1211
obtained by pairing the cohomology modular symbols
1212
$\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular
1213
symbols in $H$, or equivalently in $I$. For $1\leq i\leq d$ let
1214
$I_i$ be the $O_E$-module generated by the image of the projection
1215
of $I$ into $I\otimes E$ corresponding to $f^{(i)}$. The finite
1216
index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only
1217
by primes of congruence between $f$ and its Galois conjugates. Up
1218
to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then
1219
a product of the $d$ complex numbers obtained by pairing
1220
$\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. Bearing in
1221
mind the last line of \S 3, and ignoring divisors of $\aaa^{\pm}$,
1222
which are clearly of no importance, we see that these complex
1223
numbers are the $\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$.
1224
We have then a factorisation of the left hand side which shows it
1225
to be equal to the right hand side, to the extent claimed by the
1226
lemma.
1227
\end{proof}
1228
1229
\begin{remar}
1230
The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
1231
modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma
1232
and our computations
1233
say nothing about whether or not $17$ divides
1234
$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$.
1235
\end{remar}
1236
1237
1238
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and
1239
weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
1240
$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
1241
or $k=12$ and $N\leq 49$. Given $f\in \mathcal{S}$, let~$B$ be
1242
defined as follows:
1243
\begin{enumerate}
1244
\item Let $L_1$ be the numerator of the
1245
rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$.
1246
If $L_1=0$ let $B=1$ and terminate.
1247
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1248
\item Let $L_3$ be the part of $L_2$ that is coprime to
1249
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1250
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1251
of any prime of
1252
congruence between~$f$ and a form of weight~$k$ and
1253
lower level. (By congruence here, we mean a congruence for coefficients
1254
$a_n$ with $n$ coprime to the level of~$f$.)
1255
\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
1256
of any prime of congruence
1257
between~$f$ and an Eisenstein series. (This eliminates
1258
residue characteristics of reducible representations.)
1259
\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
1260
of any prime of congruence between $f$ and any one of its Galois
1261
conjugates.
1262
\end{enumerate}
1263
Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1264
$\ord_p(B)
1265
> 0$ then, according
1266
to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$. We
1267
have left the congruence primes in $B$ in the starred examples
1268
since the squares are still suggestive.
1269
1270
We computed~$B$ for every newform in~$\mathcal{S}$. There are
1271
many examples in which $L_3$ is large, but~$B$ is not, and this is
1272
because of Tamagawa factors. For example, {\bf 39k4C} has
1273
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1274
level~$13$; in this case we must have $19\mid c_{3}(2)$, where
1275
$c_{3}(2)$ is as in Section~\ref{sec:bkconj}. See
1276
Section~\ref{sec:other_ex} for more details. Also note that in
1277
every example~$B$ is a perfect square, which is as predicted by
1278
the existence of Flach's generalised Cassels-Tate pairing
1279
\cite{Fl2}. (Note that for $\lambda\mid l$, a non-congruence prime
1280
for $f$, the lattice $T_{\lambda}$ is self-dual, so the pairing
1281
shows that the order of the $\lambda$-part of $\Sha$, if finite,
1282
is a square.) That our computed value of~$B$ should be a square is
1283
not {\it a priori} obvious.
1284
1285
For simplicity, we discard residue characteristics instead of primes
1286
of rings of integers, so our definition of~$B$ is overly conservative.
1287
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1288
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1289
some prime above~$5$, but the prime of congruences of
1290
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1291
Eisenstein.
1292
1293
1294
The newforms for which $B>1$ are given in
1295
Table~\ref{tab:invisforms}. The second column of the table records
1296
the degree of the field generated by the Fourier coefficients
1297
of~$f$. The third contains~$B$. Let~$W$ be the intersection of
1298
the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1299
$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1300
$S_k(\Gamma_0(N),\ZZ)$. The fourth column contains the odd
1301
prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1302
are exactly the possible primes of congruence for~$f$. We place a
1303
$*$ next to the four entries of Table~\ref{tab:invisforms} that
1304
also occur in Table~\ref{tab:newforms}.
1305
1306
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1307
We have some other examples where forms of
1308
different levels are congruent.
1309
However, Remark~\ref{sign} does not
1310
apply, so that one of the forms could have an odd functional
1311
equation, and the other could have an even functional equation.
1312
For instance, we have a $19$-congruence between the
1313
newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1314
coefficients coprime to $39$.
1315
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1316
has {\it odd} functional equation.
1317
Here~$f$ fails the condition about not being congruent
1318
to a form of lower level, so in Lemma~\ref{local1} it is possible that
1319
$\ord_{\qq}(c_{3}(2))>0$. In fact this does happen. Because
1320
$V'_{\qq}$ (attached to~$g$ of level $13$) is unramified at $p=3$,
1321
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1322
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1323
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1324
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1325
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1326
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1327
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1328
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1329
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1330
our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1331
nontrivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1332
does not work. This is just as well, since had it
1333
worked we would have expected
1334
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1335
shows not to be the case.
1336
1337
In the following example, the divisibility between the levels is
1338
the other way round. There is a $7$-congruence between
1339
$g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even
1340
functional equation, and $L(g,3)=0$. In the proof of
1341
Theorem~\ref{local}, there is a problem with the local condition
1342
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1343
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1344
kernel is at most one dimensional, so we still get the
1345
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1346
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1347
elements of $\Sha$ for \nf{61k6B} (assuming all along the strong
1348
Beilinson-Bloch conjecture). In particular, these elements of
1349
$\Sha$ are {\it invisible} at level 61. When the levels are
1350
different we are no longer able to apply Theorem 2.1 of \cite{FJ}.
1351
However, we still have the congruences of integral modular symbols
1352
required to make the proof of Proposition \ref{div} go through.
1353
Indeed, as noted above, the congruences of modular forms were
1354
found by producing congruences of modular symbols. Despite these
1355
congruences of modular symbols, Remark~\ref{sign} does not apply,
1356
since there is no reason to suppose that $w_N=w_{N'}$, where $N$
1357
and $N'$ are the distinct levels.
1358
1359
Finally, there are two examples where we have a form $g$ with even
1360
functional equation such that $L(g,k/2)=0$, and a congruent form
1361
$f$ which has odd functional equation; these are a 23-congruence
1362
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1363
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1364
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1365
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1366
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1367
the appropriate modification of Theorem \ref{local} (with strong
1368
Beilinson-Bloch conjecture) does not necessarily provide us with
1369
nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
1370
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has
1371
$\FF_{\qq}$-rank at least $1$. It could all be in the image of
1372
$H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula
1373
for the first derivative of the complex $L$ function, evaluated at
1374
$s=k/2$, but in combination with a regulator that we have no way
1375
of calculating.
1376
1377
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1378
associated with $f$ and $g$ by the construction of Mazur, Tate and
1379
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1380
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1381
quite clear what to make of this. This divisibility may be proved
1382
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1383
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1384
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1385
congruence between the modular symbols out of which they are
1386
constructed. Integrating an appropriate function against these
1387
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1388
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1389
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1390
where the signs in the functional equations of $L(g,s)$ and
1391
$L_q(g,s)$ are the same, positive in this instance. (According to
1392
the proposition in Section 18 of \cite{MTT}, the signs differ
1393
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1394
1395
We also found some examples for which the conditions of
1396
Theorem~\ref{local} were not met. For example, we have a
1397
$7$-congruence between \nf{639k4B} and \nf{639k4H}, but
1398
$w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a
1399
similar problem with a $7$-congruence between \nf{260k6A} and
1400
\nf{260k6E} --- here $w_{13}=1$ so that $13\equiv
1401
-w_{13}\pmod{7}$. According to Propositions \ref{div} and
1402
\ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
1403
is non-trivial in these examples. Finally, there is a
1404
$5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the
1405
prime~$5$ is less than the weight~$6$ so Propositions \ref{div}
1406
and \ref{sha} do not even apply.
1407
1408
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