CoCalc Shared Fileswww / papers / motive_visibility / dsw_13.texOpen in CoCalc with one click!
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{28 January 2003}
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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, Be} 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 congruence 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 the
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congruence 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}). We thank the referee for
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many constructive comments.
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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~$\ell N$, 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 $\ell$-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 $p\neq \ell$ (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 $p=\ell$ let
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$$
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H^1_f(\QQ_{\ell},V_{\lambda}(j))=\ker
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(H^1(D_{\ell},V_{\lambda}(j))\rightarrow
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H^1(D_{\ell},V_{\lambda}(j)\otimes_{\QQ_{\ell}} B_{\cris}))
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$$
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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 $\ell=2$. 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 a (self-dual) isogeny class of
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abelian varieties over $\QQ$, with endomorphism algebra
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containing~$E$. Choose an abelian variety~$B$ in the isogeny class in
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such a way that the endomorphism ring of~$B$ contains the full ring of
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integers $O_E$. If one takes all the $T_{\lambda}(1)$ to be
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$\lambda$-adic Tate modules, then what we have defined above coincides
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with the usual Shafarevich-Tate group of~$B$ (assuming finiteness of
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the latter, or just taking the quotient by its maximal divisible
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subgroup). To see this one uses 3.11 of \cite{BK}, for $\ell=p$. For
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$\ell\neq p$, $H^1_f(\QQ_p,V_{\ell})=0$. Considering the formal
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group, every class in $B(\QQ_p)/\ell B(\QQ_p)$ is represented by an
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$\ell$-power torsion point in $B(\QQ_p)$, so maps to zero in
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$H^1(\QQ_p,A_{\ell})$.
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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, in the
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case $E=\QQ$.
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\begin{lem}\label{vol}
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$\vol_{\infty}/\aaa^{\pm}=c(2\pi i)^{k/2}\aaa^{\pm}\Omega_{\pm}$, with $c\in E$ and
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$\ord_{\lambda}(c)=0$ for $\lambda\nmid Nk!$.
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\end{lem}
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\begin{proof}
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We note that $\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ times 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. Meanwhile, $\Omega_{\pm}$ is the determinant of the same map with
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respect to the lattices $F^{k/2}T_{\dR}$ and $O_E\delta_f^{\pm}$.
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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}
445
\begin{remar} Note that the ``quantities'' $\aaa^{\pm}\Omega_{\pm}$ and
446
$\vol_{\infty}/\aaa^{\pm}$ are independent of the choice of $\delta_f^{\pm}$.
447
\end{remar}
448
\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
449
Then the fractional ideal $c_p(j)$ is supported at most on
450
divisors of~$p$.
451
\end{lem}
452
\begin{proof}
453
As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
454
$\ord_{\lambda}(c_p(j))$ is the length of the finite
455
$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
456
where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
457
trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
458
torsion free.
459
\end{proof}
460
461
\begin{lem}\label{local1}
462
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
463
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
464
$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
465
that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not
466
congruent modulo $\qq$ (for Fourier coefficients of index coprime
467
to $Nq$) to any newform of weight~$k$, trivial character, and
468
level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$ for all
469
integers~$j$.
470
\end{lem}
471
\begin{proof}
472
There is a natural injective map from
473
$V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A_{\qq}(j))$
474
(i.e., $A_{\qq}(j)^{I_p}$). Consideration of $\qq$-torsion shows
475
that
476
$$
477
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}}
478
H^0(I_p,V_{\qq}(j)).
479
$$ To prove the lemma it suffices to show that
480
$$
481
\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
482
$$
483
since this ensures that $H^0(I_p,A_{\qq}(j))=
484
V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that
485
$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$.
486
If the dimensions differ then, given that $f$ is not congruent
487
modulo $\qq$ to a newform of level dividing $N/p$, Condition (b) of
488
Proposition~2.3 of \cite{L} is satisfied. If Condition (a) was not
489
satisfied then Proposition~2.2 of \cite{L} would imply that $f$
490
was congruent modulo $\qq$ to a twist of level dividing $N/p$.
491
Since Condition (c) is clearly also satisfied, we are in a situation
492
covered by one of the three cases in Proposition~2.3 of \cite{L}.
493
Since $p\not\equiv -1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded,
494
so $A[\qq](j)$ is unramified at $p$ and $\ord_p(N)=1$. (Here we
495
are using Carayol's result that $N$ is the prime-to-$q$ part of
496
the conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem~1 of
497
\cite{JL} (which uses the condition $q>k$) implies the existence
498
of a newform of weight~$k$, trivial character and level dividing
499
$N/p$, congruent to~$g$ modulo $\qq$, for Fourier coefficients of
500
index coprime to $Nq$. This contradicts our hypotheses.
501
\end{proof}
502
503
\begin{remar}
504
For an example of what can be done when~$f$ is congruent to
505
a form of lower level, see the first example in Section~\ref{sec:other_ex}
506
below.
507
\end{remar}
508
509
\begin{lem}\label{at q}
510
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
511
$\ord_{\qq}(c_q)=0$.
512
\end{lem}
513
\begin{proof}
514
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
515
application, at the end of Section~2.2, of the results of
516
\cite{Fa1}) that $T_{\qq}$ is the
517
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
518
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
519
$\mathbb{V}$. (This property is part of the definition of an
520
$S$-integral premotivic structure given in Section~1.2 of
521
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
522
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
523
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
524
2(h) of \cite{Fa1}.)
525
\end{proof}
526
527
\begin{lem}
528
If $A[\lambda]$ is an
529
irreducible representation of $\Gal(\Qbar/\QQ)$,
530
then
531
$$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$
532
\end{lem}
533
\begin{proof}
534
This follows trivially from the definition.
535
\end{proof}
536
537
Putting together the above lemmas we arrive at the following:
538
\begin{prop}\label{sha}
539
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that
540
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$,
541
where $\qq\mid q$. Assume the same hypotheses as in Lemma
542
\ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which
543
locally at $\qq$ are as in the previous section. If
544
$L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato
545
conjecture predicts that
546
$$
547
\ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}).
548
$$
549
\end{prop}
550
551
\section{Congruences of special values}
552
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
553
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
554
large enough to contain all the coefficients $a_n$ and $b_n$.
555
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
556
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume
557
that $A[\qq]$ is an irreducible representation of
558
$\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose
559
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
560
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
561
$T_B^{\pm}$ locally at $\qq$. Make two further assumptions:
562
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
563
564
\begin{prop} \label{div}
565
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
566
\end{prop}
567
\begin{proof} This is based on some of the ideas used in Section 1 of
568
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
569
which presumably should refer to ``Condition 2''. Since
570
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
571
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
572
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
573
$$
574
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
575
$$
576
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
577
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
578
where the integral is taken along the positive imaginary axis,
579
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
580
Thus we are reduced
581
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
582
583
Let $\mathcal{D}_0$ be the group of divisors of degree zero
584
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
585
integer $r\geq 0$, let $P_r(R)$ be the additive group of
586
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
587
groups have a natural action of $\Gamma_1(N)$. Let
588
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
589
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
590
591
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
592
the argument in 1.7 of \cite{V}, the cohomology class
593
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
594
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
595
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are
596
now dealing with cohomology over $X_1(N)$ rather than $M(N)$,
597
which is why we insist that $q\nmid \phi(N)$. It follows from the
598
last line of Section~4.2 of \cite{St} that, up to some small
599
factorials which do not matter locally at $\qq$,
600
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
601
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
602
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
603
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
604
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
605
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
606
show is divisible by $\qq$.
607
Similarly
608
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
609
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
610
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
611
Therefore it would suffice to show that, for some $\mu\in O_E$,
612
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
613
$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,
614
for some $\mu\in O_E$, the element
615
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
616
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
617
non-constant coefficients. This would be the case if
618
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
619
one-dimensional subspace upon reduction modulo~$\qq$. But this is
620
a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need
621
the irreducibility of $A[\qq]$).
622
\end{proof}
623
\begin{remar}\label{sign}
624
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
625
equal. They are determined by the eigenvalue of the
626
Atkin-Lehner involution~$W_N$,
627
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
628
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
629
characteristic coprime to $2N$. The common sign in the functional
630
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
631
$W_N$ acting on~$f$ and~$g$.
632
\end{remar}
633
634
This is analogous to the remark at the end of Section~3 of \cite{CM},
635
which shows that if~$\qq$ has odd residue characteristic and
636
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
637
at least two at $s=k/2$. Note that Maeda's conjecture
638
implies that there are no examples of~$g$ of
639
level one with positive sign in their functional equation such that
640
$L(g,k/2)=0$ (see \cite{CF}).
641
642
\section{Constructing elements of the Shafarevich-Tate group}
643
Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the
644
previous section. In the previous section we showed how the
645
congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$
646
to the divisibility by $\qq$ of an ``algebraic part'' of
647
$L(f,k/2)$. Conjecturally the former is associated with the
648
existence of certain algebraic cycles (for $M_g$) while the latter
649
is associated with the existence of certain elements of the
650
Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this
651
section we show how the congruence, interpreted in terms of Galois
652
representations, provides a direct link between algebraic cycles
653
and the Shafarevich-Tate group.
654
655
For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
656
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
657
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
658
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
659
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
660
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
661
662
Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in
663
the functional equation for $L(g,s)$ is positive (this follows
664
from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of
665
vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the
666
Beilinson-Bloch conjecture \cite{B,Be}, the order of vanishing of
667
$L(g,s)$ at $s=k/2$ is the rank of the group
668
$\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence
669
classes of null-homologous, algebraic cycles of codimension $k/2$
670
on the motive $M_g$. (This generalises the part of the
671
Birch--Swinnerton-Dyer conjecture which says that for an elliptic
672
curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is
673
equal to the rank of the Mordell-Weil group $E(\QQ)$.)
674
675
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
676
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
677
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
678
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
679
get (assuming also the Beilinson-Bloch conjecture) a subspace of
680
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
681
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
682
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
683
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
684
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
685
Sections~1 and~6.5 of \cite{Fo2}. We shall call it the ``strong''
686
Beilinson-Bloch conjecture.
687
688
Similarly, if $L(f,k/2)\neq 0$ then we expect that
689
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
690
coincides with the $\qq$-part of $\Sha$.
691
\begin{thm}\label{local}
692
Let $q\nmid N$ be a prime satisfying $q>k$. Let~$r$ be the
693
dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$. Suppose that $A[\qq]$ is
694
an irreducible representation of $\Gal(\Qbar/\QQ)$ and that for no
695
prime $p\mid N$ is $f$ congruent modulo $\qq$ (for Fourier
696
coefficients of index coprime to $Nq$) to a newform of weight~$k$,
697
trivial character and level dividing $N/p$. Suppose that, for all
698
primes $p\mid N$, $\,p\not\equiv -w_p\pmod{q}$, with $p\not\equiv
699
-1\pmod{q}$ if $p^2\mid N$. (Here $w_p$ is the common eigenvalue
700
of the Atkin-Lehner involution $W_p$ acting on $f$ and $g$.) Then
701
the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
702
$\FF_{\qq}$-rank at least $r$.
703
\end{thm}
704
705
\begin{proof}
706
The theorem is trivially true if $r=0$, so we assume that $r>0$.
707
It follows easily from our hypothesis that the rank of the free
708
part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is~$r$. The natural map from
709
$H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to
710
$H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in
711
the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in
712
H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the
713
$\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence
714
$$
715
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
716
$$
717
where~$\pi$ is multiplication by a uniformising element of
718
$O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial.
719
Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
720
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
721
we get a nonzero, $\qq$-torsion class $\gamma\in
722
H^1(\QQ,A_{\qq}(k/2))$.
723
724
Our aim is to show that $\res_p(\gamma)\in
725
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
726
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
727
728
\vspace{1em}
729
\noindent{\bf Case (1)} $p\nmid qN$:
730
731
Consider the $I_p$-cohomology of the short exact sequence above.
732
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
733
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
734
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
735
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
736
follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that
737
the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is
738
zero, hence that the restriction of~$c$ to
739
$H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence
740
the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also
741
zero. By line~3 of p.~125 of \cite{Fl2},
742
$H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in)
743
the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to
744
$H^1(I_p,A_{\qq}(k/2))$, so we have shown that $\res_p(\gamma)\in
745
H^1_f(\QQ_p,A_{\qq}(k/2))$.
746
747
\vspace{1em}
748
\noindent{\bf Case (2)} $p\mid N$:
749
750
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
751
It suffices to show that
752
$$\hspace{3.5em}
753
\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),
754
$$
755
since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to
756
$H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in
757
the proof of Lemma \ref{local1}. It follows as above that the
758
image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is
759
zero. Then $\res_p(c)$ comes from
760
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
761
order of this group is the same as the order of the group
762
$H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we
763
claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$
764
is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that
765
$V_{\qq}(k/2)$ is ramified at $p$, hence $\dim
766
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
767
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
768
consider the case where this common dimension is $1$. The
769
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
770
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication
771
by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It
772
follows from Theor\'eme A of \cite{Ca1} that this is the same as
773
the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of
774
\cite{AL}, it then follows that $p^2\nmid N$ and
775
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
776
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
777
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as
778
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
779
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
780
$\res_p(\gamma)=0$ and certainly lies in
781
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
782
783
\vspace{1em}
784
\noindent{\bf Case (3)} $p=q$:
785
786
Since $q\nmid N$ is a prime of good reduction for the motive
787
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
788
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
789
$V'_{\qq}$ have the same dimension, where
790
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
791
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
792
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
793
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
794
filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may
795
now prove, in the same manner as Proposition 9.2 of \cite{Du3},
796
that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the
797
convenience of the reader, we give some details.
798
799
In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq
800
0}$ is constructed on the Fontaine-Lafaille category of filtered
801
Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and
802
all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where
803
$1_{FD}$ is the ``unit'' filtered Dieudonn\'e module.
804
805
Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T'_{\dR}\otimes
806
O_{\qq}$. By Lemma 4.5 (c) of \cite{BK},
807
$$
808
\hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}),
809
$$
810
where
811
$$
812
\hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow
813
H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq}))
814
$$
815
and
816
$$
817
\hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow
818
H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})).
819
$$ Likewise
820
$h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of
821
\cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules,
822
forgetting the $O_{\qq}$-structure.
823
824
For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration
825
shifted by $j$. Then
826
$$\hspace{3.5em}
827
h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j))
828
$$
829
(as long as $k-p+1<j<p-1$, so that $D(j)$ satisfies the hypotheses
830
of Lemma 4.5 of \cite{BK}). By Corollary 3.8.4 of \cite{BK},
831
$$
832
\hspace{3.5em}
833
H^1_f(\QQ_q,V_{\qq}(j))/H^1_e(\QQ_q,V_{\qq}(j))\simeq
834
(D(j)\otimes_{\ZZ_q}\QQ_q)/(1-f)(D(j)\otimes_{\ZZ_q}\QQ_q),
835
$$
836
where $f$ is the Frobenius operator on crystalline cohomology. By
837
1.2.4(ii) of \cite{Sc}, and the Weil conjectures,
838
$H^1_e(\QQ_q,V_{\qq}(j))=H^1_f(\QQ_q,V_{\qq}(j))$, since $j\neq
839
(k-1)/2$. Similarly
840
$H^1_e(\QQ_q,V'_{\qq}(j))=H^1_f(\QQ_q,V'_{\qq}(j))$.
841
842
We have
843
$$\hspace{3.5em}h^1(D(k/2))\simeq H^1_f(\QQ_q,T_{\qq}(k/2))\quad\text{and}\quad
844
h^1(D'(k/2))\simeq H^1_f(\QQ_q,T'_{\qq}(k/2)).$$
845
The exact sequence in the middle of page 366 of \cite{BK} gives us a
846
commutative diagram.
847
$$\hspace{3.5em}\begin{CD}
848
h^1(D'(k/2))@>\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\
849
@VVV@VVV@VVV\\
850
H^1(\QQ_q,T'_{\qq}(k/2))@>\pi
851
>>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)).
852
\end{CD}$$
853
The vertical arrows are all inclusions and we know that the image
854
of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly
855
$H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is
856
surjective since $h^2(D'(k/2))=0$.
857
858
The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image
859
of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is
860
in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and
861
exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem
862
4.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to
863
$D(k/2)/\qq D(k/2)$.
864
865
It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is
866
in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in
867
the exact sequence analogous to the above. Since the map from
868
$h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective,
869
$\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From
870
this it follows that $\res_q(\gamma)\in
871
H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired.
872
\end{proof}
873
874
Theorem~2.7 of \cite{AS} is concerned with verifying local
875
conditions in the case $k=2$, where~$f$ and~$g$ are associated
876
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
877
abelian varieties over number fields.) Our restriction outlawing
878
congruences modulo $\qq$ with cusp forms of lower level is
879
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
880
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
881
curve with $\ord_l(j(A))<0$, consideration of a Tate
882
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
883
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
884
at~$l$.)
885
886
In this paper we have encountered two technical problems which we
887
dealt with in quite similar ways:
888
\begin{enumerate}
889
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
890
\item proving local conditions at primes $p\mid N$, for an element
891
of $\qq$-torsion.
892
\end{enumerate}
893
If our only interest was in testing the Bloch-Kato conjecture at
894
$\qq$, we could have made these problems cancel out, as in Lemma
895
8.11 of \cite{DFG}, by weakening the local conditions. However, we
896
have chosen not to do so, since we are also interested in the
897
Shafarevich-Tate group, and since the hypotheses we had to assume
898
are not particularly strong. Note that, since $A[\qq]$ is
899
irreducible, the $\qq$-part of $\Sha$ does not depend on the
900
choice of $T_{\qq}$.
901
902
\section{Examples and Experiments}
903
\label{sec:examples} This section contains tables and numerical
904
examples that illustrate the main themes of this paper. In
905
Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},
906
which contains~$16$ examples of pairs $f,g$ such that the strong
907
Beilinson-Bloch conjecture and Theorem~\ref{local} together imply
908
the existence of nontrivial elements of the Shafarevich-Tate group
909
of the motive attached to~$f$. Section~\ref{sec:howdone} outlines
910
the higher-weight modular symbol computations that were used in
911
making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses
912
Table~\ref{tab:invisforms}, which summarizes the results of an
913
extensive computation of conjectural orders of Shafarevich-Tate
914
groups for modular motives of low level and weight.
915
Section~\ref{sec:other_ex} gives specific examples in which
916
various hypotheses fail. Note that in \S 7 ``modular symbol'' has
917
a different meaning from in \S 5, being related to homology rather
918
than cohomology. For precise definitions see \cite{SV}.
919
920
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
921
\begin{table}
922
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
923
924
$$
925
\begin{array}{|c|c|c|c|c|}\hline
926
g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline
927
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
928
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
929
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
930
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
931
\vspace{-2ex} & & & & \\
932
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
933
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
934
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
935
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
936
\vspace{-2ex} & & & & \\
937
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
938
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
939
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
940
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
941
\vspace{-2ex} & & & & \\
942
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
943
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
944
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
945
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
946
\hline
947
\end{array}
948
$$
949
\end{table}
950
951
952
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
953
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
954
along with at least one prime~$q$ such that there is a prime
955
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
956
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
957
The notation is as follows.
958
The first column contains a label whose structure is
959
\begin{center}
960
{\bf [Level]k[Weight][GaloisOrbit]}
961
\end{center}
962
This label determines a newform $g=\sum a_n q^n$, up to Galois
963
conjugacy. For example, \nf{127k4C} denotes a newform in the third
964
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
965
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
966
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
967
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
968
being first in the event that the two absolute values are equal,
969
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
970
and so on. The second column contains the degree of the field
971
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
972
contain~$f$ and its degree, respectively. The fifth column
973
contains at least one prime~$q$ such that there is a prime
974
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
975
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
976
satisfied for~$f$,~$g$, and~$\qq$.
977
978
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
979
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
980
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
981
at the end of 7.3 below.
982
983
984
We describe the first line of Table~\ref{tab:newforms}
985
in more detail. See the next section for further details
986
on how the computations were performed.
987
988
Using modular symbols, we find that there is a newform
989
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
990
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
991
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
992
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
993
coefficients generate a number field~$K$ of degree~$17$, and by
994
computing the image of the modular symbol $XY\{0,\infty\}$ under
995
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
996
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
997
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
998
both equal to
999
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
1000
+ \cdots\in \FF_{43}[[q]].$$
1001
1002
There is no form in the Eisenstein subspaces of
1003
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
1004
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
1005
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
1006
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
1007
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
1008
of Theorem~\ref{local}, so if $r$ is the dimension of
1009
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
1010
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
1011
1012
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
1013
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
1014
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
1015
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
1016
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
1017
Bloch-Kato conjecture.
1018
1019
For particular examples of elliptic curves one can often find and
1020
write down rational points predicted by the Birch and
1021
Swinnerton-Dyer conjecture. It would be nice if likewise one could
1022
explicitly produce algebraic cycles predicted by the
1023
Beilinson-Bloch conjecture in the above examples. Since
1024
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
1025
0.3.2 of \cite{Z}), so ought to be trivial in
1026
$\CH_0^{k/2}(M_g)\otimes\QQ$.
1027
1028
\subsection{How the computation was performed}\label{sec:howdone}
1029
We give a brief summary of how the computation was performed. The
1030
algorithms that we used were implemented by the second author, and
1031
most are a standard part of MAGMA (see \cite{magma}).
1032
1033
Let~$g$,~$f$, and~$q$ be some data from a line of
1034
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
1035
verified the existence of a congruence modulo~$q$, that
1036
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
1037
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
1038
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
1039
1040
To prove there is a congruence, we showed that the corresponding
1041
{\em integral} spaces of modular symbols satisfy an appropriate
1042
congruence, which forces the existence of a congruence on the
1043
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
1044
irreducible by computing a set that contains all possible residue
1045
characteristics of congruences between~$g$ and any Eisenstein
1046
series of level dividing~$N$, where by congruence, we mean a
1047
congruence for all Fourier coefficients of index~$n$ with
1048
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
1049
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
1050
listing a basis of such~$h$ and finding the possible congruences,
1051
where again we disregard the Fourier coefficients of index not
1052
coprime to~$N$.
1053
1054
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
1055
modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$
1056
under a map with the same kernel as the period mapping, and found that the
1057
image was~$0$. The period mapping sends the modular
1058
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
1059
so that ${\mathbf e}$ maps to~$0$ implies that
1060
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
1061
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
1062
=(-1)^{k/2} g$ which, because of the functional equation, implies
1063
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
1064
independent interest because it includes examples of modular forms
1065
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
1066
the functional equation. We found no such examples of weights
1067
$\geq 8$.
1068
1069
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
1070
In this section we apply some of the results of
1071
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
1072
of Shafarevich-Tate groups of many modular motives. The results of
1073
this section suggest that~$\Sha$ of a modular motive is usually not
1074
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
1075
which agrees with the observations of \cite{CM} and \cite{AS}. For
1076
example, when $k>6$ we find many examples of conjecturally
1077
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
1078
1079
For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}
1080
L(f^{(i)},s)$ where $f^{(i)}$ runs over the
1081
$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus
1082
$\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
1083
is defined by integrating integral cuspidal modular symbols (for
1084
$\Gamma_0(N)$) against the conjugates of~$f$. Let
1085
$\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$
1086
eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$
1087
for complex conjugation on~$T$.
1088
1089
\newpage
1090
{\begin{table}
1091
\vspace{-2ex}
1092
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
1093
\vspace{-4ex}
1094
1095
$$
1096
\begin{array}{|c|c|c|c|}\hline
1097
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1098
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
1099
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
1100
\nf{263k4B} & 39 & 41^{2} & 263 \\
1101
\nf{269k4C} & 39 & 23^{2} & 269 \\
1102
\nf{271k4B} & 39 & 29^{2} & 271 \\
1103
\nf{281k4B} & 40 & 29^{2} & 281 \\
1104
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
1105
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
1106
%\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
1107
% 319k4C removed since Lemma not satisfied.
1108
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
1109
\hline
1110
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
1111
\nf{101k6B} & 24 & 17^{2} & 101 \\
1112
\nf{103k6B} & 24 & 23^{2} & 103 \\
1113
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
1114
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
1115
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
1116
\nf{157k6B} & 34 & 251^{2} & 157 \\
1117
\nf{167k6B} & 40 & 41^{2} & 167 \\
1118
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
1119
\nf{173k6B} & 39 & 71^{2} & 173 \\
1120
\nf{181k6B} & 40 & 107^{2} & 181 \\
1121
\nf{191k6B} & 46 & 85091^{2} & 191 \\
1122
\nf{193k6B} & 41 & 31^{2} & 193 \\
1123
\nf{199k6B} & 46 & 200329^2 & 199 \\
1124
\hline
1125
\nf{47k8B} & 16 & 19^{2} & 47 \\
1126
\nf{59k8B} & 20 & 29^{2} & 59 \\
1127
\nf{67k8B} & 20 & 29^{2} & 67 \\
1128
\nf{71k8B} & 24 & 379^{2} & 71 \\
1129
\nf{73k8B} & 22 & 197^{2} & 73 \\
1130
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
1131
\nf{79k8B} & 25 & 307^{2} & 79 \\
1132
\nf{83k8B} & 27 & 1019^{2} & 83 \\
1133
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
1134
\nf{89k8B} & 29 & 44491^{2} & 89 \\
1135
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
1136
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
1137
\nf{103k8B} & 32 & 75367^{2} & 103 \\
1138
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
1139
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
1140
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
1141
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
1142
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
1143
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
1144
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
1145
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
1146
\hline
1147
\end{array}
1148
$$
1149
\end{table}
1150
\begin{table}
1151
$$
1152
\begin{array}{|c|c|c|c|}\hline
1153
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
1154
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
1155
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
1156
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
1157
\nf{125k8D} & 16 & 179^{2} & 5 \\
1158
\nf{127k8B} & 39 & 59^{2} & 127 \\
1159
\nf{128k8F} & 4 & 11^{2} & 1 \\
1160
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
1161
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
1162
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
1163
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
1164
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
1165
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
1166
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,
1167
639259, 17440535
1168
97287 \\
1169
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,
1170
769, 36583
1171
\\
1172
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,
1173
518737, 9837145
1174
699 \\
1175
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
1176
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
1177
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
1178
1179
\hline
1180
1181
\nf{43k10B} & 17 & 449^{2} & 43 \\
1182
\nf{47k10B} & 20 & 2213^{2} & 47 \\
1183
\nf{53k10B} & 21 & 673^{2} & 53 \\
1184
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
1185
\nf{59k10B} & 25 & 37^{2} & 59 \\
1186
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
1187
\nf{64k10K} & 2 & 19^{2} & 3 \\
1188
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
1189
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
1190
\nf{71k10B} & 30 & 1103^{2} & 71 \\
1191
1192
\hline
1193
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
1194
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
1195
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
1196
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
1197
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
1198
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
1199
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
1200
\nf{49k12H} & 12 & 271^{2} & 7 \\
1201
\hline
1202
\end{array}
1203
$$
1204
\end{table}
1205
1206
\begin{lem}\label{lem:lrat}
1207
Suppose that $p\nmid Nk!$ is such that~$f$ is not congruent to any of its
1208
Galois conjugates modulo a prime dividing~$p$. Then the $p$-parts
1209
of
1210
$$
1211
\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad
1212
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)
1213
$$
1214
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1215
\end{lem}
1216
1217
\begin{proof} Let~$H$ be the $\ZZ$-module of all
1218
integral cuspidal modular symbols for $\Gamma_0(N)$. Let~$I$ be the
1219
image of~$H$ under projection into
1220
the submodule of $H\otimes\QQ$ corresponding
1221
to~$f$ and its Galois conjugates. Note that~$I$ is not necessarily
1222
contained in~$H$, but it is contained in $H\otimes \ZZ[\frac{1}{m}]$
1223
where~$m$ is divisible by the residue
1224
characteristics of any primes of congruence between~$f$ and cuspforms
1225
of weight~$k$ for $\Gamma_0(N)$ which are not Galois conjugate to~$f$.
1226
1227
The lattice $\mathcal{L}$ defined in the paragraph before
1228
the lemma is (up to divisors of $Nk!$)
1229
obtained by pairing the cohomology modular symbols
1230
$\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular
1231
symbols in~$H$; equivalently, since the pairing factors
1232
through the map $H\to I$, the lattice $\mathcal{L}$ is obtained
1233
by pairing with the elements of~$I$.
1234
For $1\leq i\leq d$ let
1235
$I_i$ be the $O_E$-module generated by the image of the projection
1236
of~$I$ into $I\otimes E$ corresponding to $f^{(i)}$.
1237
The finite
1238
index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only
1239
by primes of congruence between $f$ and its Galois conjugates. Up
1240
to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then
1241
a product of the $d$ quantities obtained by pairing
1242
$\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. (These quantities
1243
inhabit a kind of tensor product of $\CC^*$ over $E^*$ with the
1244
group of fractional
1245
ideals of $E$.) Bearing in
1246
mind the last line of \S 3, we see that these quantities are the
1247
$\aaa^{\pm}\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$.
1248
Now we may apply Lemma \ref{vol}. We have then a
1249
factorisation of the left hand side which shows it to be equal to the
1250
right hand side, to the extent claimed by the lemma. Note that
1251
$\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}$ has an interpretation in terms
1252
of integral modular symbols, as in \S 5, and just gets Galois-conjugated when
1253
one replaces $f$ by some $f^{(i)}$.
1254
\end{proof}
1255
1256
\begin{remar}
1257
The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates
1258
modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma
1259
and our computations
1260
say nothing about whether or not $17$ divides
1261
$\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$.
1262
\end{remar}
1263
1264
1265
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and
1266
weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
1267
$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
1268
or $k=12$ and $N\leq 49$. Given $f\in \mathcal{S}$, let~$B$ be
1269
defined as follows:
1270
\begin{enumerate}
1271
\item Let $L_1$ be the numerator of the
1272
rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$.
1273
If $L_1=0$ let $B=1$ and terminate.
1274
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1275
\item Let $L_3$ be the part of $L_2$ that is coprime to
1276
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1277
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1278
of any prime of
1279
congruence between~$f$ and a form of weight~$k$ and
1280
lower level. (By congruence here, we mean a congruence for coefficients
1281
$a_n$ with $n$ coprime to the level of~$f$.)
1282
\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
1283
of any prime of congruence
1284
between~$f$ and an Eisenstein series. (This eliminates
1285
residue characteristics of reducible representations.)
1286
\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
1287
of any prime of congruence between $f$ and any one of its Galois
1288
conjugates.
1289
\end{enumerate}
1290
Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if
1291
$\ord_p(B) > 0$ then, according
1292
to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$.
1293
1294
We computed~$B$ for every newform in~$\mathcal{S}$. There are
1295
many examples in which $L_3$ is large, but~$B$ is not, and this is
1296
because of Tamagawa factors. For example, {\bf 39k4C} has
1297
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
1298
level~$13$; in this case we must have $19\mid c_{3}(2)$, where
1299
$c_{3}(2)$ is as in Section~\ref{sec:bkconj}. See
1300
Section~\ref{sec:other_ex} for more details. Also note that in
1301
every example~$B$ is a perfect square, which, away from congruence
1302
primes, is as predicted by the existence of Flach's generalised
1303
Cassels-Tate pairing \cite{Fl2}. (Note that if $A[\lambda]$ is
1304
irreducible then the lattice $T_{\lambda}$ is at worst a scalar
1305
multiple of its dual, so the pairing shows that the order of the
1306
$\lambda$-part of $\Sha$, if finite, is a square.) That our
1307
computed value of~$B$ should be a square is not {\it a priori}
1308
obvious.
1309
1310
For simplicity, we discard residue characteristics instead of primes
1311
of rings of integers, so our definition of~$B$ is overly conservative.
1312
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1313
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1314
some prime above~$5$, but the prime of congruences of
1315
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1316
Eisenstein.
1317
1318
1319
The newforms for which $B>1$ are given in
1320
Table~\ref{tab:invisforms}. The second column of the table records the
1321
degree of the field generated by the Fourier coefficients of~$f$. The
1322
third contains~$B$. Let~$W$ be the intersection of the span of all
1323
conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the
1324
Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$. The
1325
fourth column contains the odd prime divisors of
1326
$\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which are exactly the
1327
possible primes of congruence between~$f$ and non-conjugate cusp forms
1328
of the same weight and level. We place a~$*$ next to the four entries
1329
of Table~\ref{tab:invisforms} that also occur in
1330
Table~\ref{tab:newforms}.
1331
1332
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1333
We have some other examples where forms of different levels are
1334
congruent (for Fourier coefficients of index coprime to the
1335
levels). However, Remark~\ref{sign} does not apply, so that one of
1336
the forms could have an odd functional equation, and the other
1337
could have an even functional equation. For instance, we have a
1338
$19$-congruence between the newforms $g=\nf{13k4A}$ and
1339
$f=\nf{39k4C}$ of Fourier coefficients of index coprime to $39$.
1340
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$ has {\it odd}
1341
functional equation. Here~$f$ fails the condition about not being
1342
congruent to a form of lower level, so in Lemma~\ref{local1} it is
1343
possible that $\ord_{\qq}(c_{3}(2))>0$. In fact this does happen.
1344
Because $V'_{\qq}$ (attached to~$g$ of level $13$) is unramified
1345
at $p=3$, $H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1346
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1347
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1348
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1349
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1350
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1351
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1352
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1353
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1354
our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1355
nontrivial when $w_p=-1$, so (2) of the proof of
1356
Theorem~\ref{local} does not work. This is just as well, since had
1357
it worked we would have expected
1358
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1359
shows not to be the case.
1360
1361
In the following example, the divisibility between the levels is
1362
the other way round. There is a $7$-congruence between
1363
$g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even
1364
functional equation, and $L(g,3)=0$. In the proof of
1365
Theorem~\ref{local}, there is a problem with the local condition
1366
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1367
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1368
kernel is at most one dimensional, so we still get the
1369
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1370
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1371
elements of $\Sha$ for \nf{61k6B} (assuming all along the strong
1372
Beilinson-Bloch conjecture). In particular, these elements of
1373
$\Sha$ are {\it invisible} at level 61. When the levels are
1374
different we are no longer able to apply Theorem 2.1 of \cite{FJ}.
1375
However, we still have the congruences of integral modular symbols
1376
required to make the proof of Proposition \ref{div} go through.
1377
Indeed, as noted above, the congruences of modular forms were
1378
found by producing congruences of modular symbols. Despite these
1379
congruences of modular symbols, Remark~\ref{sign} does not apply,
1380
since there is no reason to suppose that $w_N=w_{N'}$, where $N$
1381
and $N'$ are the distinct levels.
1382
1383
Finally, there are two examples where we have a form $g$ with even
1384
functional equation such that $L(g,k/2)=0$, and a congruent form
1385
$f$ which has odd functional equation; these are a 23-congruence
1386
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1387
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1388
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1389
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1390
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1391
the appropriate modification of Theorem \ref{local} (with strong
1392
Beilinson-Bloch conjecture) does not necessarily provide us with
1393
nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
1394
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has
1395
$\FF_{\qq}$-rank at least $1$. It could all be in the image of
1396
$H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula
1397
for the first derivative of the complex $L$ function, evaluated at
1398
$s=k/2$, but in combination with a regulator that we have no way
1399
of calculating.
1400
1401
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1402
associated with $f$ and $g$ by the construction of Mazur, Tate and
1403
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1404
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1405
quite clear what to make of this. This divisibility may be proved
1406
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1407
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1408
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1409
congruence between the modular symbols out of which they are
1410
constructed. Integrating an appropriate function against these
1411
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1412
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1413
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1414
where the signs in the functional equations of $L(g,s)$ and
1415
$L_q(g,s)$ are the same, positive in this instance. (According to
1416
the proposition in Section 18 of \cite{MTT}, the signs differ
1417
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1418
1419
We also found some examples for which the conditions of
1420
Theorem~\ref{local} were not met. For example, we have a
1421
$7$-congruence between \nf{639k4B} and \nf{639k4H}, but
1422
$w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a
1423
similar problem with a $7$-congruence between \nf{260k6A} and
1424
\nf{260k6E} --- here $w_{13}=1$ so that $13\equiv
1425
-w_{13}\pmod{7}$. According to Propositions \ref{div} and
1426
\ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
1427
is non-trivial in these examples. Finally, there is a
1428
$5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the
1429
prime~$5$ is less than the weight~$6$ so Propositions \ref{div}
1430
and \ref{sha} (and even Lemma~\ref{lem:lrat}) do not apply.
1431
1432
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1433
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1460
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1474
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1475
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1478
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1481
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1512
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\end{document}
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