%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % dsw_13.tex % % LAST MODIFIED: 11 January 2003 % % AUTHORS: Neil Dummigan, William Stein, Mark Watkins % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass{amsart} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amscd} \newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}} \theoremstyle{plain} \newtheorem{prop}{Proposition}[section] \newtheorem{conj}[prop]{Conjecture} \newtheorem{lem}[prop]{Lemma} \newtheorem{thm}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary} \theoremstyle{definition} \newtheorem{defi}[prop]{Definition} \theoremstyle{remark} \newtheorem{examp}[prop]{Example} \newtheorem{remar}[prop]{Remark} \def\id{\mathop{\mathrm{ id}}\nolimits} \DeclareMathOperator{\Ker}{\mathrm {Ker}} \DeclareMathOperator{\Aut}{{\mathrm {Aut}}} \renewcommand{\Im}{{\mathrm {Im}}} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\End}{End} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Mor}{Mor} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\Nm}{Nm} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Hol}{Hol} \DeclareMathOperator{\vol}{vol} \DeclareMathOperator{\tors}{tors} \DeclareMathOperator{\cris}{cris} \DeclareMathOperator{\length}{length} \DeclareMathOperator{\dR}{dR} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Frob}{Frob} \def\rank{\mathop{\mathrm{ rank}}\nolimits} \newcommand{\Gal}{\mathrm {Gal}} \newcommand{\Spec}{{\mathrm {Spec}}} \newcommand{\Ext}{{\mathrm {Ext}}} \newcommand{\res}{{\mathrm {res}}} \newcommand{\Cor}{{\mathrm {Cor}}} \newcommand{\AAA}{{\mathbb A}} \newcommand{\CC}{{\mathbb C}} \newcommand{\RR}{{\mathbb R}} \newcommand{\QQ}{{\mathbb Q}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\EE}{{\mathbb E}} \newcommand{\TT}{{\mathbb T}} \newcommand{\HHH}{{\mathbb H}} \newcommand{\pp}{{\mathfrak p}} \newcommand{\qq}{{\mathfrak q}} \newcommand{\FF}{{\mathbb F}} \newcommand{\KK}{{\mathbb K}} \newcommand{\GL}{\mathrm {GL}} \newcommand{\SL}{\mathrm {SL}} \newcommand{\Sp}{\mathrm {Sp}} \newcommand{\Br}{\mathrm {Br}} \newcommand{\Qbar}{\overline{\mathbb Q}} \newcommand{\Xbar}{\overline{X}} \newcommand{\Ebar}{\overline{E}} \newcommand{\sbar}{\overline{s}} \newcommand{\nf}[1]{\mbox{\bf #1}} \newcommand{\fbar}{\overline{f}} % ---- SHA ---- \DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts \newcommand{\textcyr}[1]{% {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}% \selectfont #1}} \newcommand{\Sha}{{\mbox{\textcyr{Sh}}}} \newcommand{\HH}{{\mathfrak H}} \newcommand{\aaa}{{\mathfrak a}} \newcommand{\bb}{{\mathfrak b}} \newcommand{\dd}{{\mathfrak d}} \newcommand{\ee}{{\mathbf e}} \newcommand{\Fbar}{\overline{F}} \newcommand{\CH}{\mathrm {CH}} \begin{document} \title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives} \author{Neil Dummigan} \author{William Stein} \author{Mark Watkins} \date{28 January 2003} \subjclass{11F33, 11F67, 11G40.} \keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture, Shafarevich-Tate group.} \address{University of Sheffield\\ Department of Pure Mathematics\\ Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\ U.K.} \address{Harvard University\\Department of Mathematics\\ One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.} \address{Penn State Mathematics Department\\ University Park\\State College, PA 16802\\ U.S.A.} \email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu} \email{watkins@math.psu.edu} \begin{abstract} We study Shafarevich-Tate groups of motives attached to modular forms on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a criterion for the existence of nontrivial elements of these Shafarevich-Tate groups, and give $16$ examples in which a strong form of the Beilinson-Bloch conjecture implies the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich-Tate groups of modular motives of low level and weight at most $12$. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives instead of abelian varieties. \end{abstract} \maketitle \section{Introduction} Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$ be the associated $L$-function. The conjecture of Birch and Swinnerton-Dyer \cite{BSD} predicts that the order of vanishing of $L(E,s)$ at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and also gives an interpretation of the leading term in the Taylor expansion in terms of various quantities, including the order of the Shafarevich-Tate group of~$E$. Cremona and Mazur \cite{CM} look, among all strong Weil elliptic curves over $\QQ$ of conductor $N\leq 5500$, at those with nontrivial Shafarevich-Tate group (according to the Birch and Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate group has predicted elements of prime order~$p$. In most cases they find another elliptic curve, often of the same conductor, whose $p$-torsion is Galois-isomorphic to that of the first one, and which has positive rank. The rational points on the second elliptic curve produce classes in the common $H^1(\QQ,E[p])$. They show \cite{CM2} that these lie in the Shafarevich-Tate group of the first curve, so rational points on one curve explain elements of the Shafarevich-Tate group of the other curve. The Bloch-Kato conjecture \cite{BK} is the generalisation to arbitrary motives of the leading term part of the Birch and Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture \cite{B, Be} generalises the part about the order of vanishing at the central point, identifying it with the rank of a certain Chow group. This paper is a partial generalisation of \cite{CM} and \cite{AS} from abelian varieties over $\QQ$ associated to modular forms of weight~$2$ to the motives attached to modular forms of higher weight. It also does for congruences between modular forms of equal weight what \cite{Du2} did for congruences between modular forms of different weights. We consider the situation where two newforms~$f$ and~$g$, both of even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal $\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but $L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples}, we give sixteen such examples (all with $k=4$ and $k=6$), and in each example, we find that $\qq$ divides the numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain canonical period. In fact, we show how this divisibility may be deduced from the vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The point is, the congruence between$f$ and~$g$ leads to a congruence between suitable ``algebraic parts'' of the special values $L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity one result of Faltings and Jordan shows that the congruence of Fourier expansions leads to a congruence of certain associated cohomology classes. These are then identified with the modular symbols which give rise to the algebraic parts of special values. If $L(g,k/2)$ vanishes then the congruence implies that $L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$. The Bloch-Kato conjecture sometimes then implies that the Shafarevich-Tate group $\Sha$ attached to~$f$ has nonzero $\qq$-torsion. Under certain hypotheses and assumptions, the most substantial of which is the Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to the existence of algebraic cycles, we are able to construct some of the predicted elements of~$\Sha$ using the Galois-theoretic interpretation of the congruence to transfer elements from a Selmer group for~$g$ to a Selmer group for~$f$. One might say that algebraic cycles for one motive explain elements of~$\Sha$ for the other, or that we use the congruence to link the Beilinson-Bloch conjecture for one motive with the Bloch-Kato conjecture for the other. %In proving the local %conditions at primes dividing the level, and also in examining the %local Tamagawa factors at these primes, we make use of a higher weight %level-lowering result due to Jordan and Livn\'e \cite{JL}. We also compute data which, assuming the Bloch-Kato conjecture, provides lower bounds for the orders of numerous Shafarevich-Tate groups (see Section~\ref{sec:invis}). We thank the referee for many constructive comments. %Our data is consistent %with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the %congruence modulus is necessarily a perfect square (assuming that~$\Sha$ %is finite). \section{Motives and Galois representations} This section and the next provide definitions of some of the quantities appearing later in the Bloch-Kato conjecture. Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for $\Gamma_0(N)$, with coefficients in an algebraic number field~$E$, which is necessarily totally real. Let~$\lambda$ be any finite prime of~$E$, and let~$\ell$ denote its residue characteristic. A theorem of Deligne \cite{De1} implies the existence of a two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and a continuous representation $$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$ such that \begin{enumerate} \item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing~$\ell N$, and \item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the characteristic polynomial of $\Frob_p^{-1}$ acting on $V_{\lambda}$ is $x^2-a_px+p^{k-1}$. \end{enumerate} Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as the $\lambda$-adic realisation of a Grothendieck motive $M_f$. There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$, both $2$-dimensional $E$-vector spaces. For details of the construction see \cite{Sc}. The de Rham realisation has a Hodge filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset F^k=\{0\}$. The Betti realisation $V_B$ comes from singular cohomology, while $V_{\lambda}$ comes from \'etale $\ell$-adic cohomology. For each prime $\lambda$, there is a natural isomorphism $V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a $\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$. Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$. There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$), which amounts to multiplying the action of $\Frob_p$ by $p^j$. Following \cite{BK} (Section 3), for $p\neq \ell$ (including $p=\infty$) let $$ H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow H^1(I_p,V_{\lambda}(j))). $$ The subscript~$f$ stands for ``finite part'', $D_p$ is a decomposition subgroup at a prime above~$p$, $I_p$ is the inertia subgroup, and the cohomology is for continuous cocycles and coboundaries. For $p=\ell$ let $$ H^1_f(\QQ_{\ell},V_{\lambda}(j))=\ker (H^1(D_{\ell},V_{\lambda}(j))\rightarrow H^1(D_{\ell},V_{\lambda}(j)\otimes_{\QQ_{\ell}} B_{\cris})) $$ (see Section 1 of \cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and $B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$. There is a natural exact sequence $$ \begin{CD}0@>>>T_{\lambda}(j)@>>> V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}. $$ Let $H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$. Define the $\lambda$-Selmer group $H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of $H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in $H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the condition at $p=\infty$ is superfluous unless $\ell=2$. Define the Shafarevich-Tate group $$ \Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/ \pi_*H^1_f(\QQ,V_{\lambda}(j)). $$ Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any prime ideal~$\lambda$ is the length of the $\lambda$-component of $\Sha(j)$. We shall only concern ourselves with the case $j=k/2$, and write~$\Sha$ for~$\Sha(k/2)$. It depends on the choice of $\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside each $V_{\lambda}$. But if $A[\lambda]$ is irreducible then $T_{\lambda}$ is unique up to scaling and the $\lambda$-part of $\Sha$ is independent of choices. In the case $k=2$ the motive comes from a (self-dual) isogeny class of abelian varieties over $\QQ$, with endomorphism algebra containing~$E$. Choose an abelian variety~$B$ in the isogeny class in such a way that the endomorphism ring of~$B$ contains the full ring of integers $O_E$. If one takes all the $T_{\lambda}(1)$ to be $\lambda$-adic Tate modules, then what we have defined above coincides with the usual Shafarevich-Tate group of~$B$ (assuming finiteness of the latter, or just taking the quotient by its maximal divisible subgroup). To see this one uses 3.11 of \cite{BK}, for $\ell=p$. For $\ell\neq p$, $H^1_f(\QQ_p,V_{\ell})=0$. Considering the formal group, every class in $B(\QQ_p)/\ell B(\QQ_p)$ is represented by an $\ell$-power torsion point in $B(\QQ_p)$, so maps to zero in $H^1(\QQ_p,A_{\ell})$. Define the group of global torsion points $$ \Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)). $$ This is analogous to the group of rational torsion points on an elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in which the exponent of any prime ideal~$\lambda$ is the length of the $\lambda$-component of $\Gamma_{\QQ}$. \section{Canonical periods} We assume from now on for convenience that $N\geq 3$. We need to choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$ and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in \cite{DFG} using cohomology, with non-constant coefficients, of modular curves. (In \cite{DFG}, see especially Sections 2.2 and 5.4, and the paragraph preceding Lemma 2.3.) For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$ module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of $T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$-module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable. Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising generalised elliptic curves with full level-$N$ structure. Let $\mathfrak{E}$ be the universal generalised elliptic curve over $M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed using a projector on the cohomology of a desingularisation of $\mathfrak{E}^{k-2}$). Realising $M(N)(\CC)$ as the disjoint union of $\phi(N)$ copies of the quotient $\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the completed upper half plane), and letting $\tau$ be a variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to the elliptic curve with period lattice generated by $1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product. Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and naturally represents a generating element of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at primes dividing $Nk!$ so that this is the case.) We shall call this element $e(f)$. Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and $V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is a natural action of complex conjugation on $V_B$, breaking it up into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$. Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$ to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let $T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These are rank one $O_E$-modules, but not necessarily free, since the class number of $O_E$ may be greater than one. Choose nonzero elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers $\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$. \section{The Bloch-Kato conjecture}\label{sec:bkconj} In this section we extract from the Bloch-Kato conjecture for $L(f,k/2)$ a prediction about the order of the Shafarevich-Tate group, by analysing the other terms in the formula. Let $L(f,s)$ be the $L$-function attached to~$f$. For $\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with Euler product $\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but there is an analytic continuation given by an integral, as described in the next section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the following equality of fractional ideals of~$E$: $$ \frac{L(f,k/2)}{\vol_{\infty}}= \left(\prod_pc_p(k/2)\right) \frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}. $$ Here, {\bf and from this point onwards, }$\pm$ represents the parity of $(k/2)-1$. The quantity $\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ multiplied by the determinant of the isomorphism $V_B^{\pm}\otimes\CC\simeq (V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For $l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be \begin{align*} \length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}- \ord_{\lambda}(P_p(p^{-j}))\\ =&\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). \end{align*} We omit the definition of $\ord_{\lambda}(c_p(j))$ for $\lambda\mid p$, which requires one to assume Fontaine's de Rham conjecture (\cite{Fo}, Appendix A6), and depends on the choices of $T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$ is a prime of good reduction. For such primes, the de Rham conjecture follows from Theorem 5.6 of \cite{Fa1}.) Strictly speaking, the conjecture in \cite{BK} is only given for $E=\QQ$. We have taken here the obvious generalisation of a slight rearrangement of (5.15.1) of \cite{BK}. The Bloch-Kato conjecture has been reformulated and generalised by Fontaine and Perrin-Riou, who work with general $E$, though that is not really the point of their work. In Section 11 of \cite{Fo2} it is sketched how to deduce the original conjecture from theirs, in the case $E=\QQ$. \begin{lem}\label{vol} $\vol_{\infty}/\aaa^{\pm}=c(2\pi i)^{k/2}\aaa^{\pm}\Omega_{\pm}$, with $c\in E$ and $\ord_{\lambda}(c)=0$ for $\lambda\nmid Nk!$. \end{lem} \begin{proof} We note that $\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ times the determinant of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we used above in the definition of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}). We are using here natural pairings. Meanwhile, $\Omega_{\pm}$ is the determinant of the same map with respect to the lattices $F^{k/2}T_{\dR}$ and $O_E\delta_f^{\pm}$. Recall that the index of $O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed by noting that, locally away from primes dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its dual, both being equal to the ideal denoted~$\eta$ in \cite{DFG2}. \end{proof} \begin{remar} Note that the ``quantities'' $\aaa^{\pm}\Omega_{\pm}$ and $\vol_{\infty}/\aaa^{\pm}$ are independent of the choice of $\delta_f^{\pm}$. \end{remar} \begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer. Then the fractional ideal $c_p(j)$ is supported at most on divisors of~$p$. \end{lem} \begin{proof} As on p.~30 of \cite{Fl1}, for odd $l\neq p$, $\ord_{\lambda}(c_p(j))$ is the length of the finite $O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$ where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is torsion free. \end{proof} \begin{lem}\label{local1} Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent modulo $\qq$ (for Fourier coefficients of index coprime to $Nq$) to any newform of weight~$k$, trivial character, and level dividing $N/p$. Then $\ord_{\qq}(c_p(j))=0$ for all integers~$j$. \end{lem} \begin{proof} There is a natural injective map from $V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$ to $H^0(I_p,A_{\qq}(j))$ (i.e., $A_{\qq}(j)^{I_p}$). Consideration of $\qq$-torsion shows that $$ \dim_{O_E/\qq} H^0(I_p,A[\qq](j))\geq \dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)). $$ To prove the lemma it suffices to show that $$ \dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)), $$ since this ensures that $H^0(I_p,A_{\qq}(j))= V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}$, hence that $H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p})$. If the dimensions differ then, given that $f$ is not congruent modulo $\qq$ to a newform of level dividing $N/p$, Condition (b) of Proposition~2.3 of \cite{L} is satisfied. If Condition (a) was not satisfied then Proposition~2.2 of \cite{L} would imply that $f$ was congruent modulo $\qq$ to a twist of level dividing $N/p$. Since Condition (c) is clearly also satisfied, we are in a situation covered by one of the three cases in Proposition~2.3 of \cite{L}. Since $p\not\equiv -1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's result that $N$ is the prime-to-$q$ part of the conductor of $V_{\qq}$ \cite{Ca1}.) But then Theorem~1 of \cite{JL} (which uses the condition $q>k$) implies the existence of a newform of weight~$k$, trivial character and level dividing $N/p$, congruent to~$g$ modulo $\qq$, for Fourier coefficients of index coprime to $Nq$. This contradicts our hypotheses. \end{proof} \begin{remar} For an example of what can be done when~$f$ is congruent to a form of lower level, see the first example in Section~\ref{sec:other_ex} below. \end{remar} \begin{lem}\label{at q} If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then $\ord_{\qq}(c_q)=0$. \end{lem} \begin{proof} It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an application, at the end of Section~2.2, of the results of \cite{Fa1}) that $T_{\qq}$ is the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered module $T_{\dR}\otimes O_{\qq}$ by the functor they call $\mathbb{V}$. (This property is part of the definition of an $S$-integral premotivic structure given in Section~1.2 of \cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii) of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used in Theorem~4.1 of \cite{BK} follows from the first paragraph of 2(h) of \cite{Fa1}.) \end{proof} \begin{lem} If $A[\lambda]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$, then $$\ord_{\lambda}(\#\Gamma_{\QQ})=0.$$ \end{lem} \begin{proof} This follows trivially from the definition. \end{proof} Putting together the above lemmas we arrive at the following: \begin{prop}\label{sha} Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$. Assume the same hypotheses as in Lemma \ref{local1} for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$ are as in the previous section. If $L(f,k/2)\aaa^{\pm}/\vol_{\infty}\neq 0$ then the Bloch-Kato conjecture predicts that $$ \ord_{\qq}(\#\Sha)=\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty}). $$ \end{prop} \section{Congruences of special values} Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field large enough to contain all the coefficients $a_n$ and $b_n$. Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Assume that $A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$, and that $q\nmid N\phi(N)k!$. Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Make two further assumptions: $$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$ \begin{prop} \label{div} With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$. \end{prop} \begin{proof} This is based on some of the ideas used in Section 1 of \cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V}, which presumably should refer to ``Condition 2''. Since $\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that $\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm 1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that $$ \int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s). $$ Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period $$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$ where the integral is taken along the positive imaginary axis, then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$ Thus we are reduced to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$. Let $\mathcal{D}_0$ be the group of divisors of degree zero supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and integer $r\geq 0$, let $P_r(R)$ be the additive group of homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these groups have a natural action of $\Gamma_1(N)$. Let $S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$ be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$. Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with the argument in 1.7 of \cite{V}, the cohomology class $\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are now dealing with cohomology over $X_1(N)$ rather than $M(N)$, which is why we insist that $q\nmid \phi(N)$. It follows from the last line of Section~4.2 of \cite{St} that, up to some small factorials which do not matter locally at $\qq$, $$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv (k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that $$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv (k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to show is divisible by $\qq$. Similarly $$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv (k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$. Therefore it would suffice to show that, for some $\mu\in O_E$, the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by $\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that, for some $\mu\in O_E$, the element $\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$, considered as an element of $\qq$-adic cohomology of $X_1(N)$ with non-constant coefficients. This would be the case if $\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same one-dimensional subspace upon reduction modulo~$\qq$. But this is a consequence of Theorem 2.1(1) of \cite{FJ} (for which we need the irreducibility of $A[\qq]$). \end{proof} \begin{remar}\label{sign} The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are equal. They are determined by the eigenvalue of the Atkin-Lehner involution~$W_N$, which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and $b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue characteristic coprime to $2N$. The common sign in the functional equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of $W_N$ acting on~$f$ and~$g$. \end{remar} This is analogous to the remark at the end of Section~3 of \cite{CM}, which shows that if~$\qq$ has odd residue characteristic and $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order at least two at $s=k/2$. Note that Maeda's conjecture implies that there are no examples of~$g$ of level one with positive sign in their functional equation such that $L(g,k/2)=0$ (see \cite{CF}). \section{Constructing elements of the Shafarevich-Tate group} Let~$f$,~$g$ and $\qq$ be as in the first paragraph of the previous section. In the previous section we showed how the congruence between $f$ and $g$ relates the vanishing of $L(g,k/2)$ to the divisibility by $\qq$ of an ``algebraic part'' of $L(f,k/2)$. Conjecturally the former is associated with the existence of certain algebraic cycles (for $M_g$) while the latter is associated with the existence of certain elements of the Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this section we show how the congruence, interpreted in terms of Galois representations, provides a direct link between algebraic cycles and the Shafarevich-Tate group. For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$ is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules. Recall that $L(g,k/2)=0$ and $L(f,k/2)\neq 0$. Since the sign in the functional equation for $L(g,s)$ is positive (this follows from $L(f,k/2)\neq 0$, see Remark \ref{sign}), the order of vanishing of $L(g,s)$ at $s=k/2$ is at least $2$. According to the Beilinson-Bloch conjecture \cite{B,Be}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational rational equivalence classes of null-homologous, algebraic cycles of codimension $k/2$ on the motive $M_g$. (This generalises the part of the Birch--Swinnerton-Dyer conjecture which says that for an elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.) Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}. If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we get (assuming also the Beilinson-Bloch conjecture) a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in Sections~1 and~6.5 of \cite{Fo2}. We shall call it the ``strong'' Beilinson-Bloch conjecture. Similarly, if $L(f,k/2)\neq 0$ then we expect that $H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$ coincides with the $\qq$-part of $\Sha$. \begin{thm}\label{local} Let $q\nmid N$ be a prime satisfying $q>k$. Let~$r$ be the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$. Suppose that $A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ (for Fourier coefficients of index coprime to $Nq$) to a newform of weight~$k$, trivial character and level dividing $N/p$. Suppose that, for all primes $p\mid N$, $\,p\not\equiv -w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here $w_p$ is the common eigenvalue of the Atkin-Lehner involution $W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$. \end{thm} \begin{proof} The theorem is trivially true if $r=0$, so we assume that $r>0$. It follows easily from our hypothesis that the rank of the free part of $H^1_f(\QQ,T'_{\qq}(k/2))$ is~$r$. The natural map from $H^1_f(\QQ,T'_{\qq}(k/2))/\qq H^1_f(\QQ,T'_{\qq}(k/2))$ to $H^1(\QQ,A'[\qq](k/2))$ is injective. Take a nonzero class $c$ in the image, which has $\FF_{\qq}$-rank $r$. Choose $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ mapping to $c$. Consider the $\Gal(\Qbar/\QQ)$-cohomology of the short exact sequence $$ \begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD}, $$ where~$\pi$ is multiplication by a uniformising element of $O_{\qq}$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial. Hence $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so $H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and we get a nonzero, $\qq$-torsion class $\gamma\in H^1(\QQ,A_{\qq}(k/2))$. Our aim is to show that $\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$. \vspace{1em} \noindent{\bf Case (1)} $p\nmid qN$: Consider the $I_p$-cohomology of the short exact sequence above. Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in H^1_f(\QQ,T'_{\qq}(k/2))$ that the image in $H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence that the restriction of~$c$ to $H^1(I_p,A'[\qq](k/2))\simeq H^1(I_p,A[\qq](k/2))$ is zero. Hence the restriction of $\gamma$ to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$ to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that $\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$. \vspace{1em} \noindent{\bf Case (2)} $p\mid N$: First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible. It suffices to show that $$\hspace{3.5em} \dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)), $$ since then the natural map from $H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is surjective; this may be done as in the proof of Lemma \ref{local1}. It follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then $\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The order of this group is the same as the order of the group $H^0(\QQ_p,A[\qq](k/2))$ (this is Lemma 1 of \cite{W}), which we claim is trivial. By the work of Carayol \cite{Ca1}, the level $N$ is the conductor of $V_{\qq}(k/2)$, so $p\mid N$ implies that $V_{\qq}(k/2)$ is ramified at $p$, hence $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only consider the case where this common dimension is $1$. The (motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq})$. It follows from Theor\'eme A of \cite{Ca1} that this is the same as the Euler factor at $p$ of $L(f,s)$. By Theorems 3(ii) and 5 of \cite{AL}, it then follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$) as $-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in $H^1_f(\QQ_p,A_{\qq}(k/2))$. \vspace{1em} \noindent{\bf Case (3)} $p=q$: Since $q\nmid N$ is a prime of good reduction for the motive $M_g$, $\,V'_{\qq}$ is a crystalline representation of $\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and $V'_{\qq}$ have the same dimension, where $D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.) As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered module $T_{\dR}\otimes O_{\qq}$. Since also $q>k$, we may now prove, in the same manner as Proposition 9.2 of \cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$. For the convenience of the reader, we give some details. In Lemma 4.4 of \cite{BK}, a cohomological functor $\{h^i\}_{i\geq 0}$ is constructed on the Fontaine-Lafaille category of filtered Dieudonn\'e modules over $\ZZ_q$. $h^i(D)=0$ for all $i\geq2$ and all $D$, and $h^i(D)=\Ext^i(1_{FD},D)$ for all $i$ and $D$, where $1_{FD}$ is the ``unit'' filtered Dieudonn\'e module. Now let $D=T_{\dR}\otimes O_{\qq}$ and $D'=T'_{\dR}\otimes O_{\qq}$. By Lemma 4.5 (c) of \cite{BK}, $$ \hspace{3.5em} h^1(D)\simeq H^1_e(\QQ_q,T_{\qq}), $$ where $$ \hspace{3.5em}H^1_e(\QQ_q,T_{\qq})=\ker(H^1(\QQ_q,T_{\qq})\rightarrow H^1(\QQ_q,V_{\qq})/H^1_e(\QQ_q,V_{\qq})) $$ and $$ \hspace{3.5em}H^1_e(\QQ_q,V_{\qq})=\ker(H^1(\QQ_q,V_{\qq})\rightarrow H^1(\QQ_q,B_{\cris}^{f=1}\otimes_{\QQ_q} V_{\qq})). $$ Likewise $h^1(D')\simeq H^1_e(\QQ_q,T'_{\qq}).$ When applying results of \cite{BK} we view $D$, $T_{\qq}$ etc. simply as $\ZZ_q$-modules, forgetting the $O_{\qq}$-structure. For an integer $j$ let $D(j)$ be $D$ with the Hodge filtration shifted by $j$. Then $$\hspace{3.5em} h^1(D(j))\simeq H^1_e(\QQ_q,T_{\qq}(j)) $$ (as long as $k-p+1\pi >>h^1(D'(k/2))@>>>h^1(D'(k/2)/\qq D'(k/2))\\ @VVV@VVV@VVV\\ H^1(\QQ_q,T'_{\qq}(k/2))@>\pi >>H^1(\QQ_q,T'_{\qq}(k/2))@>>>H^1(\QQ_q,A'[\qq](k/2)). \end{CD}$$ The vertical arrows are all inclusions and we know that the image of $h^1(D'(k/2))$ in $H^1(\QQ_q,T'_{\qq}(k/2))$ is exactly $H^1_f(\QQ_q,T'_{\qq}(k/2))$. The top right horizontal map is surjective since $h^2(D'(k/2))=0$. The class $\res_q(c)\in H^1(\QQ_q,A'[\qq](k/2))$ is in the image of $H^1_f(\QQ_q,T'_{\qq}(k/2))$, by construction, and therefore is in the image of $h^1(D'(k/2)/\qq D'(k/2))$. By the fullness and exactness of the Fontaine-Lafaille functor \cite{FL} (see Theorem 4.3 of \cite{BK}), $D'(k/2)/\qq D'(k/2)$ is isomorphic to $D(k/2)/\qq D(k/2)$. It follows that the class $\res_q(c)\in H^1(\QQ_q,A[\qq](k/2))$ is in the image of $h^1(D(k/2)/\qq D(k/2))$ by the vertical map in the exact sequence analogous to the above. Since the map from $h^1(D(k/2))$ to $h^1(D(k/2)/\qq D(k/2))$ is surjective, $\res_q(c)$ lies in the image of $H^1_f(\QQ_q,T_{\qq}(k/2))$. From this it follows that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$, as desired. \end{proof} Theorem~2.7 of \cite{AS} is concerned with verifying local conditions in the case $k=2$, where~$f$ and~$g$ are associated with abelian varieties~$A$ and~$B$. (Their theorem also applies to abelian varieties over number fields.) Our restriction outlawing congruences modulo $\qq$ with cusp forms of lower level is analogous to theirs forbidding~$q$ from dividing Tamagawa factors $c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic curve with $\ord_l(j(A))<0$, consideration of a Tate parametrisation shows that if $q\mid c_{A,l}$, i.e., if $q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified at~$l$.) In this paper we have encountered two technical problems which we dealt with in quite similar ways: \begin{enumerate} \item dealing with the $\qq$-part of $c_p$ for $p\mid N$; \item proving local conditions at primes $p\mid N$, for an element of $\qq$-torsion. \end{enumerate} If our only interest was in testing the Bloch-Kato conjecture at $\qq$, we could have made these problems cancel out, as in Lemma 8.11 of \cite{DFG}, by weakening the local conditions. However, we have chosen not to do so, since we are also interested in the Shafarevich-Tate group, and since the hypotheses we had to assume are not particularly strong. Note that, since $A[\qq]$ is irreducible, the $\qq$-part of $\Sha$ does not depend on the choice of $T_{\qq}$. \section{Examples and Experiments} \label{sec:examples} This section contains tables and numerical examples that illustrate the main themes of this paper. In Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms}, which contains~$16$ examples of pairs $f,g$ such that the strong Beilinson-Bloch conjecture and Theorem~\ref{local} together imply the existence of nontrivial elements of the Shafarevich-Tate group of the motive attached to~$f$. Section~\ref{sec:howdone} outlines the higher-weight modular symbol computations that were used in making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses Table~\ref{tab:invisforms}, which summarizes the results of an extensive computation of conjectural orders of Shafarevich-Tate groups for modular motives of low level and weight. Section~\ref{sec:other_ex} gives specific examples in which various hypotheses fail. Note that in \S 7 ``modular symbol'' has a different meaning from in \S 5, being related to homology rather than cohomology. For precise definitions see \cite{SV}. \subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable} \begin{table} \caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex} $$ \begin{array}{|c|c|c|c|c|}\hline g & \deg(g) & f & \deg(f) & q\text{'}s \\\hline \nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ \nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ \nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ \nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ \vspace{-2ex} & & & & \\ \nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ \nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ \nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ \nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ \vspace{-2ex} & & & & \\ \nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\ \nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\ \nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\ \nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\ \vspace{-2ex} & & & & \\ \nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\ \nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ \nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ \nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\ \hline \end{array} $$ \end{table} Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels) along with at least one prime~$q$ such that there is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case, $\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$. The notation is as follows. The first column contains a label whose structure is \begin{center} {\bf [Level]k[Weight][GaloisOrbit]} \end{center} This label determines a newform $g=\sum a_n q^n$, up to Galois conjugacy. For example, \nf{127k4C} denotes a newform in the third Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois orbits are ordered first by the degree of $\QQ(\ldots, a_n, \ldots)$, then by the sequence of absolute values $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace being first in the event that the two absolute values are equal, and the first Galois orbit is denoted {\bf A}, the second {\bf B}, and so on. The second column contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The third and fourth columns contain~$f$ and its degree, respectively. The fifth column contains at least one prime~$q$ such that there is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are satisfied for~$f$,~$g$, and~$\qq$. For the two examples \nf{581k4E} and \nf{684k4K}, the square of a prime $q$ appears in the $q$-column, meaning $q^2$ divides the order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined at the end of 7.3 below. We describe the first line of Table~\ref{tab:newforms} in more detail. See the next section for further details on how the computations were performed. Using modular symbols, we find that there is a newform $$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots \in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$, the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier coefficients generate a number field~$K$ of degree~$17$, and by computing the image of the modular symbol $XY\{0,\infty\}$ under the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$ and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are both equal to $$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$ There is no form in the Eisenstein subspaces of $M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with $(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so $\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a level~$1$ form of weight~$4$. Thus we have checked the hypotheses of Theorem~\ref{local}, so if $r$ is the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$. Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions, we have constructed the $\qq$-torsion in $\Sha$ predicted by the Bloch-Kato conjecture. For particular examples of elliptic curves one can often find and write down rational points predicted by the Birch and Swinnerton-Dyer conjecture. It would be nice if likewise one could explicitly produce algebraic cycles predicted by the Beilinson-Bloch conjecture in the above examples. Since $L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary 0.3.2 of \cite{Z}), so ought to be trivial in $\CH_0^{k/2}(M_g)\otimes\QQ$. \subsection{How the computation was performed}\label{sec:howdone} We give a brief summary of how the computation was performed. The algorithms that we used were implemented by the second author, and most are a standard part of MAGMA (see \cite{magma}). Let~$g$,~$f$, and~$q$ be some data from a line of Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We verified the existence of a congruence modulo~$q$, that $L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does not arise from any $S_k(\Gamma_0(N/p))$, as follows: To prove there is a congruence, we showed that the corresponding {\em integral} spaces of modular symbols satisfy an appropriate congruence, which forces the existence of a congruence on the level of Fourier expansions. We showed that $\rho_{g,\qq}$ is irreducible by computing a set that contains all possible residue characteristics of congruences between~$g$ and any Eisenstein series of level dividing~$N$, where by congruence, we mean a congruence for all Fourier coefficients of index~$n$ with $(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by listing a basis of such~$h$ and finding the possible congruences, where again we disregard the Fourier coefficients of index not coprime to~$N$. To verify that $L(g,\frac{k}{2})=0$, we computed the image of the modular symbol ${\mathbf e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map with the same kernel as the period mapping, and found that the image was~$0$. The period mapping sends the modular symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$, so that ${\mathbf e}$ maps to~$0$ implies that $L(g,\frac{k}{2})=0$. In a similar way, we verified that $L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) =(-1)^{k/2} g$ which, because of the functional equation, implies that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of independent interest because it includes examples of modular forms of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by the functional equation. We found no such examples of weights $\geq 8$. \subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis} In this section we apply some of the results of Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders of Shafarevich-Tate groups of many modular motives. The results of this section suggest that~$\Sha$ of a modular motive is usually not ``visible at level~$N$'', i.e., explained by congruences at level~$N$, which agrees with the observations of \cite{CM} and \cite{AS}. For example, when $k>6$ we find many examples of conjecturally nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$. For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d} L(f^{(i)},s)$ where $f^{(i)}$ runs over the $\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus $\CC^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by integrating integral cuspidal modular symbols (for $\Gamma_0(N)$) against the conjugates of~$f$. Let $\Omega_{M_f/\QQ}$ denote the volume of the $(-1)^{(k/2)-1}$ eigenspace $T^{\pm}=\{z \in T : \overline{z}=(-1)^{(k/2)-1}z\}$ for complex conjugation on~$T$. \newpage {\begin{table} \vspace{-2ex} \caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)} \vspace{-4ex} $$ \begin{array}{|c|c|c|c|}\hline f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline \nf{127k4C}* & 17 & 43^{2} & 43, 127 \\ \nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\ \nf{263k4B} & 39 & 41^{2} & 263 \\ \nf{269k4C} & 39 & 23^{2} & 269 \\ \nf{271k4B} & 39 & 29^{2} & 271 \\ \nf{281k4B} & 40 & 29^{2} & 281 \\ \nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\ \nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\ %\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\ % 319k4C removed since Lemma not satisfied. \nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\ \hline \nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\ \nf{101k6B} & 24 & 17^{2} & 101 \\ \nf{103k6B} & 24 & 23^{2} & 103 \\ \nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\ \nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\ \nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\ \nf{157k6B} & 34 & 251^{2} & 157 \\ \nf{167k6B} & 40 & 41^{2} & 167 \\ \nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\ \nf{173k6B} & 39 & 71^{2} & 173 \\ \nf{181k6B} & 40 & 107^{2} & 181 \\ \nf{191k6B} & 46 & 85091^{2} & 191 \\ \nf{193k6B} & 41 & 31^{2} & 193 \\ \nf{199k6B} & 46 & 200329^2 & 199 \\ \hline \nf{47k8B} & 16 & 19^{2} & 47 \\ \nf{59k8B} & 20 & 29^{2} & 59 \\ \nf{67k8B} & 20 & 29^{2} & 67 \\ \nf{71k8B} & 24 & 379^{2} & 71 \\ \nf{73k8B} & 22 & 197^{2} & 73 \\ \nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\ \nf{79k8B} & 25 & 307^{2} & 79 \\ \nf{83k8B} & 27 & 1019^{2} & 83 \\ \nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\ \nf{89k8B} & 29 & 44491^{2} & 89 \\ \nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\ \nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\ \nf{103k8B} & 32 & 75367^{2} & 103 \\ \nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\ \nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\ \nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\ \nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\ \nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\ \nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\ \nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\ \nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\ \hline \end{array} $$ \end{table} \begin{table} $$ \begin{array}{|c|c|c|c|}\hline f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline \nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\ \nf{121k8G} & 12 & 13^{2} & 3, 11 \\ \nf{121k8H} & 12 & 19^{2} & 5, 11 \\ \nf{125k8D} & 16 & 179^{2} & 5 \\ \nf{127k8B} & 39 & 59^{2} & 127 \\ \nf{128k8F} & 4 & 11^{2} & 1 \\ \nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\ \nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\ \nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\ \nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\ \nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\ \nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\ \nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397, 639259, 17440535 97287 \\ \nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241, 769, 36583 \\ \nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577, 518737, 9837145 699 \\ \nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\ \nf{148k8B} & 11 & 19^{2} & 3, 37 \\ \nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\ \hline \nf{43k10B} & 17 & 449^{2} & 43 \\ \nf{47k10B} & 20 & 2213^{2} & 47 \\ \nf{53k10B} & 21 & 673^{2} & 53 \\ \nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\ \nf{59k10B} & 25 & 37^{2} & 59 \\ \nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\ \nf{64k10K} & 2 & 19^{2} & 3 \\ \nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\ \nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\ \nf{71k10B} & 30 & 1103^{2} & 71 \\ \hline \nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\ \nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\ \nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\ \nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\ \nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\ \nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\ \nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\ \nf{49k12H} & 12 & 271^{2} & 7 \\ \hline \end{array} $$ \end{table} \begin{lem}\label{lem:lrat} Suppose that $p\nmid Nk!$ is such that~$f$ is not congruent to any of its Galois conjugates modulo a prime dividing~$p$. Then the $p$-parts of $$ \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right) $$ are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}. \end{lem} \begin{proof} Let~$H$ be the $\ZZ$-module of all integral cuspidal modular symbols for $\Gamma_0(N)$. Let~$I$ be the image of~$H$ under projection into the submodule of $H\otimes\QQ$ corresponding to~$f$ and its Galois conjugates. Note that~$I$ is not necessarily contained in~$H$, but it is contained in $H\otimes \ZZ[\frac{1}{m}]$ where~$m$ is divisible by the residue characteristics of any primes of congruence between~$f$ and cuspforms of weight~$k$ for $\Gamma_0(N)$ which are not Galois conjugate to~$f$. The lattice $\mathcal{L}$ defined in the paragraph before the lemma is (up to divisors of $Nk!$) obtained by pairing the cohomology modular symbols $\Phi_{f^{(i)}}^{\pm}$ (as in \S 5) with the homology modular symbols in~$H$; equivalently, since the pairing factors through the map $H\to I$, the lattice $\mathcal{L}$ is obtained by pairing with the elements of~$I$. For $1\leq i\leq d$ let $I_i$ be the $O_E$-module generated by the image of the projection of~$I$ into $I\otimes E$ corresponding to $f^{(i)}$. The finite index of $I\otimes O_E$ in $\oplus_{i=1}^d I_i$ is divisible only by primes of congruence between $f$ and its Galois conjugates. Up to these primes, $\Omega_{M_f/\QQ}/(2\pi i)^{((k/2)-1)d}$ is then a product of the $d$ quantities obtained by pairing $\Phi_{f^{(i)}}^{\pm}$ with $I_i$, for $1\leq i\leq d$. (These quantities inhabit a kind of tensor product of $\CC^*$ over $E^*$ with the group of fractional ideals of $E$.) Bearing in mind the last line of \S 3, we see that these quantities are the $\aaa^{\pm}\Omega^{\pm}_{f^{(i)}}$, up to divisors of $Nk!$. Now we may apply Lemma \ref{vol}. We have then a factorisation of the left hand side which shows it to be equal to the right hand side, to the extent claimed by the lemma. Note that $\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}$ has an interpretation in terms of integral modular symbols, as in \S 5, and just gets Galois-conjugated when one replaces $f$ by some $f^{(i)}$. \end{proof} \begin{remar} The newform $f=\nf{319k4C}$ is congruent to one of its Galois conjugates modulo~$17$ and $17\mid \frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}$ so the lemma and our computations say nothing about whether or not $17$ divides $\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\aaa^{\pm}\right)$. \end{remar} Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq 49$. Given $f\in \mathcal{S}$, let~$B$ be defined as follows: \begin{enumerate} \item Let $L_1$ be the numerator of the rational number $L(M_f/\QQ,k/2)/\Omega_{M_f/\QQ}$. If $L_1=0$ let $B=1$ and terminate. \item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$. \item Let $L_3$ be the part of $L_2$ that is coprime to $p+1$ for every prime~$p$ such that $p^2\mid N$. \item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic of any prime of congruence between~$f$ and a form of weight~$k$ and lower level. (By congruence here, we mean a congruence for coefficients $a_n$ with $n$ coprime to the level of~$f$.) \item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic of any prime of congruence between~$f$ and an Eisenstein series. (This eliminates residue characteristics of reducible representations.) \item Let $B$ be the part of $L_5$ coprime to the residue characteristic of any prime of congruence between $f$ and any one of its Galois conjugates. \end{enumerate} Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if $\ord_p(B) > 0$ then, according to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) > 0$. We computed~$B$ for every newform in~$\mathcal{S}$. There are many examples in which $L_3$ is large, but~$B$ is not, and this is because of Tamagawa factors. For example, {\bf 39k4C} has $L_3=19$, but $B=1$ because of a $19$-congruence with a form of level~$13$; in this case we must have $19\mid c_{3}(2)$, where $c_{3}(2)$ is as in Section~\ref{sec:bkconj}. See Section~\ref{sec:other_ex} for more details. Also note that in every example~$B$ is a perfect square, which, away from congruence primes, is as predicted by the existence of Flach's generalised Cassels-Tate pairing \cite{Fl2}. (Note that if $A[\lambda]$ is irreducible then the lattice $T_{\lambda}$ is at worst a scalar multiple of its dual, so the pairing shows that the order of the $\lambda$-part of $\Sha$, if finite, is a square.) That our computed value of~$B$ should be a square is not {\it a priori} obvious. For simplicity, we discard residue characteristics instead of primes of rings of integers, so our definition of~$B$ is overly conservative. For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at some prime above~$5$, but the prime of congruences of characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not Eisenstein. The newforms for which $B>1$ are given in Table~\ref{tab:invisforms}. The second column of the table records the degree of the field generated by the Fourier coefficients of~$f$. The third contains~$B$. Let~$W$ be the intersection of the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$. The fourth column contains the odd prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which are exactly the possible primes of congruence between~$f$ and non-conjugate cusp forms of the same weight and level. We place a~$*$ next to the four entries of Table~\ref{tab:invisforms} that also occur in Table~\ref{tab:newforms}. \subsection{Examples in which hypotheses fail}\label{sec:other_ex} We have some other examples where forms of different levels are congruent (for Fourier coefficients of index coprime to the levels). However, Remark~\ref{sign} does not apply, so that one of the forms could have an odd functional equation, and the other could have an even functional equation. For instance, we have a $19$-congruence between the newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier coefficients of index coprime to $39$. Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$ has {\it odd} functional equation. Here~$f$ fails the condition about not being congruent to a form of lower level, so in Lemma~\ref{local1} it is possible that $\ord_{\qq}(c_{3}(2))>0$. In fact this does happen. Because $V'_{\qq}$ (attached to~$g$ of level $13$) is unramified at $p=3$, $H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is two-dimensional. As in (2) of the proof of Theorem~\ref{local}, one of the eigenvalues of $\Frob_p^{-1}$ acting on this two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where $W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that $\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that $\Frob_p^{-1}$ acts as $-w_p$ on the quotient of $H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$. Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is nontrivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local} does not work. This is just as well, since had it worked we would have expected $\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation shows not to be the case. In the following example, the divisibility between the levels is the other way round. There is a $7$-congruence between $g=\nf{122k6A}$ and $f=\nf{61k6B}$, both $L$-functions have even functional equation, and $L(g,3)=0$. In the proof of Theorem~\ref{local}, there is a problem with the local condition at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its kernel is at most one dimensional, so we still get the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having $\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get elements of $\Sha$ for \nf{61k6B} (assuming all along the strong Beilinson-Bloch conjecture). In particular, these elements of $\Sha$ are {\it invisible} at level 61. When the levels are different we are no longer able to apply Theorem 2.1 of \cite{FJ}. However, we still have the congruences of integral modular symbols required to make the proof of Proposition \ref{div} go through. Indeed, as noted above, the congruences of modular forms were found by producing congruences of modular symbols. Despite these congruences of modular symbols, Remark~\ref{sign} does not apply, since there is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels. Finally, there are two examples where we have a form $g$ with even functional equation such that $L(g,k/2)=0$, and a congruent form $f$ which has odd functional equation; these are a 23-congruence between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If $\ord_{s=2}L(f,s)=1$, it ought to be the case that $\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately the appropriate modification of Theorem \ref{local} (with strong Beilinson-Bloch conjecture) does not necessarily provide us with nontrivial $\qq$-torsion in $\Sha$. It only tells us that the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$ appears in the conjectural formula for the first derivative of the complex $L$ function, evaluated at $s=k/2$, but in combination with a regulator that we have no way of calculating. Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions associated with $f$ and $g$ by the construction of Mazur, Tate and Teitelbaum \cite{MTT}, each divided by a suitable canonical period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not quite clear what to make of this. This divisibility may be proved as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably normalised) are congruent $\bmod{\,\qq}$, as a result of the congruence between the modular symbols out of which they are constructed. Integrating an appropriate function against these measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$ to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$, since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case where the signs in the functional equations of $L(g,s)$ and $L_q(g,s)$ are the same, positive in this instance. (According to the proposition in Section 18 of \cite{MTT}, the signs differ precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.) We also found some examples for which the conditions of Theorem~\ref{local} were not met. For example, we have a $7$-congruence between \nf{639k4B} and \nf{639k4H}, but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar problem with a $7$-congruence between \nf{260k6A} and \nf{260k6E} --- here $w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. According to Propositions \ref{div} and \ref{sha}, Bloch-Kato still predicts that the $\qq$-part of $\Sha$ is non-trivial in these examples. Finally, there is a $5$-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime~$5$ is less than the weight~$6$ so Propositions \ref{div} and \ref{sha} (and even Lemma~\ref{lem:lrat}) do not apply. \begin{thebibliography}{AL} \bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on $\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160. \bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of Shafarevich-Tate groups of abelian varieties, preprint. \bibitem[BS-D]{BSD} B. J. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I and II. {\em J. reine angew. Math. }{\bf 212 }(1963), 7--25, {\bf 218 }(1965), 79--108. \bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions, {\em J. reine angew. Math. }{\bf 350 }(1984), 94--108. \bibitem[BCP]{magma} W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}. {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4, 235--265, Computational algebra and number theory (London, 1993). \bibitem[Be]{Be} A. Beilinson, Height pairing between algebraic cycles, {\em in }Current trends in arithmetical algebraic geometry (K. Ribet, ed.) {\em Contemp. Math. }{\bf 67 }(1987), 1--24. \bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift Volume I, 333--400, Progress in Mathematics, 86, Birkh\"auser, Boston, 1990. \bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci. \'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468. \bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em Duke Math. J. }{\bf 59 }(1989), 785--801. \bibitem[CM1]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000), 13--28. \bibitem[CM2]{CM2} J. E. Cremona, B. Mazur, Appendix to A. Agashe, W. Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of rank zero, preprint. \bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of $L_f(s)$ at the center of the critical strip, preprint. \bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations $\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math. {\bf 179}, 139--172, Springer, 1969. \bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979), part 2, 313--346. \bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives of modular forms and the Tamagawa number conjecture, preprint. {{\sf http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}} \bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato conjecture for adjoint motives of modular forms, {\em Math. Res. Lett. }{\bf 8 }(2001), 437--442. \bibitem[Du1]{Du3} N. Dummigan, Symmetric square $L$-functions and Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001), 383--400. \bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494. \bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic Galois representations, {\em in }Algebraic analysis, geometry and number theory (J. Igusa, ed.), 25--80, Johns Hopkins University Press, Baltimore, 1989. \bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66. \bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127. \bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations, S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David, ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel Boston Berlin, 1993. \bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de repr\'esentations $p$-adiques du groupe de Galois d'un corps local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math. }{\bf 115 }(1982), 529--577. \bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249. \bibitem[FL]{FL} J.-M. Fontaine, G. Lafaille, Construction de repr\'esentations $p$-adiques, {\em Ann. Sci. E.N.S. }{\bf 15 }(1982), 547--608. \bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon'' for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989), 51--56. \bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois representations coming from modular forms, {\em J. Number Theory }{\bf 31 }(1989), 133--141. \bibitem[MTT]{MTT} B. Mazur, J. Tate, J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, {\em Invent. Math. }{\bf 84 }(1986), 1--48. \bibitem[Ne]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic heights. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math. Soc., Providence, RI, 2000. \bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms, {\em Invent. Math. }{\bf 100 }(1990), 419--430. \bibitem[SV]{SV} W. A. Stein, H. A. Verrill, Cuspidal modular symbols are transportable, {\em L.M.S. Journal of Computational Mathematics }{\bf 4 }(2001), 170--181. \bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of half-integral weight and a $\Lambda$-adic Shintani lifting. Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math., {\bf 174}, Amer. Math. Soc., Providence, RI, 1994. \bibitem[V]{V} V. Vatsal, Canonical periods and congruence formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419. \bibitem[W]{W} L. C. Washington, Galois cohomology, {\em in }Modular Forms and Fermat's Last Theorem, (G. Cornell, J. H. Silverman, G. Stevens, eds.),101--120, Springer-Verlag, New York, 1997. \bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series, {\em Invent. Math. }{\bf 130 }(1997), 99--152. \end{thebibliography} \end{document}