%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1%2% motive_visibility.tex3%4% 25 August 20025%6% Project of William Stein, Neil Dummigan, Mark Watkins7%8%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%910\documentclass{amsart}11\usepackage{amssymb}12\usepackage{amsmath}13\usepackage{amscd}1415\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}1617\newtheorem{prop}{Proposition}[section]18\newtheorem{defi}[prop]{Definition}19\newtheorem{conj}[prop]{Conjecture}20\newtheorem{lem}[prop]{Lemma}21\newtheorem{thm}[prop]{Theorem}22\newtheorem{cor}[prop]{Corollary}23\newtheorem{examp}[prop]{Example}24\newtheorem{remar}[prop]{Remark}25\def\id{\mathop{\mathrm{ id}}\nolimits}26\DeclareMathOperator{\Ker}{\mathrm {Ker}}27\DeclareMathOperator{\Aut}{{\mathrm {Aut}}}28\renewcommand{\Im}{{\mathrm {Im}}}29\DeclareMathOperator{\ord}{ord}30\DeclareMathOperator{\End}{End}31\DeclareMathOperator{\Hom}{Hom}32\DeclareMathOperator{\Mor}{Mor}33\DeclareMathOperator{\Norm}{Norm}34\DeclareMathOperator{\Nm}{Nm}35\DeclareMathOperator{\tr}{tr}36\DeclareMathOperator{\Tor}{Tor}37\DeclareMathOperator{\Sym}{Sym}38\DeclareMathOperator{\Hol}{Hol}39\DeclareMathOperator{\vol}{vol}40\DeclareMathOperator{\tors}{tors}41\DeclareMathOperator{\cris}{cris}42\DeclareMathOperator{\length}{length}43\DeclareMathOperator{\dR}{dR}44\DeclareMathOperator{\lcm}{lcm}45\DeclareMathOperator{\Frob}{Frob}46\def\rank{\mathop{\mathrm{ rank}}\nolimits}47\newcommand{\Gal}{\mathrm {Gal}}48\newcommand{\Spec}{{\mathrm {Spec}}}49\newcommand{\Ext}{{\mathrm {Ext}}}50\newcommand{\res}{{\mathrm {res}}}51\newcommand{\Cor}{{\mathrm {Cor}}}52\newcommand{\AAA}{{\mathbb A}}53\newcommand{\CC}{{\mathbb C}}54\newcommand{\RR}{{\mathbb R}}55\newcommand{\QQ}{{\mathbb Q}}56\newcommand{\ZZ}{{\mathbb Z}}57\newcommand{\NN}{{\mathbb N}}58\newcommand{\EE}{{\mathbb E}}59\newcommand{\TT}{{\mathbb T}}60\newcommand{\HHH}{{\mathbb H}}61\newcommand{\pp}{{\mathfrak p}}62\newcommand{\qq}{{\mathfrak q}}63\newcommand{\FF}{{\mathbb F}}64\newcommand{\KK}{{\mathbb K}}65\newcommand{\GL}{\mathrm {GL}}66\newcommand{\SL}{\mathrm {SL}}67\newcommand{\Sp}{\mathrm {Sp}}68\newcommand{\Br}{\mathrm {Br}}69\newcommand{\Qbar}{\overline{\mathbb Q}}70\newcommand{\Xbar}{\overline{X}}71\newcommand{\Ebar}{\overline{E}}72\newcommand{\sbar}{\overline{s}}73\newcommand{\nf}[1]{\mbox{\bf #1}}74\newcommand{\fbar}{\overline{f}}7576% ---- SHA ----77\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts78\newcommand{\textcyr}[1]{%79{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%80\selectfont #1}}81\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}8283\newcommand{\HH}{{\mathfrak H}}84\newcommand{\aaa}{{\mathfrak a}}85\newcommand{\bb}{{\mathfrak b}}86\newcommand{\dd}{{\mathfrak d}}87\newcommand{\ee}{{\mathbf e}}88\newcommand{\Fbar}{\overline{F}}89\newcommand{\CH}{\mathrm {CH}}9091\begin{document}92\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}93\author{Neil Dummigan}94\author{William Stein}95\author{Mark Watkins}96\date{August 24th, 2002}97\subjclass{11F33, 11F67, 11G40.}9899\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,100Shafarevich-Tate group.}101102\address{University of Sheffield\\ Department of Pure103Mathematics\\104Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\105U.K.}106\address{Harvard University\\Department of Mathematics\\107One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}108\address{Penn State Mathematics Department\\109University Park\\State College, PA 16802\\ U.S.A.}110111\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}112\email{watkins@math.psu.edu}113114115\begin{abstract}116117We study Shafarevich-Tate groups of motives attached to modular forms118on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a criterion for119the existence of nontrivial elements of these Shafarevich-Tate groups,120and give $16$ examples in which the Beilinson-Bloch conjecture implies121the existence of such elements. We also use modular symbols and122observations about Tamagawa numbers to compute nontrivial conjectural123lower bounds on the orders of the Shafarevich-Tate groups of modular124motives of low level and weight at most $12$. Our methods build upon125Mazur's idea of visibility, but in the context of motives instead of126abelian varieties.127\end{abstract}128129\maketitle130131\section{Introduction}132Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$133be the associated $L$-function. The conjecture of Birch and134Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$135at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and136also gives an interpretation of the leading term in the Taylor137expansion in terms of various quantities, including the order of138the Shafarevich-Tate group of $E$.139140Cremona and Mazur \cite{CM} look, among all strong Weil elliptic141curves over $\QQ$ of conductor $N\leq 5500$, at those with142non-trivial Shafarevich-Tate group (according to the Birch and143Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate144group has predicted elements of prime order $m$. In most cases145they find another elliptic curve, often of the same conductor,146whose $m$-torsion is Galois-isomorphic to that of the first one,147and which has positive rank. The rational points on the second elliptic148curve produce classes in the common $H^1(\QQ,E[m])$. They show149\cite{CM2} that these lie in the Shafarevich-Tate group of the150first curve, so rational points on one curve explain elements of151the Shafarevich-Tate group of the other curve.152153The Bloch-Kato conjecture \cite{BK} is the generalisation to154arbitrary motives of the leading term part of the Birch and155Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture156\cite{B} generalises the part about the order of vanishing at the157central point, identifying it with the rank of a certain Chow158group.159160This paper is a partial generalisation of \cite{CM} and \cite{AS}161from abelian varieties over $\QQ$ associated to modular forms of162weight~$2$ to the motives attached to modular forms of higher weight.163It also does for congruences between modular forms of equal weight164what \cite{Du2} did for congruences between modular forms of different165weights.166167We consider the situation where two newforms~$f$ and~$g$, both of168even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal169$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but170$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish171to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},172we give sixteen173examples (all with $k=4$ and $k=6$), and in each $\qq$ divides the174numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where175$\vol_{\infty}$ is a certain canonical period.176177In fact, we show how this divisibility may be deduced from the178vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The179point is, the congruence between$f$ and~$g$ leads to a congruence180between suitable ``algebraic parts'' of the special values181$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity182one result of Faltings and Jordan shows that the congruence of183Fourier expansions leads to a congruence of certain associated184cohomology classes. These are then identified with the modular185symbols which give rise to the algebraic parts of special values.186If $L(g,k/2)$ vanishes then the congruence implies that187$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.188189The Bloch-Kato conjecture sometimes then implies that the190Shafarevich-Tate group attached to~$f$ has nonzero $\qq$-torsion.191Under certain hypotheses and assumptions, the most substantial of192which is the Beilinson-Bloch conjecture relating the vanishing of193$L(g,k/2)$ to the existence of algebraic cycles, we are able to194construct some of the predicted elements of~$\Sha$ using the195Galois-theoretic interpretation of the congruences to transfer196elements from a Selmer group for~$g$ to a Selmer group for~$f$.197One might say that algebraic cycles for one motive explain198elements of~$\Sha$ for the other, or that we use congruences to199link the Beilinson-Bloch conjecture for one motive with the200Bloch-Kato conjecture for the other.201%In proving the local202%conditions at primes dividing the level, and also in examining the203%local Tamagawa factors at these primes, we make use of a higher weight204%level-lowering result due to Jordan and Livn\'e \cite{JL}.205206We also compute data which, assuming the Bloch-Kato conjecture,207provides lower bounds for the orders of numerous Shafarevich-Tate208groups (see Section~\ref{sec:invis}).209%Our data is consistent210%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the211%congruence modulus is necessarily a perfect square (assuming that~$\Sha$212%is finite).213214\section{Motives and Galois representations}215This section and the next provide definitions of some of the216quantities appearing later in the Bloch-Kato conjecture. Let217$f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for218$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,219which is necessarily totally real. Let~$\lambda$ be any finite220prime of~$E$, and let~$\ell$ denote its residue characteristic. A221theorem of Deligne \cite{De1} implies the existence of a222two-dimensional vector space $V_{\lambda}$ over $E_{\lambda}$, and223a continuous representation224$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$225such that226\begin{enumerate}227\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing228$lN$, and229\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the230characteristic polynomial of $\Frob_p^{-1}$ acting on231$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.232\end{enumerate}233234Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as235the $\lambda$-adic realisation of a Grothendieck motive $M_f$.236There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,237both $2$-dimensional $E$-vector spaces. For details of the238construction see \cite{Sc}. The de Rham realisation has a Hodge239filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset240F^k=\{0\}$. The Betti realisation $V_B$ comes from singular241cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic242cohomology.243For each prime $\lambda$, there is a natural isomorphism244$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a245$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside246each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.247Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.248There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),249which amounts to multiplying the action of $\Frob_p$ by $p^j$.250251Following \cite{BK} (Section 3), for $p\neq l$ (including252$p=\infty$) let253$$254H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow255H^1(I_p,V_{\lambda}(j))).256$$257The subscript~$f$ stands for ``finite258part'', $D_p$ is a decomposition subgroup at a prime above~$p$,259$I_p$ is the inertia subgroup, and the cohomology is for260continuous cocycles and coboundaries. For $p=l$ let261$$262H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow263H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))264$$265(see Section 1 of266\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and267$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of268elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie269in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.270271There is a natural exact sequence272$$273\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.274$$275Let276$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.277Define the $\lambda$-Selmer group278$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of279$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in280$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the281condition at $p=\infty$ is superfluous unless $l=2$. Define the282Shafarevich-Tate group283$$284\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/285\pi_*H^1_f(\QQ,V_{\lambda}(j)).286$$287Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any288prime ideal~$\lambda$ is the length of the $\lambda$-component of289$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,290and write~$\Sha$ for~$\Sha(j)$.291292Define the group of global torsion points293$$294\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).295$$296This is analogous to the group of rational torsion points on an297elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in298which the exponent of any prime ideal~$\lambda$ is the length of299the $\lambda$-component of $\Gamma_{\QQ}$.300301\section{Canonical periods}302We assume from now on for convenience that $N\geq 3$. We need to303choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti304and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this305in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$306agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$307and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in308\cite{DFG} using cohomology, with non-constant coefficients, of309modular curves. (In \cite{DFG}, see especially Sections 2.2 and3105.4, and the paragraph preceding Lemma 2.3.)311312For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$313module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of314$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes315E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$ module316$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.317318Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising319generalised elliptic curves with full level-$N$ structure. Let320$\mathfrak{E}$ be the universal generalised elliptic curve over321$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product322of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed323using a projector on the cohomology of a desingularisation of324$\mathfrak{E}^{k-2}$. Realising $M(N)(\CC)$ as the disjoint union325of $\phi(N)$ copies of the quotient326$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the327completed upper half plane), and letting $\tau$ be a variable on328$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to329the elliptic curve with period lattice generated by $1$ and330$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on331the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.332Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$333is a well-defined differential form on (a desingularisation of)334$\mathfrak{E}^{k-2}$ and naturally represents a generating element335of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at336primes dividing $Nk!$ so that this is the case.) We shall call337this element $e(f)$.338339Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and340$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is341a natural action of complex conjugation on $V_B$, breaking it up342into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.343Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$344to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let345$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These346are rank one $O_E$-modules, but not necessarily free, since the347class number of $O_E$ may be greater than one. Choose nonzero348elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be349the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers350$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.351352\section{The Bloch-Kato conjecture}\label{sec:bkconj}353In this section we extract from the Bloch-Kato conjecture for354$L(f,k/2)$ a prediction about the order of the Shafarevich-Tate355group, by analysing the other terms in the formula.356357Let $L(f,s)$ be the $L$-function attached to~$f$. For358$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with359Euler product360$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but361there is an analytic continuation given by an integral, as362described in the next section. Suppose that $L(f,k/2)\neq 0$. The363Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the364following equality of fractional ideals of~$E$:365$$366\frac{L(f,k/2)}{\vol_{\infty}}=367\left(\prod_pc_p(k/2)\right)368\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.369$$370(Strictly speaking, the conjecture in \cite{BK}371is only given for $E=\QQ$.) Here, $\pm$372represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal373to $(2\pi i)^{k/2}$ multiplied by the determinant of the374isomorphism $V_B^{\pm}\otimes\CC\simeq375(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the376lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For377$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be378\begin{align*}379\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-380\ord_{\lambda}(P_p(p^{-j}))\\381=&\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).382\end{align*}383384We omit the definition of $\ord_{\lambda}(c_p(j))$ for385$\lambda\mid p$, which requires one to assume Fontaine's de Rham386conjecture (\cite{Fo}, Appendix A6), and depends on the choices of387$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be388concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$389is a prime of good reduction. For such primes, the de Rham390conjecture follows from Theorem 5.6 of \cite{Fa1}.)391392\begin{lem}\label{vol}393$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and394$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.395\end{lem}396\begin{proof}397$\vol_{\infty}$ is also equal to the determinant of the period map398from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with399respect to lattices dual to those we used above in the definition400of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}).401We are using here natural pairings. Recall that the index of402$O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then403the proof is completed by noting that, locally away from primes404dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the405index of $T_B$ in its dual, both being equal to the ideal406denoted~$\eta$ in \cite{DFG2}.407\end{proof}408\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.409Then the fractional ideal $c_p(j)$ is supported at most on410divisors of~$p$.411\end{lem}412\begin{proof}413As on p.~30 of \cite{Fl1}, for odd $l\neq p$,414$\ord_{\lambda}(c_p(j))$ is the length of the finite415$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$416where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a417trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is418torsion free.419\end{proof}420421\begin{lem}\label{local1}422Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$423is an irreducible representation of $\Gal(\Qbar/\QQ)$, where424$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose425that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent426modulo $\qq$ to any newform of weight~$k$, trivial character and427level dividing $N/p$. Then any integer~$j$,428$\ord_{\qq}(c_p(j))=0$.429\end{lem}430\begin{proof}431It suffices to show that432$$433\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),434$$435since this ensures that436$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that437$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If438the dimensions differ then, given that $f$ is not congruent modulo439$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of440\cite{L} shows that we are in the situation covered by one of the441three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv442-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is443unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's444result that $N$ is the prime-to-$q$ part of the conductor of445$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses446the condition $q>k$) implies the existence of a newform of weight447$k$, trivial character and level dividing $N/p$, congruent to~$g$448modulo $\qq$. This contradicts our hypotheses.449\end{proof}450451\begin{remar}452For an example of what can be done when~$f$ is congruent to453a form of lower level, see the first example in Section~\ref{sec:other_ex}454below.455\end{remar}456457\begin{lem}\label{at q}458If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then459$\ord_{\qq}(c_q)=0$.460\end{lem}461\begin{proof}462It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an463application, at the end of Section~2.2, of the results of464\cite{Fa1}) that $T_{\qq}$ is the465$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered466module $T_{\dR}\otimes O_{\qq}$ by the functor they call467$\mathbb{V}$. (This property is part of the definition of an468$S$-integral premotivic structure given in Section~1.2 of469\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)470of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used471in Theorem~4.1 of \cite{BK} follows from the first paragraph of4722(h) of \cite{Fa1}.)473\end{proof}474475\begin{lem}476If $A[\lambda]$ is an477irreducible representation of $\Gal(\Qbar/\QQ)$,478then479$$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0.$$480\end{lem}481This follows trivially from the definition.482483Putting together the above lemmas we arrive at the following:484\begin{prop}\label{sha}485Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$486is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.487Assume the same hypotheses as in Lemma \ref{local1},488for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$489are as in the previous section. If490$$491\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0492$$493(with numerator nonzero) then the Bloch-Kato conjecture494predicts that495$$496\ord_{\qq}(\#\Sha)>0.497$$498\end{prop}499500\section{Congruences of special values}501Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal502weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field503large enough to contain all the coefficients $a_n$ and $b_n$.504Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv505g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$, and assume506that $q\nmid N\phi(N)k!$.507Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that508$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates509$T_B^{\pm}$ locally at $\qq$.510Make two further assumptions:511$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$512513\begin{prop} \label{div}514With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.515\end{prop}516\begin{proof} This is based on some of the ideas used in Section 1 of517\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},518which presumably should refer to ``Condition 2''. Since519$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that520$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm5211=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that522$$523\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).524$$525Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period526$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$527where the integral is taken along the positive imaginary axis,528then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$529Thus we are reduced530to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.531532Let $\mathcal{D}_0$ be the group of divisors of degree zero533supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and534integer $r\geq 0$, let $P_r(R)$ be the additive group of535homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these536groups have a natural action of $\Gamma_1(N)$. Let537$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$538be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.539540Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with541the argument in 1.7 of \cite{V}, the cohomology class542$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in543S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an544element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_{E,\qq})$. We are545now dealing with cohomology over $X_1(N)$ rather than $M(N)$,546which is why we insist that $q\nmid \phi(N)$. It follows from the547last line of Section~4.2 of \cite{St} that, up to some small548factorials which do not matter locally at $\qq$,549$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv550(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since551$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that552$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv553(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The554coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to555show is divisible by $\qq$.556Similarly557$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv558(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of559$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.560Therefore it would suffice to show that, for some $\mu\in O_E$,561the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by562$\qq$ in $S_{\Gamma_1(N)}(k,O_{E,\qq})$. It suffices to show that,563for some $\mu\in O_E$, the element564$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,565considered as an element of $\qq$-adic cohomology of $X_1(N)$ with566non-constant coefficients. This would be the case if567$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same568one-dimensional subspace upon reduction $\pmod{\qq}$. But this is569a consequence of Theorem 2.1(1) of \cite{FJ}.570\end{proof}571\begin{remar}\label{sign}572The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are573equal. They are determined by the eigenvalue of the involution $W_N$,574which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and575$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue576characteristic coprime to $2N$. The common sign in the functional577equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of578$W_N$ acting on~$f$ and~$g$.579\end{remar}580581This is analogous to the remark at the end of Section~3 of \cite{CM},582which shows that if~$\qq$ has odd residue characteristic and583$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order584at least two at $s=k/2$. Note that Maeda's conjecture585implies that there are no examples of~$g$ of586level one with positive sign in their functional equation such that587$L(g,k/2)=0$ (see \cite{CF}).588589\section{Constructing elements of the Shafarevich-Tate group}590Let~$f$ and~$g$ be as in the first paragraph of the previous591section. In the previous section we showed how the congruence592between $f$ and $g$ relates the vanishing of $L(g,k/2)$ to the593divisibility by $\qq$ of an ``algebraic part'' of $L(f,k/2)$.594Conjecturally the former is associated with the existence of595certain algebraic cycles (for $M_g$) while the latter is596associated with the existence of certain elements of the597Shafarevich-Tate group (for $M_f$, as we saw in \S 4). In this598section we show how the congruence, interpreted in terms of Galois599representations, provides a direct link between algebraic cycles600and the Shafarevich-Tate group.601602For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and603$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and604$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$605is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from606the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if607irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.608609Suppose that $L(g,k/2)=0$. If the sign in the functional equation610is positive (as it must be if $L(f,k/2)\neq 0$, see Remark611\ref{sign}), this implies that the order of vanishing of $L(g,s)$612at $s=k/2$ is at least $2$. According to the Beilinson-Bloch613conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$614is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational615rational equivalence classes of null-homologous,616algebraic cycles of codimension $k/2$617on the motive $M_g$. (This generalises the part618of the Birch--Swinnerton-Dyer conjecture which says that for an619elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at620$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)621622Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps623to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the624subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.625If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we626get (assuming also the Beilinson-Bloch conjecture) a subspace of627$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of628vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply629conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is630equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would631follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in632Sections~1 and~6.5 of \cite{Fo2}.633634Similarly, if $L(f,k/2)\neq 0$ then we expect that635$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$636coincides with the $\qq$-part of $\Sha$.637\begin{thm}\label{local}638Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that639$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that640$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and641that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a642newform of weight~$k$, trivial character and level dividing $N/p$.643Suppose that, for all primes $p\mid N$, $\,p\not\equiv644-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here645$w_p$ is the common eigenvalue of the Atkin-Lehner involution646$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of647$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.648\end{thm}649650\begin{proof}651Take a nonzero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By652continuity and rescaling we may assume that it lies in653$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq654H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that655$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a656nonzero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq657H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$658is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so659$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and660we get a nonzero, $\qq$-torsion class $\gamma\in661H^1(\QQ,A_{\qq}(k/2))$.662663Our aim is to show that $\res_p(\gamma)\in664H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We665consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.666667\begin{enumerate}668\item {\bf $p\nmid qN$. }669670Consider the $I_p$-cohomology of the short exact sequence671$$672\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},673$$674where~$\pi$ is multiplication by a uniformising element of675$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at676$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is677$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,678remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into679$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in680H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in681$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence682that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq683H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$684to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of685\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just686contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$687to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that688$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.689690\item {\bf $p\mid N$. }691692First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.693It suffices to show that694$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$695since then the natural map from696$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is697surjective; this may be done as in the proof of Lemma698\ref{local1}. It follows as above that the image of $c\in699H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then700$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by701inflation-restriction. The order of this group is the same as the702order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is703trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that704$V_{\qq}(k/2)$ is ramified at $p$, so $\dim705H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim706H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only707consider the case where this common dimension is $1$. The708(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha709p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$710on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It711follows from Theor\'eme A of \cite{Ca1} that this is the same as712the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and713Lehner \cite{AL}, it then follows that $p^2\nmid N$ and714$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that715$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on716$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as717$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that718$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so719$\res_p(\gamma)=0$ and certainly lies in720$H^1_f(\QQ_p,A_{\qq}(k/2))$.721722\item {\bf $p=q$. }723724Since $q\nmid N$ is a prime of good reduction for the motive725$M_g$, $\,V'_{\qq}$ is a crystalline representation of726$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and727$V'_{\qq}$ have the same dimension, where728$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}729B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)730As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is731the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the732filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we733may now prove, in the same manner as Proposition 9.2 of734\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.735\end{enumerate}736\end{proof}737738Theorem~2.7 of \cite{AS} is concerned with verifying local739conditions in the case $k=2$, where~$f$ and~$g$ are associated740with abelian varieties~$A$ and~$B$. (Their theorem also applies to741abelian varieties over number fields.) Our restriction outlawing742congruences modulo $\qq$ with cusp forms of lower level is743analogous to theirs forbidding~$q$ from dividing Tamagawa factors744$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic745curve with $\ord_l(j(A))<0$, consideration of a Tate746parametrisation shows that if $q\mid c_{A,l}$, i.e., if747$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified748at~$l$.)749750In this paper we have encountered two technical problems which we751dealt with in quite similar ways:752\begin{enumerate}753\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;754\item proving local conditions at primes $p\mid N$, for an element755of $\qq$-torsion.756\end{enumerate}757If our only interest was in testing the Bloch-Kato conjecture at758$\qq$, we could have made these problems cancel out, as in Lemma7598.11 of \cite{DFG}, by weakening the local conditions. However, we760have chosen not to do so, since we are also interested in the761Shafarevich-Tate group, and since the hypotheses we had to assume762are not particularly strong. Note that, since $A[\qq]$ is763irreducible, the $\qq$-part of $\Sha$ does not depend on the764choice of $T_{\qq}$.765766\section{Examples and Experiments}767\label{sec:examples} This section contains tables and numerical768examples that illustrate the main themes of this paper. In769Section~\ref{sec:vistable}, we explain Table~\ref{tab:newforms},770which contains~$16$ examples of pairs $f,g$ such that the771Beilinson-Bloch conjecture and Theorem~\ref{local} together imply772the existence of nontrivial elements of the Shafarevich-Tate group773of the motive attached to~$f$. Section~\ref{sec:howdone} outlines774the higher-weight modular symbol computations that were used in775making Table~\ref{tab:newforms}. Section~\ref{sec:invis} discusses776Table~\ref{tab:invisforms}, which summarizes the results of an777extensive computation of conjectural orders of Shafarevich-Tate778groups for modular motives of low level and weight.779Section~\ref{sec:other_ex} gives specific examples in which780various hypotheses fail. Note that in this section ``modular781symbol'' has a different meaning from in \S 5, being related to782homology rather than cohomology. For precise definitions see783\cite{SV}.784785\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}786\begin{table}787\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}788789$$790\begin{array}{|c|c|c|c|c|}\hline791g & \deg(g) & f & \deg(f) & q's \\\hline792\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\793\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\794\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\795\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\796\vspace{-2ex} & & & & \\797\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\798\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\799\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\800\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\801\vspace{-2ex} & & & & \\802\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\803\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\804\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\805\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\806\vspace{-2ex} & & & & \\807\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\808\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\809\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\810\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\811\hline812\end{array}813$$814\end{table}815816817Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists818sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)819along with at least one prime~$q$ such that there is a prime820$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,821$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.822The notation is as follows.823The first column contains a label whose structure is824\begin{center}825{\bf [Level]k[Weight][GaloisOrbit]}826\end{center}827This label determines a newform $g=\sum a_n q^n$, up to Galois828conjugacy. For example, \nf{127k4C} denotes a newform in the third829Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois830orbits are ordered first by the degree of $\QQ(\ldots, a_n,831\ldots)$, then by the sequence of absolute values $|\mbox{\rm832Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace833being first in the event that the two absolute values are equal,834and the first Galois orbit is denoted {\bf A}, the second {\bf B},835and so on. The second column contains the degree of the field836$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns837contain~$f$ and its degree, respectively. The fifth column838contains at least one prime~$q$ such that there is a prime839$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the840hypotheses of Theorem~\ref{local} (except possibly $r>0$) are841satisfied for~$f$,~$g$, and~$\qq$.842843For the two examples \nf{581k4E} and \nf{684k4K}, the square of a844prime $q$ appears in the $q$-column, meaning $q^2$ divides the845order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined846at the end of 7.3 below.847848849We describe the first line of Table~\ref{tab:newforms}850in more detail. See the next section for further details851on how the computations were performed.852853Using modular symbols, we find that there is a newform854$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots855\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,856the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We857also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier858coefficients generate a number field~$K$ of degree~$17$, and by859computing the image of the modular symbol $XY\{0,\infty\}$ under860the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$861and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue862characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are863both equal to864$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7865+ \cdots\in \FF_{43}[[q]].$$866867There is no form in the Eisenstein subspaces of868$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with869$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so870$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is871prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a872level~$1$ form of weight~$4$. Thus we have checked the hypotheses873of Theorem~\ref{local}, so if $r$ is the dimension of874$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of875$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.876877Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that878$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the879$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to880the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,881we have constructed the $\qq$-torsion in $\Sha$ predicted by the882Bloch-Kato conjecture.883884For particular examples of elliptic curves one can often find and885write down rational points predicted by the Birch and886Swinnerton-Dyer conjecture. It would be nice if likewise one could887explicitly produce algebraic cycles predicted by the888Beilinson-Bloch conjecture in the above examples. Since889$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary8900.3.2 of \cite{Z}), so ought to be trivial in891$\CH_0^{k/2}(M_g)\otimes\QQ$.892893\subsection{How the computation was performed}\label{sec:howdone}894We give a brief summary of how the computation was performed. The895algorithms that we used were implemented by the second author, and896most are a standard part of MAGMA (see \cite{magma}).897898Let~$g$,~$f$, and~$q$ be some data from a line of899Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We900verified the existence of a congruence modulo~$q$, that901$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq9020$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does903not arise from any $S_k(\Gamma_0(N/p))$, as follows:904905To prove there is a congruence, we showed that the corresponding906{\em integral} spaces of modular symbols satisfy an appropriate907congruence, which forces the existence of a congruence on the908level of Fourier expansions. We showed that $\rho_{g,\qq}$ is909irreducible by computing a set that contains all possible residue910characteristics of congruences between~$g$ and any Eisenstein911series of level dividing~$N$, where by congruence, we mean a912congruence for all Fourier coefficients of index~$n$ with913$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any914form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by915listing a basis of such~$h$ and finding the possible congruences,916where again we disregard the Fourier coefficients of index not917coprime to~$N$.918919To verify that $L(g,\frac{k}{2})=0$, we computed the image of the920modular symbol ${\mathbf921e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map922with the same kernel as the period mapping, and found that the923image was~$0$. The period mapping sends the modular924symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,925so that ${\mathbf e}$ maps to~$0$ implies that926$L(g,\frac{k}{2})=0$. In a similar way, we verified that927$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)928=(-1)^{k/2} g$ which, because of the functional equation, implies929that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of930independent interest because it includes examples of modular forms931of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by932the functional equation. We found no such examples of weights933$\geq 8$.934935\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}936In this section we apply some of the results of937Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders938of Shafarevich-Tate groups of many modular motives. The results of939this section suggest that~$\Sha$ of a modular motive is usually not940``visible at level~$N$'', i.e., explained by congruences at level~$N$,941which agrees with the observations of \cite{CM} and \cite{AS}. For942example, when $k>6$ we find many examples of conjecturally943nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.944945For any newform~$f$, let $L(M_f/\QQ,s) = \prod_{i=1}^{d}946L(f^{(i)},s)$ where $f^{(i)}$ runs over the947$\Gal(\Qbar/\QQ)$-conjugates of~$f$. Let~$T$ be the complex torus948$\CC^d/\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by949integrating integral cuspidal modular symbols against the950conjugates of~$f$. Let $\Omega_{M_f/\QQ}$ denote the volume of951the $-1$ eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for952complex conjugation on~$T$.953954955{\begin{table}956\vspace{-2ex}957\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}958\vspace{-4ex}959960$$961\begin{array}{|c|c|c|c|}\hline962f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline963\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\964\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\965\nf{263k4B} & 39 & 41^{2} & 263 \\966\nf{269k4C} & 39 & 23^{2} & 269 \\967\nf{271k4B} & 39 & 29^{2} & 271 \\968\nf{281k4B} & 40 & 29^{2} & 281 \\969\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\970\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\971\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\972\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\973\hline974\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\975\nf{101k6B} & 24 & 17^{2} & 101 \\976\nf{103k6B} & 24 & 23^{2} & 103 \\977\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\978\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\979\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\980\nf{157k6B} & 34 & 251^{2} & 157 \\981\nf{167k6B} & 40 & 41^{2} & 167 \\982\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\983\nf{173k6B} & 39 & 71^{2} & 173 \\984\nf{181k6B} & 40 & 107^{2} & 181 \\985\nf{191k6B} & 46 & 85091^{2} & 191 \\986\nf{193k6B} & 41 & 31^{2} & 193 \\987\nf{199k6B} & 46 & 200329^2 & 199 \\988\hline989\nf{47k8B} & 16 & 19^{2} & 47 \\990\nf{59k8B} & 20 & 29^{2} & 59 \\991\nf{67k8B} & 20 & 29^{2} & 67 \\992\nf{71k8B} & 24 & 379^{2} & 71 \\993\nf{73k8B} & 22 & 197^{2} & 73 \\994\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\995\nf{79k8B} & 25 & 307^{2} & 79 \\996\nf{83k8B} & 27 & 1019^{2} & 83 \\997\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\998\nf{89k8B} & 29 & 44491^{2} & 89 \\999\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\1000\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\1001\nf{103k8B} & 32 & 75367^{2} & 103 \\1002\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\1003\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\1004\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\1005\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\1006\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\1007\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\1008\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\1009\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\1010\hline1011\end{array}1012$$1013\end{table}1014\begin{table}1015$$1016\begin{array}{|c|c|c|c|}\hline1017f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline1018\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\1019\nf{121k8G} & 12 & 13^{2} & 3, 11 \\1020\nf{121k8H} & 12 & 19^{2} & 5, 11 \\1021\nf{125k8D} & 16 & 179^{2} & 5 \\1022\nf{127k8B} & 39 & 59^{2} & 127 \\1023\nf{128k8F} & 4 & 11^{2} & 1 \\1024\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\1025\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\1026\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\1027\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\1028\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\1029\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\1030\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397,1031639259, 17440535103297287 \\1033\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241,1034769, 365831035\\1036\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577,1037518737, 98371451038699 \\1039\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\1040\nf{148k8B} & 11 & 19^{2} & 3, 37 \\1041\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\10421043\hline10441045\nf{43k10B} & 17 & 449^{2} & 43 \\1046\nf{47k10B} & 20 & 2213^{2} & 47 \\1047\nf{53k10B} & 21 & 673^{2} & 53 \\1048\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\1049\nf{59k10B} & 25 & 37^{2} & 59 \\1050\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\1051\nf{64k10K} & 2 & 19^{2} & 3 \\1052\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\1053\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\1054\nf{71k10B} & 30 & 1103^{2} & 71 \\10551056\hline1057\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\1058\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\1059\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\1060\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\1061\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\1062\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\1063\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\1064\nf{49k12H} & 12 & 271^{2} & 7 \\1065\hline1066\end{array}1067$$1068\end{table}10691070The following lemma may be proved by tensoring the space of1071integral modular symbols with $O_E$, then decomposing it in such a1072way that $\Omega_{M_f/\QQ}$ becomes the determinant of a diagonal1073matrix with $d$ non-zero entries. We omit the somewhat awkward1074details.1075\begin{lem}\label{lem:lrat}1076If $p\nmid Nk!$ is a non-congruence prime for $f$ then the1077$p$-parts of1078$$1079\frac{L(M_f/\QQ,k/2)}{\Omega_{M_f/\QQ}}\qquad\text{and}\qquad1080\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right)1081$$1082are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.1083\end{lem}1084For the rest of this section, {\em we officially assume the1085Bloch-Kato conjecture.}10861087Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$1088satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or1089$k=8$ and $N\leq 133$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq109049$. Given $f\in \mathcal{S}$, let~$B$ be the lower bound on $\#\Sha$ defined1091as follows:1092\begin{enumerate}1093\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.1094If $L_1=0$ let $B=1$ and terminate.1095\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.1096\item Let $L_3$ be the part of $L_2$ that is coprime to1097$p+1$ for every prime~$p$ such that $p^2\mid N$.1098\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic1099of any prime of1100congruence between~$f$ and a form of weight~$k$ and1101lower level. (By congruence here, we mean a congruence for coefficients1102$a_n$ with $n$ coprime to the level of~$f$.)1103\item Let $B$ be the part of $L_4$ coprime to the residue characteristic1104of any prime of congruence1105between~$f$ and an Eisenstein series. (This eliminates1106residue characteristics of reducible representations.)1107\end{enumerate}1108Proposition~\ref{sha} and Lemma~\ref{lem:lrat} imply that if1109$\ord_p(B)1110> 0$, and $p$ is not a congruence prime for $f$, then1111$\ord_p(\#\Sha) > 0$. We have left the congruence primes in $B$ in1112the starred examples since the squares are still suggestive.11131114We computed~$B$ for every newform in~$\mathcal{S}$. There are1115many examples in which $L_3$ is large, but~$B$ is not, and this is1116because of Tamagawa factors. For example, {\bf 39k4C} has1117$L_3=19$, but $B=1$ because of a $19$-congruence with a form of1118level~$13$; in this case we must have $19\mid c_{13}(2)$, where1119$c_{13}(2)$ is as in Section~\ref{sec:bkconj}. See1120Section~\ref{sec:other_ex} for more details. Also note that in1121every example~$B$ is a perfect square, which is consistent with1122the fact \cite{Fl2} that the order of $\Sha$ (if finite) is1123necessarily a perfect square. That our computed value of~$B$1124should be a square is not a priori obvious.11251126For simplicity, we discard residue characteristics instead of primes1127of rings of integers, so our definition of~$B$ is overly conservative.1128For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not1129in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at1130some prime above~$5$, but the prime of congruences of1131characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not1132Eisenstein.113311341135The newforms for which $B>1$ are given in1136Table~\ref{tab:invisforms}. The second column of the table records1137the degree of the field generated by the Fourier coefficients1138of~$f$. The third contains~$B$. Let~$W$ be the intersection of1139the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and1140$W^{\perp}$ the Petersson orthogonal complement of~$W$ in1141$S_k(\Gamma_0(N),\ZZ)$. Then the fourth column contains the odd1142prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which1143are exactly the possible primes of congruence for~$f$. We place a1144$*$ next to the four entries of Table~\ref{tab:invisforms} that1145also occur in Table~\ref{tab:newforms}.11461147\subsection{Examples in which hypotheses fail}\label{sec:other_ex}1148We have some other examples where forms of1149different levels are congruent.1150However, Remark~\ref{sign} does not1151apply, so that one of the forms could have an odd functional1152equation, and the other could have an even functional equation.1153For instance, we have a $19$-congruence between the1154newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier1155coefficients coprime to $39$.1156Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$1157has {\it odd} functional equation.1158Here~$f$ fails the condition about not being congruent1159to a form of lower level, so in Lemma~\ref{local1} it is possible that1160$\ord_{\qq}(c_{19}(2))>0$. In fact this does happen. Because1161$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=19$,1162$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is1163two-dimensional. As in (2) of the proof of Theorem~\ref{local},1164one of the eigenvalues of $\Frob_p^{-1}$ acting on this1165two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where1166$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that1167$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that1168$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of1169$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.1170Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in1171our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is1172non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}1173does not work. This is just as well, since had it1174worked we would have expected1175$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation1176shows not to be the case.11771178In the following example, the divisibility between the levels is the1179other way round. There is a $7$-congruence between $g=\nf{122k6A}$1180and $f=\nf{61k6B}$, both $L$-functions have even functional equation,1181and $L(g,3)=0$. In the proof of Theorem~\ref{local},1182there is a problem with the local condition at $p=2$. The map from1183$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily1184injective, but its kernel is at most one dimensional, so we still get1185the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having1186$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get1187(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these1188elements of $\Sha$ are {\it invisible} at level 61. When the levels1189are different we are no longer able to apply Theorem 2.1 of1190\cite{FJ}. However, we still have the congruences of integral modular1191symbols required to make the proof of Proposition \ref{div} go1192through. Indeed, as noted above, the congruences of modular forms were1193found by producing congruences of modular symbols. Despite these1194congruences of modular symbols, Remark 5.3 does not apply, since there1195is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the1196distinct levels.11971198Finally, there are two examples where we have a form $g$ with even1199functional equation such that $L(g,k/2)=0$, and a congruent form1200$f$ which has odd functional equation; these are a 23-congruence1201between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence1202between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If1203$\ord_{s=2}L(f,s)=1$, it ought to be the case that1204$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and1205similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately1206the appropriate modification of Theorem \ref{local} does not1207necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.1208It only tells us that the $\qq$-torsion subgroup of1209$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It1210could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$1211appears in the conjectural formula for the first derivative of the1212complex $L$ function, evaluated at $s=k/2$, but in combination1213with a regulator that we have no way of calculating.12141215Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions1216associated with $f$ and $g$ by the construction of Mazur, Tate and1217Teitelbaum \cite{MTT}, each divided by a suitable canonical1218period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not1219quite clear what to make of this. This divisibility may be proved1220as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit1221times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably1222normalised) are congruent $\bmod{\,\qq}$, as a result of the1223congruence between the modular symbols out of which they are1224constructed. Integrating an appropriate function against these1225measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$1226to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,1227since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case1228where the signs in the functional equations of $L(g,s)$ and1229$L_q(g,s)$ are the same, positive in this instance. (According to1230the proposition in Section 18 of \cite{MTT}, the signs differ1231precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)12321233We also found some examples for which the conditions of Theorem~\ref{local}1234were not met. 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Math. }{\bf 130 }(1997), 99--152.1333\end{thebibliography}133413351336\end{document}1337127k4A 43 127k4C 17 [43]1338159k4A 5,23 159k4E 8 [5]x[23]1339365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)1340369k4A 5,13 369k4J 9 [5]x[13]x[2]1341453k4A 5,17 453k4E 23 [5]x[17]1342453k4A 23 151k4A 30 Odd func eq for g1343465k4A 11 465k4H 7 [11]x[5]x[2]1344477k4A 73 477k4M 12 [73]x[2]1345567k4A 23 567k4I 8 [23]x[3]134681k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.1347581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)1348639k4A 7 639k4H 12 [7]1349657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)1350657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)1351681k4A 43 227k4A 23 Odd func eq for g1352681k4A 59 681k4D 30 [59]x[3]x[2]1353684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)135495k6A 31,59 95k6D 9 [31]x[59]1355116k6A 5 116k6D 6 [5]x[29]x[2]1356122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)1357122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)1358260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP1359136013611362136313641365136613671368136913701371\nf{263k4B} & & 41^2 & \\1372\nf{269k4C} & & 23^2 & \\1373\nf{271k4B} & & 29^2 &\\1374\nf{281k4B} & & 29^2\\1375\hline1376\nf{101k6B} & & 17^2 & 101\\1377\nf{103k6B} & & 23^2\\1378\nf{111k6C} & & 11^2\\1379\nf{153k6G} & & 7^2\\1380\nf{157k6B} & & 252^2\\1381\nf{167k6B} & & 41^2\\1382\nf{172k6B} & & 7^2\\1383\nf{173k6B} & & 71^2\\1384\nf{181k6B} & & 107^2\\1385\hline1386\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\13871388