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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1%2% motive_visibility_v2.tex3%4% August, 20015%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\newcommand{\Ker}{\mathrm {Ker}}26\newcommand{\Aut}{{\mathrm {Aut}}}27\def\id{\mathop{\mathrm{ id}}\nolimits}28\renewcommand{\Im}{{\mathrm {Im}}}29\newcommand{\ord}{{\mathrm {ord}}}30\newcommand{\End}{{\mathrm {End}}}31\newcommand{\Hom}{{\mathrm {Hom}}}32\newcommand{\Mor}{{\mathrm {Mor}}}33\newcommand{\Norm}{{\mathrm {Norm}}}34\newcommand{\Nm}{{\mathrm {Nm}}}35\newcommand{\tr}{{\mathrm {tr}}}36\newcommand{\Tor}{{\mathrm {Tor}}}37\newcommand{\Sym}{{\mathrm {Sym}}}38\newcommand{\Hol}{{\mathrm {Hol}}}39\newcommand{\vol}{{\mathrm {vol}}}40\newcommand{\tors}{{\mathrm {tors}}}41\newcommand{\cris}{{\mathrm {cris}}}42\newcommand{\length}{{\mathrm {length}}}43\newcommand{\dR}{{\mathrm {dR}}}44\newcommand{\lcm}{{\mathrm {lcm}}}45\newcommand{\Frob}{{\mathrm {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{\Sha}{\underline{III}}74%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}75% ---- SHA ----76\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts77\newcommand{\textcyr}[1]{%78{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%79\selectfont #1}}80\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}8182\newcommand{\HH}{{\mathfrak H}}83\newcommand{\aaa}{{\mathfrak a}}84\newcommand{\bb}{{\mathfrak b}}85\newcommand{\dd}{{\mathfrak d}}86\newcommand{\ee}{{\mathbf e}}87\newcommand{\Fbar}{\overline{F}}88\newcommand{\CH}{\mathrm {CH}}8990\begin{document}91\title{Constructing elements in92Shafarevich-Tate groups of modular motives\\93{\sc (NOT FOR DISTRIBUTION!)}}94\author{Neil Dummigan}95\author{William Stein}96\author{Mark Watkins}97\date{March 8th, 2002}98\subjclass{11F33, 11F67, 11G40.}99100\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,101Shafarevich-Tate group.}102103\address{University of Sheffield\\ Department of Pure104Mathematics\\105Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\106U.K.}107\address{Harvard University\\Department of Mathematics\\108One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}109\address{Penn State Mathematics Department\\110University Park\\State College, PA 16802\\ U.S.A.}111112\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}113\email{watkins@math.psu.edu}114115\maketitle116\section{Introduction}117Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$118be the associated $L$-function. The conjecture of Birch and119Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$120at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and121also gives an interpretation of the leading term in the Taylor122expansion in terms of various quantities, including the order of123the Shafarevich-Tate group.124125Cremona and Mazur [2000] look, among all strong Weil elliptic126curves over $\QQ$ of conductor $N\leq 5500$, at those with127non-trivial Shafarevich-Tate group (according to the Birch and128Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate129group has predicted elements of order $m$. In most cases they find130another elliptic curve, often of the same conductor, whose131$m$-torsion is Galois-isomorphic to that of the first one, and132which has rank two. The rational points on the second elliptic133curve produce classes in the common $H^1(\QQ,E[m])$. They expect134that these lie in the Shafarevich-Tate group of the first curve,135so rational points on one curve explain elements of the136Shafarevich-Tate group of the other curve.137138The Bloch-Kato conjecture \cite{BK} is the generalisation to139arbitrary motives of the leading term part of the Birch and140Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture141\cite{B} generalises the part about the order of vanishing at the142central point, identifying it with the rank of a certain Chow143group.144145The present work may be considered as a partial generalisation of146the work of Cremona and Mazur, from elliptic curves over $\QQ$147(with which are associated modular forms of weight $2$) to the148motives attached to modular forms of higher weight. (See \cite{AS}149for a different generalisation, to modular abelian varieties of150higher dimension.) It may also be regarded as doing, for151congruences between modular forms of equal weight, what \cite{Du2}152did for congruences between modular forms of different weights.153154We consider the situation where two newforms $f$ and $g$, both of155weight $k>2$ and level $N$, are congruent modulo some $\qq$, and156$L(g,s)$ vanishes to order at least $2$ at $s=k/2$. We are able to157find eleven examples (all with $k=4$ and $k=6$), and in each case158$\qq$ appears in the numerator of the algebraic number159$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain160canonical period. In fact, we show how this divisibility may be161deduced from the vanishing of $L(g,k/2)$ using recent work of162Vatsal \cite{V}. The point is, the congruence between $f$ and $g$163leads to a congruence between suitable ``algebraic parts'' of the164special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the165other is divisible by $\qq$. Under certain hypotheses, the166Bloch-Kato conjecture then implies that the Shafarevich-Tate group167attached to $f$ has non-zero $\qq$-torsion. Under certain168hypotheses and assumptions, the most substantial of which is the169Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to170the existence of algebraic cycles, we are able to construct the171predicted elements of $\Sha$, using the Galois-theoretic172interpretation of the congruences to transfer elements from a173Selmer group for $g$ to a Selmer group for $f$. In proving the174local conditions at primes dividing the level, and also in175examining the local Tamagawa factors at these primes, we make use176of a higher weight level-lowering result due to Jordan and Livn\'e177\cite{JL}.178179One might say that algebraic cycles for one motive explain180elements of $\Sha$ for the other. A main point of \cite{CM} was to181observe the frequency with which those elements of $\Sha$182predicted to exist for one elliptic curve may be explained by183finding a congruence with another elliptic curve containing points184of infinite order. One shortcoming of our work, compared to the185elliptic curve case, is that, due to difficulties with local186factors in the Bloch-Kato conjecture, we are unable to predict the187exact order of $\Sha$. We have to start with modular forms between188which there exists a congruence. However, Vatsal's work allows us189to explain how the vanishing of one $L$-function leads, via the190congruence, to the divisibility by $\qq$ of (an algebraic part of)191another, independent of observations of computational data. The192computational data does however show that there exist examples to193which our results apply. Moreover, it displays factors of $\qq^2$,194whose existence we cannot prove theoretically, but which are195predicted by Bloch-Kato, given the existence of a non-degenerate196alternating pairing on $\Sha$ \cite{Fl2}.197198\section{Motives and Galois representations}199Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for200$\Gamma_0(N)$, with coefficients in an algebraic number field $E$,201which is necessarily totally real. A theorem of Deligne \cite{De1}202implies the existence, for each (finite) prime $\lambda$ of $E$,203of a two-dimensional vector space $V_{\lambda}$ over204$E_{\lambda}$, and a continuous representation205$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$206such that207\begin{enumerate}208\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing209$lN$ (where $\lambda \mid l$);210\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the211characteristic polynomial of $\Frob_p^{-1}$ acting on212$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.213\end{enumerate}214215Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as216the $\lambda$-adic realisation of a Grothendieck motive $M_f$.217There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,218both $2$-dimensional $E$-vector spaces. For details of the219construction see \cite{Sc}. The de Rham realisation has a Hodge220filtration $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset221F^k=\{0\}$. The Betti realisation $V_B$ comes from singular222cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic223cohomology. There are natural isomorphisms $V_B\otimes224E_{\lambda}\simeq V_{\lambda}$. We may choose a225$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside226each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.227There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),228which amounts to multiplying the action of $\Frob_p$ by $p^j$.229230Following \cite{BK} (Section 3), for $p\neq l$ (including231$p=\infty$) let232$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow233H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for ``finite234part''. $D_p$ is a decomposition subgroup at a prime above $p$,235$I_p$ is the inertia subgroup, and the cohomology is for236continuous cocycles and coboundaries. For $p=l$ let237$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow238H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of239\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and240$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of241elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie242in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.243244There is a natural exact sequence245$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$246247Let248$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.249Define the $\lambda$-Selmer group \newline250$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of251$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in252$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the253condition at $p=\infty$ is superfluous unless $l=2$. Define the254Shafarevich-Tate group255$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$256The length of its $\lambda$-component may be taken for the257exponent of $\lambda$ in an ideal of $O_E$, which we call258$\#\Sha(j)$. We shall only concern ourselves with the case259$j=k/2$, and write $\Sha$ for $\Sha(j)$.260261Define the set of global points262$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$263This is analogous to the group of rational torsion points on an264elliptic curve. The length of its $\lambda$-component may be taken265for the exponent of $\lambda$ in an ideal of $O_E$, which we call266$\#\Gamma_{\QQ}$.267268\section{Canonical periods}269From now on we assume for convenience that $N\geq 3$. We need to270choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti271and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this272in any way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and273$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the274$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and275$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially276Sections 2.2 and 5.4 of \cite{DFG}.)277278For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$279module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of280$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes281E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the282$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,283since it comes from $\ell$-adic cohomology with $O_{\lambda}$284coefficients. We may assume that $T_{\lambda}$ is285$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting286$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.287288Let $X_0(N)$ be the modular curve over $\ZZ[1/N]$ parametrising289elliptic curves with cyclic $N$-isogenies. Let $\mathfrak{E}$ be290the universal elliptic curve over $X_0(N)$. Let291$\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product of292$\mathfrak{E}$ over $X_0(N)$. Realising $X_0(N)$ as the quotient293$\Gamma_0(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a294variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is295isomorphic to the elliptic curve with period lattice generated by296$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a297variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the298fibre product. Then $2\pi i f(\tau)\,d\tau\wedge299dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential300form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and301naturally represents a generating element of $F^{k-1}T_{\dR}$. (At302least, we can make our choices locally at primes dividing $Nk!$ so303that this is the case.) We shall call this element $e(f)$.304305Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and306$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is307a natural action of complex conjugation on $V_B$, breaking it up308into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.309Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$310to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let311$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These312are rank one $O_E$-modules, but not necessarily free, since the313class number of $O_E$ may be greater than one. Choose non-zero314elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be315the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers316$\Omega_f^{\pm}$ by $\omega_f^{\pm}=2\pi i317\Omega_f^{\pm}\delta_f^{\pm}$.318\section{The Bloch-Kato conjecture}319Let $L(f,s)$ be the $L$-function attached to $f$. For320$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series321$\sum_{n=1}^{\infty}a_nn^{-s}$, but there is an analytic322continuation given by an integral, as described in the next323section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture324for the motive $M_f(k/2)$ predicts that325$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\aaa^{\pm}\#\Sha\over (\#\Gamma_{\QQ})^2}.$$326Here, $\pm$ represents the parity of $(k/2)-1$, and327$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}\Omega_{\pm}$. For328$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be329$$\length H^1_f(\QQ_p,T_{\lambda}(j))_{\tors.}-\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j})).$$330We omit the definition of $\ord_{\lambda}(c_p(j))$ for331$\lambda\mid p$, which requires one to assume Fontaine's de Rham332conjecture (\cite{Fo}, Appendix A6), and depends on the choices of333$T_{\dR}$ and $T_B$, locally at $\lambda$. (We shall mainly be334concerned with the $q$-part of the Bloch-Kato conjecture, where335$q$ is a prime of good reduction. For such primes, the de Rham336conjecture follows from Theorem 5.6 of \cite{Fa1}.) The above337formula is to be interpreted as an equality of fractional ideals338of $E$. (Strictly speaking, the conjecture in \cite{BK} is only339given for $E=\QQ$.)340341\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.342Then the fractional ideal $c_p(j)$ is supported at most on343divisors of $p$.344\end{lem}345\begin{proof}346As on p. 30 of \cite{Fl1}, for odd $l\neq p$,347$\ord_{\lambda}(c_p(j))$ is the length of the finite348$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$349where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a350trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is351torsion-free.352\end{proof}353\begin{lem}\label{local1}354Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$355is an irreducible representation of $\Gal(\Qbar/\QQ)$, where356$\qq\mid q$, and that, for all primes $p\mid N$, $\,p\neq \pm3571\pmod{q}$. Suppose also that $f$ is not congruent modulo $\qq$ to358any newform of weight~$k$, trivial character and level dividing359$N/p$, with~$p$ any prime that exactly divides~$N$. Then for any360$p\mid N$, and $j$ an integer, $\ord_{\qq}(c_p(j))=0$.361\end{lem}362\begin{proof} Bearing in mind that363$$\ord_{\qq}((1-a_pp^{-j}+p^{k-1-2j}))=\#H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}),$$364and also the long exact sequence in $I_p$-cohomology arising from365$$\begin{CD}0@>>>T_{\qq}(j)@>>>V_{\qq}(j)@>\pi>>A_{\qq}(j)@>>>0\end{CD},$$366it suffices to show that367$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$368If the dimensions differ then, given that $f$ is not congruent369modulo $\qq$ to a newform of level strictly dividing $N$, and370since $p\neq \pm 1\pmod{q}$, Propositions 3.1 and 2.3 of \cite{L}371tell us that $A[\qq](j)$ is unramified at $p$ and that372$\ord_p(N)=1$. (Here we are using Carayol's result that $N$ is the373prime-to-$q$ part of the conductor of $V_{\qq}$ \cite{Ca1}.) But374then Theorem 1 of \cite{JL} (which uses the condition $q>k$)375implies the existence of a newform of weight $k$, trivial376character and level dividing $N/p$, congruent to $g$ modulo $\qq$.377This contradicts our hypotheses.378\end{proof}379\begin{lem} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid380Nk!$ Then $\ord_{\qq}(c_q)=0$.381\end{lem}382\begin{proof} It follows from the isomorphism at the end of383Section 2.2 of \cite{DFG} (an application of the results of384\cite{Fa1}) that $T_{\qq}$ is the385$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the386filtered module $T_{\dR}\otimes O_{\lambda}$ by the functor they387call $\mathbb{V}$. Given this, the lemma follows from Theorem3884.1(iii) of \cite{BK}.389\end{proof}390391\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an392irreducible representation of $\Gal(\Qbar/\QQ)$.393\end{lem}394This follows trivially from the definition.395396Putting together the above lemmas we arrive at the following:397\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime such that $q>k$.398Let $\qq\mid q$ be a prime of $E$ such that $A[\qq]$ is an399irreducible representation of $\Gal(\Qbar/\QQ)$. Choose $T_{\dR}$400and $T_B$ which locally at $\qq$ are as in the previous section.401If $$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with402numerator non-zero) then the Bloch-Kato conjecture predicts that403$\ord_{\qq}(\#\Sha)>0$.404\end{prop}405406\section{Congruences of special values}407Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal408weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field409large enough to contain all the coefficients $a_n$ and $b_n$.410Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv411g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. It is412easy to see that we may choose the $\delta_f^{\pm}\in T_B^{\pm}$413in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e.414$\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Let us415suppose that such a choice has been made.416417We shall now make two further assumptions:418\begin{enumerate}419\item $L(f,k/2)\neq 0$;420\item $L(g,k/2)=0$.421\end{enumerate}422\begin{prop}423With assumptions as above,424$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$.425\end{prop}426\begin{proof} This is based on some of the ideas used in Section 1 of427\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},428which presumably should refer to ``Condition 2''. Since429$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that430$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm4311=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that432$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$433Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period434$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$435where the integral is taken along the positive imaginary axis,436then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$437Thus we are reduced438to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.439440Let $\mathcal{D}_0$ be the group of divisors of degree zero441supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and442integer $r\geq 0$, let $P_r(R)$ be the additive group of443homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these444groups have a natural action of $\Gamma_0(N)$. Let445$S_{\Gamma_0(N)}(k,R):=\Hom_{\Gamma_0(N)}(\mathcal{D}_0,P_{k-2}(R))$446be the $R$-module of weight $k$ modular symbols for $\Gamma_0(N)$.447448Via the isomorphism (8) in Section 1.5 of \cite{V},449$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in450S_{\Gamma_0(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an451element $\Delta_f^{\pm}\in S_{\Gamma_0(N)}(k,O_E)$.452$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv453(k/2)-1\pmod{2}}^{k-2}r_f(j)X^jY^{k-2-j}.$$ Since454$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that455$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv456(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The457coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to458show is divisible by $\qq$.459Similarly460$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv461(k/2)-1\pmod{2}}^{k-2}r_g(j)X^jY^{k-2-j}.$$ The coefficient of462$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.463Therefore it would suffice to show that, for some $\mu\in O_E$,464the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by465$\qq$ in $S_{\Gamma_0(N)}(k,O_E)$. It suffices to show that, for466some $\mu\in O_E$, the element $\delta_f^{\pm}-\mu\delta_g^{\pm}$467is divisible by $\qq$, considered as an element of $\qq$-adic468cohomology with coefficients. This would be the case if469$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same470one-dimensional subspace upon reduction $\pmod{\qq}$. But this is471a consequence of Theorem 2.1(1) of \cite{FJ}.472\end{proof}473\begin{remar}474By Proposition \ref{sha}, the Bloch-Kato conjecture now predicts475that $\ord_{\qq}(\#\Sha)>0$. The next section provides a476conditional construction of the required elements of $\Sha$.477\end{remar}478\begin{remar}\label{sign}479The signs in the functional equations of $L(f,s)$ and $L(g,s)$480have to be equal, since they are determined by the action of the481involution $W_N$ on the common subspace generated by the482reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.483\end{remar}484This is analogous to the remark at the end of Section 3 of485\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then486$L(g,s)$ must vanish to order at least two, as in all the examples487below. It is worth pointing out that there are no examples of $g$488of level one such that $L(g,k/2)=0$, unless Maeda's conjecture489(that all the normalised cuspidal eigenforms are Galois conjugate)490is false. See \cite{CF}.491492\section{Constructing elements of the Shafarevich-Tate group}493For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and494$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and495$A'_{\lambda}$ be the corresponding objects for $g$. Let496$A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$. Since497$a_p$ is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows498from the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if499irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.500501Suppose that $L(g,k/2)=0$. If the sign in the functional equation502is positive (as it must be if $L(f,k/2)\neq 0$, see Remark503\ref{sign}), this implies that the order of vanishing of $L(g,s)$504at $s=k/2$ is at least $2$. According to the Beilinson-Bloch505conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$506is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of507$\QQ$-rational, null-homologous, codimension $k/2$ algebraic508cycles on the motive $M_g$, modulo rational equivalence. (This509generalises the part of the Birch-Swinnerton-Dyer conjecture which510says that for an elliptic curve $E/\QQ$, the order of vanishing of511$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group512$E(\QQ)$.)513514Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps515to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the516subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.517If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we518get (assuming also the Beilinson-Bloch conjecture) a subspace of519$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of520vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply521conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is522equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would523follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in524Sections 1 and 6.5 of \cite{Fo2}.525526Similarly, if $L(f,k/2)\neq 0$ then we expect that527$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$528coincides with the $\qq$-part of $\Sha$.529\begin{thm}\label{local}530Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that531$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that532$A[\qq]$ and $A'[\qq]$ are irreducible representations of533$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm5341\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent535modulo $\qq$ to any newform of weight~$k$, trivial character and536level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.537Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has538$\FF_{\qq}$-rank at least $r-1$.539\end{thm}540\begin{proof}541Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By542continuity and rescaling we may assume that it lies in543$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq544H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero545class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By546irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so547$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and548we get a non-zero, $\qq$-torsion class $\gamma\in549H^1(\QQ,A_{\qq}(k/2))$.550551Our aim is to show that $\res_p(\gamma)\in552H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We553consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.554The case $p=q$ does not quite work, which is the reason for the555``$r-1$'' in the statement of the theorem.556557\begin{enumerate}558\item {\bf $p\nmid qN$. }559560Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,561A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore562$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as563$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It564follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of565$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125566of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just567contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$568to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that569$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.570571\item {\bf $p\mid N$. }572573First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is574$\qq$-divisible. It suffices to show that575$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$576but this may be done as in the proof of Lemma\ref{local1}. It577follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in578$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from579$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The580order of this group is the same as the order of the group581$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work582of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is583ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As584above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim585H^0(I_p,A[\qq](k/2))$, so we need only consider the case where586this common dimension is $1$. The (motivic) Euler factor at $p$587for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts588as multiplication by $\alpha$ on the one-dimensional space589$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of590\cite{Ca1} that this is the same as the Euler factor at $p$ of591$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then592follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where593$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,594$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on595$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm5961\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence597$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in598$H^1_f(\QQ_p,A_{\qq}(k/2))$.599600\item {\bf $p=q$. }601602Since $q\nmid N$ is a prime of good reduction for the motive603$M_g$, $\,V'_{\qq}$ is a crystalline representation of604$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and605$V'_{\qq}$ have the same dimension, where606$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}607B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)608It follows from Theorem 4.1(ii) of \cite{BK} that609$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$610via their exponential map.611612Since $p$ is a prime of good reduction, the de Rham conjecture is613a consequence of the crystalline conjecture, which follows from614Theorem 5.6 of \cite{Fa1}. Hence615$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$616Observe that the dimension of the left-hand-side is $1$. Hence617there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at618least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It619follows that our $r$-dimensional $\FF_{\qq}$-subspace of the620$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an621$(r-1)$-dimensional subspace landing in622$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).623624\end{enumerate}625\end{proof}626627Theorem 2.7 of \cite{AS} is concerned with verifying local628conditions in the case $k=2$, where $f$ and $g$ are associated629with abelian varieties $A$ and $B$. (Their theorem also applies to630abelian varieties over number fields.) Our restriction outlawing631congruences modulo $\qq$ with cusp forms of lower level is632analogous to theirs forbidding $q$ from dividing Tamagawa factors633$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic634curve with $\ord_l(j(A))<0$, consideration of a Tate635parametrisation shows that if $q\mid c_{A,l}$, i.e. if636$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified637at $l$.)638639In this paper we have encountered two technical problems which we640dealt with in quite similar ways:641\begin{enumerate}642\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;643\item proving local conditions at primes $p\mid N$, for an element644of $\qq$-torsion.645\end{enumerate}646If our only interest was in testing the Bloch-Kato conjecture at647$\qq$, we could have made these problems cancel out, as in Lemma6488.11 of \cite{DFG}, by weakening the local conditions. However, we649have chosen not to do so, since we are also interested in the650Shafarevich-Tate group, and since the hypotheses we had to assume651are not particularly strong.652653\section{Eleven examples}654\newcommand{\nf}[1]{\mbox{\bf #1}}655\begin{figure}656\caption{\label{fig:newforms}Newforms Relevant to657Theorem~\ref{local}}658$$659\begin{array}{|ccccccc|}\hline660g & \deg(g) & f & \deg(f) & q's \\\hline661\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\662\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\663\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\664\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\665\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\666\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\667\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\668\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\669\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\670\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\671\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\672\hline673\end{array}674$$675\end{figure}676677Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists678eleven pairs of newforms~$f$ and~$g$ (of equal weights and levels)679along with at least one prime~$q$ such that there is a prime680$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,681$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.682\subsection{Notation}683Table~\ref{fig:newforms} is laid out as follows.684The first column contains a label whose structure is685\begin{center}686{\bf [Level]k[Weight][GaloisOrbit]}687\end{center}688This label determines a newform $g=\sum a_n q^n$, up to Galois689conjugacy. For example, \nf{127k4C} denotes a newform in the third690Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois691orbits are ordered first by the degree of $\QQ(\ldots, a_n,692\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing693the level, with positive trace being first in the event that the694two absolute values are equal, and the first Galois orbit is695denoted {\bf A}, the second {\bf B}, and so on. The second column696contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The697third and fourth columns contain~$f$ and its degree, respectively.698The fifth column contains at least one prime~$q$ such that there699is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that700the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are701satisfied for~$f$,~$g$, and~$\qq$.702703\subsection{The first example in detail}704\newcommand{\fbar}{\overline{f}}705We describe the first line of Table~\ref{fig:newforms}706in more detail. See the next section for further details707on how the computations were performed.708709Using modular symbols, we find that there is a newform710$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots711\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,712the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We713also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier714coefficients generate a number field~$K$ of degree~$17$, and by715computing the image of the modular symbol $XY\{0,\infty\}$ under716the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$717and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue718characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are719both equal to720$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$721722There is no form in the Eisenstein subspaces of723$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with724$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so725$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is726prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a727level~$1$ form of weight~$4$. Thus we have checked the hypotheses728of Theorem~\ref{local}, so if $r$ is the dimension of729$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of730$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.731732Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,733since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup734of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion735subgroup of $\Sha$. Admitting these assumptions, we have736constructed the $\qq$-torsion in $\Sha$ predicted by the737Bloch-Kato conjecture.738739For particular examples of elliptic curves one can often find and740write down rational points predicted by the Birch and741Swinnerton-Dyer conjecture. It would be nice if likewise one could742explicitly produce algebraic cycles predicted by the743Beilinson-Bloch conjecture in the above examples. Since744$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary7450.3.2 of \cite{Z}), so ought to be trivial in746$\CH_0^{k/2}(M_g)\otimes\QQ$.747748\subsection{Some remarks on how the computation was performed}749We give a brief summary of how the computation was performed. The750algorithms that we used were implemented by the second author, and751most are a standard part of the MAGMA V2.8 (see \cite{magma}).752753Let~$g$,~$f$, and~$q$ be some data from a line of754Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We755verified the existence of a congruence modulo~$q$, that756$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq7570$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does758not arise from any $S_k(\Gamma_0(N/p))$, as follows:759760To prove there is a congruence, we showed that the corresponding761{\em integral} spaces of modular symbols satisfy an appropriate762congruence, which forces the existence of a congruence on the763level of Fourier expansions. We showed that $\rho_{g,\qq}$ is764irreducible by computing a set that contains all possible residue765characteristics of congruences between~$g$ and any Eisenstein766series of level dividing~$N$, where by congruence, we mean a767congruence for all Fourier coefficients of index~$n$ with768$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any769form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by770listing a basis of such~$h$ and finding the possible congruences,771where again we disregard the Fourier coefficients of index not772coprime to~$N$.773774To verify that $L(g,\frac{k}{2})=0$, we computed the image of the775modular symbol ${\mathbf776e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a777map with the same kernel as the period mapping, and found that the778image was~$0$. The period mapping sends the modular779symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,780so that ${\mathbf e}$ maps to~$0$ implies that781$L(g,\frac{k}{2})=0$. In a similar way, we verified that782$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$783which, because of the functional equation, implies that784$L'(g,\frac{k}{2})=0$.785786In the course of our search, we found~$17$ plausible examples and787had to discard~$6$ of them because either the representation788$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv789\pm 1\pmod{q}$ for some $p\mid N$. Table~\ref{fig:newforms} is790of independent interest because it includes examples of modular791forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$792that is not forced by the functional equation. We found no such793examples of weights $\geq 8$.794795796797\begin{thebibliography}{AL}798\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on799$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.800\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of801Shafarevich-Tate groups of abelian varieties: evidence for the802Birch and Swinnerton-Dyer conjecture, in preparation.803\bibitem[B]{B} S. 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Math. }{\bf 130 }(1997), 99--152.880\end{thebibliography}881882883\end{document}884\subsection{Plan for finishing the computation}885886What I (William) have to do with the below stuff:\\887\begin{enumerate}888\item Fill in the missing blanks at these levels:889$$453, 639, 657, 681, 684, 95, 116, 122, 260$$890\item Make a MAGMA program that has a table of forms to try.891\item First, unfortunately, for some reason the892labels are sometimes wrong, e.g., \nf{159k4A} should893have been \nf{159k4B}, so I have to check all the894rational forms to see which has $L(1)=0$.895\item Try each level using the MAGMA program (this will take a long time to run).896\item Finish the ``Table of examples'' above.897\end{enumerate}898899900\begin{verbatim}901902903(This stuff below is being integrated into the above, as I do904the required (rather time consuming) computations.)905**************************************************906FROM Mark Watkins:907908Here's a list of stuff; the left-hand (sinister) column909contains forms with a double zero at the central point,910whilst the right-hand (dexter) column contains forms which911have a large square factor in LRatio at 2. The middle column912is the large prime factor in ModularDegree of the LHS,913and/or the lpf in the LRatio at 2 of the RHS.914915It seems that 567k4L has invisible Sha possibly.916The ModularDegree for 639k4B has no large factors.917918127k4A 43 127k4C919159k4A 23 159k4E920365k4A 29 365k4E921369k4A 13 369k4I922453k4A 17 453k4E923453k4A 23924465k4A 11 465k4H925477k4A 73 477k4L926567k4A 23 567k4G92713 567k4L928581k4A 19 581k4E929639k4B --930931932Forms with spurious zeros to do:933934657k4A935681k4A936684k4B93795k6A 31,59938116k6A --939122k6A 73940260k6A941942If we allow 5 and 7 to be small primes, then we get more943visibility info.944945159k4A 5 159k4E946369k4A 5 369k4I947453k4A 5 453k4E948639k4B 7 639k4H949950Maybe the general theorem does not apply to primes which are so small,951but we might be able to show that they are OK in these specific cases.952953I checked the first three weight 6 examples with AbelianIntersection,954but actually computing the LRatio bogs down too much. Will need955new version.95695795k6A 31 95k6D95895k6A 59 95k6D959116k6A 5 116k6D960122k6A 73 122k6D961962Incidentally, the LRatio for 581k4E has a power of 19^4.963Perhaps not surprising, as the ModularDegree of 581k4A964has a factor of 19^2. But maybe it means a bigger Sha.965\end{verbatim}966967