12%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%3%4% motive_visibility.tex5%6% April 11, 20027%8% Project of William Stein, Neil Dummigan, Mark Watkins9%10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1112\documentclass{amsart}13\usepackage{amssymb}14\usepackage{amsmath}15\usepackage{amscd}1617\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}1819\newtheorem{prop}{Proposition}[section]20\newtheorem{defi}[prop]{Definition}21\newtheorem{conj}[prop]{Conjecture}22\newtheorem{lem}[prop]{Lemma}23\newtheorem{thm}[prop]{Theorem}24\newtheorem{cor}[prop]{Corollary}25\newtheorem{examp}[prop]{Example}26\newtheorem{remar}[prop]{Remark}27\newcommand{\Ker}{\mathrm {Ker}}28\newcommand{\Aut}{{\mathrm {Aut}}}29\def\id{\mathop{\mathrm{ id}}\nolimits}30\renewcommand{\Im}{{\mathrm {Im}}}31\newcommand{\ord}{{\mathrm {ord}}}32\newcommand{\End}{{\mathrm {End}}}33\newcommand{\Hom}{{\mathrm {Hom}}}34\newcommand{\Mor}{{\mathrm {Mor}}}35\newcommand{\Norm}{{\mathrm {Norm}}}36\newcommand{\Nm}{{\mathrm {Nm}}}37\newcommand{\tr}{{\mathrm {tr}}}38\newcommand{\Tor}{{\mathrm {Tor}}}39\newcommand{\Sym}{{\mathrm {Sym}}}40\newcommand{\Hol}{{\mathrm {Hol}}}41\newcommand{\vol}{{\mathrm {vol}}}42\newcommand{\tors}{{\mathrm {tors}}}43\newcommand{\cris}{{\mathrm {cris}}}44\newcommand{\length}{{\mathrm {length}}}45\newcommand{\dR}{{\mathrm {dR}}}46\newcommand{\lcm}{{\mathrm {lcm}}}47\newcommand{\Frob}{{\mathrm {Frob}}}48\def\rank{\mathop{\mathrm{ rank}}\nolimits}49\newcommand{\Gal}{\mathrm {Gal}}50\newcommand{\Spec}{{\mathrm {Spec}}}51\newcommand{\Ext}{{\mathrm {Ext}}}52\newcommand{\res}{{\mathrm {res}}}53\newcommand{\Cor}{{\mathrm {Cor}}}54\newcommand{\AAA}{{\mathbb A}}55\newcommand{\CC}{{\mathbb C}}56\newcommand{\RR}{{\mathbb R}}57\newcommand{\QQ}{{\mathbb Q}}58\newcommand{\ZZ}{{\mathbb Z}}59\newcommand{\NN}{{\mathbb N}}60\newcommand{\EE}{{\mathbb E}}61\newcommand{\TT}{{\mathbb T}}62\newcommand{\HHH}{{\mathbb H}}63\newcommand{\pp}{{\mathfrak p}}64\newcommand{\qq}{{\mathfrak q}}65\newcommand{\FF}{{\mathbb F}}66\newcommand{\KK}{{\mathbb K}}67\newcommand{\GL}{\mathrm {GL}}68\newcommand{\SL}{\mathrm {SL}}69\newcommand{\Sp}{\mathrm {Sp}}70\newcommand{\Br}{\mathrm {Br}}71\newcommand{\Qbar}{\overline{\mathbb Q}}72\newcommand{\Xbar}{\overline{X}}73\newcommand{\Ebar}{\overline{E}}74\newcommand{\sbar}{\overline{s}}75%\newcommand{\Sha}{\underline{III}}76%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}77% ---- SHA ----78\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts79\newcommand{\textcyr}[1]{%80{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%81\selectfont #1}}82\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}8384\newcommand{\HH}{{\mathfrak H}}85\newcommand{\aaa}{{\mathfrak a}}86\newcommand{\bb}{{\mathfrak b}}87\newcommand{\dd}{{\mathfrak d}}88\newcommand{\ee}{{\mathbf e}}89\newcommand{\Fbar}{\overline{F}}90\newcommand{\CH}{\mathrm {CH}}9192\begin{document}93\title{Constructing elements in94Shafarevich-Tate groups of modular motives}95\author{Neil Dummigan}96\author{William Stein}97\author{Mark Watkins}98\date{May 29th, 2002}99\subjclass{11F33, 11F67, 11G40.}100101\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,102Shafarevich-Tate group.}103104\address{University of Sheffield\\ Department of Pure105Mathematics\\106Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\107U.K.}108\address{Harvard University\\Department of Mathematics\\109One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}110\address{Penn State Mathematics Department\\111University Park\\State College, PA 16802\\ U.S.A.}112113\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}114\email{watkins@math.psu.edu}115116\maketitle {\bf Not for distribution}117\section{Introduction}118Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$119be the associated $L$-function. The conjecture of Birch and120Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$121at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and122also gives an interpretation of the leading term in the Taylor123expansion in terms of various quantities, including the order of124the Shafarevich-Tate group of $E$.125126Cremona and Mazur [2000] look, among all strong Weil elliptic127curves over $\QQ$ of conductor $N\leq 5500$, at those with128non-trivial Shafarevich-Tate group (according to the Birch and129Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate130group has predicted elements of prime order $m$. In most cases131they find another elliptic curve, often of the same conductor,132whose $m$-torsion is Galois-isomorphic to that of the first one,133and which has rank two. The rational points on the second elliptic134curve produce classes in the common $H^1(\QQ,E[m])$. They show135\cite{CM2} that these lie in the Shafarevich-Tate group of the136first curve, so rational points on one curve explain elements of137the Shafarevich-Tate group of the other curve.138139The Bloch-Kato conjecture \cite{BK} is the generalisation to140arbitrary motives of the leading term part of the Birch and141Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture142\cite{B} generalises the part about the order of vanishing at the143central point, identifying it with the rank of a certain Chow144group.145146The present work may be considered as a partial generalisation of147the work of Cremona and Mazur, from elliptic curves over $\QQ$148(which are associated to modular forms of weight $2$) to the149motives attached to modular forms of higher weight. (See \cite{AS}150for a different generalisation, to modular abelian varieties of151higher dimension.) It may also be regarded as doing, for152congruences between modular forms of equal weight, what \cite{Du2}153did for congruences between modular forms of different weights.154155We consider the situation where two newforms $f$ and $g$, both of156weight $k>2$ and level $N$, are congruent modulo a maximal ideal157$\qq$ of odd residue characteristic, $L(g,k/2)=0$ but158$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish159to order at least $2$ at $s=k/2$. We are able to find sixteen160examples (all with $k=4$ and $k=6$), and in each case $\qq$161divides the numerator of the algebraic number162$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain163canonical period. In fact, we show how this divisibility may be164deduced from the vanishing of $L(g,k/2)$ using recent work of165Vatsal \cite{V}. The point is, the congruence between $f$ and $g$166leads to a congruence between suitable ``algebraic parts'' of the167special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the168other is divisible by $\qq$. Under certain hypotheses, the169Bloch-Kato conjecture then implies that the Shafarevich-Tate group170attached to $f$ has non-zero $\qq$-torsion. Under certain171hypotheses and assumptions, the most substantial of which is the172Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to173the existence of algebraic cycles, we are able to construct the174predicted elements of $\Sha$, using the Galois-theoretic175interpretation of the congruences to transfer elements from a176Selmer group for $g$ to a Selmer group for $f$. In proving the177local conditions at primes dividing the level, and also in178examining the local Tamagawa factors at these primes, we make use179of a higher weight level-lowering result due to Jordan and Livn\'e180\cite{JL}.181182One might say that algebraic cycles for one motive explain183elements of $\Sha$ for the other. A main point of \cite{CM} was to184observe the frequency with which those elements of $\Sha$185predicted to exist for one elliptic curve may be explained by186finding a congruence with another elliptic curve containing points187of infinite order. One shortcoming of our work, compared to the188elliptic curve case, is that, due to difficulties with local189factors in the Bloch-Kato conjecture, we are unable to compute the190exact order of $\Sha$ predicted by the Bloch-Kato conjecture. We191have to start with modular forms between which there exists a192congruence. However, Vatsal's work allows us to explain how the193vanishing of one $L$-function leads, via the congruence, to the194divisibility by $\qq$ of (an algebraic part of) another,195independent of observations of computational data. The196computational data does however show that there exist examples to197which our results apply. Moreover, it displays factors of $\qq^2$,198whose existence we do not prove theoretically, but which are199predicted by Bloch-Kato.200201\section{Motives and Galois representations}202Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for203$\Gamma_0(N)$, with coefficients in an algebraic number field $E$,204which is necessarily totally real. A theorem of Deligne \cite{De1}205implies the existence, for each (finite) prime $\lambda$ of $E$,206of a two-dimensional vector space $V_{\lambda}$ over207$E_{\lambda}$, and a continuous representation208$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$209such that210\begin{enumerate}211\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing212$lN$ (where $\lambda \mid l$);213\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the214characteristic polynomial of $\Frob_p^{-1}$ acting on215$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.216\end{enumerate}217218Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as219the $\lambda$-adic realisation of a Grothendieck motive $M_f$.220There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,221both $2$-dimensional $E$-vector spaces. For details of the222construction see \cite{Sc}. The de Rham realisation has a Hodge223filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset224F^k=\{0\}$. The Betti realisation $V_B$ comes from singular225cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic226cohomology. There are natural isomorphisms $V_B\otimes227E_{\lambda}\simeq V_{\lambda}$. We may choose a228$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside229each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.230Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.231There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),232which amounts to multiplying the action of $\Frob_p$ by $p^j$.233234Following \cite{BK} (Section 3), for $p\neq l$ (including235$p=\infty$) let236$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow237H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for ``finite238part'', $D_p$ is a decomposition subgroup at a prime above $p$,239$I_p$ is the inertia subgroup, and the cohomology is for240continuous cocycles and coboundaries. For $p=l$ let241$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow242H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of243\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and244$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of245elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie246in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.247248There is a natural exact sequence249$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$250Let251$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.252Define the $\lambda$-Selmer group \newline253$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of254$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in255$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the256condition at $p=\infty$ is superfluous unless $l=2$. Define the257Shafarevich-Tate group258$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$259Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any260prime ideal $\lambda$ is the length of the $\lambda$-component of261$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,262and write $\Sha$ for $\Sha(j)$.263264Define the group of global torsion points265$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$266This is analogous to the group of rational torsion points on an267elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in268which the exponent of any prime ideal $\lambda$ is the length of269the $\lambda$-component of $\Gamma_{\QQ}$.270271\section{Canonical periods}272We assume from now on for convenience that $N\geq 3$. We need to273choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti274and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this275in a way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and276$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the277$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and278$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially279Sections 2.2 and 5.4 of \cite{DFG}.)280281For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$282module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of283$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes284E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the285$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,286since it comes from $\ell$-adic cohomology with $O_{\lambda}$287coefficients. We may assume that $T_{\lambda}$ is288$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting289$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.290291Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising292generalised elliptic curves with full level-$N$ structure. Let293$\mathfrak{E}$ be the universal generalised elliptic curve over294$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product295of $\mathfrak{E}$ over $M(N)$. Realising $M(N)(\CC)$ as the296disjoint union of $\phi(N)$ copies of the quotient297$\Gamma(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a298variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is299isomorphic to the elliptic curve with period lattice generated by300$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a301variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the302fibre product. Then $2\pi i f(\tau)\,d\tau\wedge303dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential304form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and305naturally represents a generating element of $F^{k-1}T_{\dR}$. (At306least, we can make our choices locally at primes dividing $Nk!$ so307that this is the case.) We shall call this element $e(f)$.308309Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and310$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is311a natural action of complex conjugation on $V_B$, breaking it up312into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.313Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$314to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let315$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These316are rank one $O_E$-modules, but not necessarily free, since the317class number of $O_E$ may be greater than one. Choose non-zero318elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be319the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers320$\Omega_f^{\pm}$ by $\omega_f^{\pm}=2\pi i321\Omega_f^{\pm}\delta_f^{\pm}$.322\section{The Bloch-Kato conjecture}323Let $L(f,s)$ be the $L$-function attached to $f$. For324$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with325Euler product326$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but327there is an analytic continuation given by an integral, as328described in the next section. Suppose that $L(f,k/2)\neq 0$. The329Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the330following equality of fractional ideals of $E$:331$${L(f,k/2)\over \vol_{\infty}}=332{\left(\prod_pc_p(k/2)\right)\#\Sha\over333\aaa^{\pm}(\#\Gamma_{\QQ})^2}.$$ (Strictly speaking, the334conjecture in \cite{BK} is only given for $E=\QQ$.) Here, $\pm$335represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal336to $(2\pi i)^{k/2}$ multiplied by the determinant of the337isomorphism $V_B^{\pm}\otimes\CC\simeq338(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the339lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For340$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be341$$\length\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-342\ord_{\lambda}(P_p(p^{-j}))$$343$$=\length\>\> (H^0(\QQ_p,A_{\lambda}(j))/H^0(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p})).$$344We omit the definition of $\ord_{\lambda}(c_p(j))$ for345$\lambda\mid p$, which requires one to assume Fontaine's de Rham346conjecture (\cite{Fo}, Appendix A6), and depends on the choices of347$T_{\dR}$ and $T_B$, locally at $\lambda$. (We shall mainly be348concerned with the $q$-part of the Bloch-Kato conjecture, where349$q$ is a prime of good reduction. For such primes, the de Rham350conjecture follows from Theorem 5.6 of \cite{Fa1}.)351352\begin{lem}\label{vol}353$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and354$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.355\end{lem}356\begin{proof} $\vol_{\infty}$ is also equal to the determinant357of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to358$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we359used above in the definition of $\vol_{\infty}$ (c.f. the last360paragraph of 1.7 of \cite{De2}). We are using here the natural361pairings. Recall that the index of $O_E\delta_f^{\pm}$ in362$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed363by noting that, locally away from primes dividing $Nk!$, the index364of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its365dual, both being equal to the ideal denoted $\eta$ in \cite{DFG2}.366\end{proof}367\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.368Then the fractional ideal $c_p(j)$ is supported at most on369divisors of $p$.370\end{lem}371\begin{proof}372As on p.~30 of \cite{Fl1}, for odd $l\neq p$,373$\ord_{\lambda}(c_p(j))$ is the length of the finite374$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$375where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a376trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is377torsion-free.378\end{proof}379\begin{lem}\label{local1}380Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$381is an irreducible representation of $\Gal(\Qbar/\QQ)$, where382$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose383that $\,p\not\equiv -1\pmod{q}$. Suppose also that $f$ is not congruent384modulo $\qq$ to any newform of weight~$k$, trivial character and385level dividing $N/p$. Then for $j$ any integer,386$\ord_{\qq}(c_p(j))=0$.387\end{lem}388\begin{proof}389It suffices to show that390$$\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),$$391since this ensures that392$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that393$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If394the dimensions differ then, given that $f$ is not congruent modulo395$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of396\cite{L} shows that we are in the situation covered by one of the397three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv398-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is399unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's400result that $N$ is the prime-to-$q$ part of the conductor of401$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses402the condition $q>k$) implies the existence of a newform of weight403$k$, trivial character and level dividing $N/p$, congruent to $g$404modulo $\qq$. This contradicts our hypotheses.405\end{proof}406\begin{remar}407For an example of what can be done when $f$ {\em is } congruent to408a form of lower level, see the first example in 7.4 below.409\end{remar}410\begin{lem}\label{at q} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid411Nk!$ Then $\ord_{\qq}(c_q)=0$.412\end{lem}413\begin{proof} It follows from Lemma 5.7 of \cite{DFG} (whose proof relies on an414application, at the end of section 2.2, of the results of415\cite{Fa1}) that $T_{\qq}$ is the416$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the417filtered module $T_{\dR}\otimes O_{\qq}$ by the functor they call418$\mathbb{V}$. (This property is part of the definition of an419$S$-integral premotivic structure given in Section 1.2 of420\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)421of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used422in Theorem 4.1 of \cite{BK} follows from the first paragraph of4232(h) of \cite{Fa1}.)424\end{proof}425426\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an427irreducible representation of $\Gal(\Qbar/\QQ)$.428\end{lem}429This follows trivially from the definition.430431Putting together the above lemmas we arrive at the following:432\begin{prop}\label{sha} Assume the same hypotheses as in Lemma \ref{local1},433for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$434are as in the previous section. If435$$\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0$$ (with numerator436non-zero) then the Bloch-Kato conjecture predicts that437$$\ord_{\qq}(\#\Sha)>0.$$438\end{prop}439440\section{Congruences of special values}441Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal442weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field443large enough to contain all the coefficients $a_n$ and $b_n$.444Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv445g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose446that $q\nmid N\phi(N)k!$ It is easy to see that we may choose the447$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that448$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates449$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice450has been made.451452We shall now make two further assumptions:453\begin{enumerate}454\item $L(f,k/2)\neq 0$;455\item $L(g,k/2)=0$.456\end{enumerate}457\begin{prop} \label{div}458With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.459\end{prop}460\begin{proof} This is based on some of the ideas used in Section 1 of461\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},462which presumably should refer to ``Condition 2''. Since463$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that464$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm4651=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that466$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$467Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period468$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$469where the integral is taken along the positive imaginary axis,470then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$471Thus we are reduced472to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.473474Let $\mathcal{D}_0$ be the group of divisors of degree zero475supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and476integer $r\geq 0$, let $P_r(R)$ be the additive group of477homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these478groups have a natural action of $\Gamma_1(N)$. Let479$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$480be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.481482Via the isomorphism (8) in Section 1.5 of \cite{V},483$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in484S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an485element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.486(See also Section 4.2 of \cite{St}.) Inverting $N\phi(N)$ takes487into account the fact that we are now dealing with $X_1(N)$ rather488that $M(N)$. Up to some small factorials which do not matter489locally at $\qq$,490$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv491(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since492$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that493$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv494(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The495coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to496show is divisible by $\qq$.497Similarly498$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv499(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient500of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.501Therefore it would suffice to show that, for some $\mu\in O_E$,502the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by503$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show504that, for some $\mu\in O_E$, the element505$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,506considered as an element of $\qq$-adic cohomology of $X_1(N)$ with507non-constant coefficients. This would be the case if508$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same509one-dimensional subspace upon reduction $\pmod{\qq}$. But this is510a consequence of Theorem 2.1(1) of \cite{FJ}.511\end{proof}512\begin{remar}513By Proposition \ref{sha} (assuming, for all $p\mid N$ the same514hypotheses as in Lemma \ref{local1}, together with515$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that516$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional517construction of the required elements of $\Sha$.518\end{remar}519\begin{remar}\label{sign}520The signs in the functional equations of $L(f,s)$ and $L(g,s)$521have to be equal, since they are determined by the action of the522involution $W_N$ on the common subspace generated by the523reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.524Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the525common eigenvalue of $W_N$ acting on $f$ and $g$.526\end{remar}527This is analogous to the remark at the end of Section 3 of528\cite{CM}, which shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$529then $L(g,s)$ must vanish to order at least two, as in all the530examples below. It is worth pointing out that there are no531examples of $g$ of level one, and positive sign in the functional532equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that533all the normalised cuspidal eigenforms of weight $k$ and level one534are Galois conjugate) is false. See \cite{CF}.535536\section{Constructing elements of the Shafarevich-Tate group}537Let $f$ and $g$ be as in the first paragraph of the previous538section. For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and539$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and540$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$541is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from542the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if543irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.544545Suppose that $L(g,k/2)=0$. If the sign in the functional equation546is positive (as it must be if $L(f,k/2)\neq 0$, see Remark547\ref{sign}), this implies that the order of vanishing of $L(g,s)$548at $s=k/2$ is at least $2$. According to the Beilinson-Bloch549conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$550is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational551rational equivalence classes of null-homologous, codimension $k/2$552algebraic cycles on the motive $M_g$. (This generalises the part553of the Birch--Swinnerton-Dyer conjecture which says that for an554elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at555$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)556557Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps558to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the559subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.560If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we561get (assuming also the Beilinson-Bloch conjecture) a subspace of562$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of563vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply564conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is565equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would566follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in567Sections 1 and 6.5 of \cite{Fo2}.568569Similarly, if $L(f,k/2)\neq 0$ then we expect that570$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$571coincides with the $\qq$-part of $\Sha$.572\begin{thm}\label{local}573Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that574$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that575$A[\qq]$ is an irreducible representations of $\Gal(\Qbar/\QQ)$576and that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a577newform of weight~$k$, trivial character and level dividing $N/p$.578Suppose that, for all primes $p\mid N$, $\,p\not\equiv579-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here580$w_p$ is the common eigenvalue of the Atkin-Lehner involution581$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of582$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.583\end{thm}584\begin{proof}585Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By586continuity and rescaling we may assume that it lies in587$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq588H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that589$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a590non-zero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq591H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$592is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so593$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and594we get a non-zero, $\qq$-torsion class $\gamma\in595H^1(\QQ,A_{\qq}(k/2))$.596597Our aim is to show that $\res_p(\gamma)\in598H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We599consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.600601\begin{enumerate}602\item {\bf $p\nmid qN$. }603604Consider the $I_p$-cohomology of the short exact sequence605$$\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},$$606where $\pi$ is multiplication by a uniformising element of607$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at608$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is609$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,610remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into611$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in612H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in613$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence614that the restriction of $c$ (to $H^1(I_p,A'[\qq](k/2))\simeq615H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$616to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line 3 of p.~125 of617\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just618contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$619to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that620$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.621622\item {\bf $p\mid N$. }623624First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.625It suffices to show that626$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$627since this would imply that the natural map from628$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is629surjective, but this may be done as in the proof of Lemma630\ref{local1}. It follows as above that the image of $c\in631H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then632$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by633inflation-restriction. The order of this group is the same as the634order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is635trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that636$V_{\qq}(k/2)$ is ramified at $p$, so $\dim637H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim638H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only639consider the case where this common dimension is $1$. The640(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha641p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by642$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It643follows from Theor\'eme A of \cite{Ca1} that this is the same as644the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and645Lehner \cite{AL}, it then follows that $p^2\nmid N$ and646$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that647$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on648$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as649$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that650$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so651$\res_p(\gamma)=0$ and certainly lies in652$H^1_f(\QQ_p,A_{\qq}(k/2))$.653654\item {\bf $p=q$. }655656Since $q\nmid N$ is a prime of good reduction for the motive657$M_g$, $\,V'_{\qq}$ is a crystalline representation of658$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and659$V'_{\qq}$ have the same dimension, where660$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}661B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)662As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is663the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the664filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we665may now prove, in the same manner as Proposition 9.2 of666\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.667\end{enumerate}668\end{proof}669670Theorem 2.7 of \cite{AS} is concerned with verifying local671conditions in the case $k=2$, where $f$ and $g$ are associated672with abelian varieties $A$ and $B$. (Their theorem also applies to673abelian varieties over number fields.) Our restriction outlawing674congruences modulo $\qq$ with cusp forms of lower level is675analogous to theirs forbidding~$q$ from dividing Tamagawa factors676$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic677curve with $\ord_l(j(A))<0$, consideration of a Tate678parametrisation shows that if $q\mid c_{A,l}$, i.e., if679$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified680at $l$.)681682In this paper we have encountered two technical problems which we683dealt with in quite similar ways:684\begin{enumerate}685\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;686\item proving local conditions at primes $p\mid N$, for an element687of $\qq$-torsion.688\end{enumerate}689If our only interest was in testing the Bloch-Kato conjecture at690$\qq$, we could have made these problems cancel out, as in Lemma6918.11 of \cite{DFG}, by weakening the local conditions. However, we692have chosen not to do so, since we are also interested in the693Shafarevich-Tate group, and since the hypotheses we had to assume694are not particularly strong.695696\section{Sixteen examples}697\newcommand{\nf}[1]{\mbox{\bf #1}}698\begin{figure}699\caption{\label{fig:newforms}Newforms Relevant to700Theorem~\ref{local}}701$$702\begin{array}{|ccccc|}\hline703g & \deg(g) & f & \deg(f) & q's \\\hline704\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\705\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\706\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\707\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\708\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\709\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\710\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\711\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\712\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\713\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\714\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\715\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\716\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\717\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\718\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\719\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\720\hline721\end{array}722$$723\end{figure}724725Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists726sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)727along with at least one prime~$q$ such that there is a prime728$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,729$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.730\subsection{Notation}731Table~\ref{fig:newforms} is laid out as follows.732The first column contains a label whose structure is733\begin{center}734{\bf [Level]k[Weight][GaloisOrbit]}735\end{center}736This label determines a newform $g=\sum a_n q^n$, up to Galois737conjugacy. For example, \nf{127k4C} denotes a newform in the third738Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois739orbits are ordered first by the degree of $\QQ(\ldots, a_n,740\ldots)$, then by the sequence of absolute values $|\mbox{\rm741Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace742being first in the event that the two absolute values are equal,743and the first Galois orbit is denoted {\bf A}, the second {\bf B},744and so on. The second column contains the degree of the field745$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns746contain~$f$ and its degree, respectively. The fifth column747contains at least one prime~$q$ such that there is a prime748$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the749hypotheses of Theorem~\ref{local} (except possibly $r>0$) are750satisfied for~$f$,~$g$, and~$\qq$.751752\subsection{The first example in detail}753\newcommand{\fbar}{\overline{f}}754We describe the first line of Table~\ref{fig:newforms}755in more detail. See the next section for further details756on how the computations were performed.757758Using modular symbols, we find that there is a newform759$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots760\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,761the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We762also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier763coefficients generate a number field~$K$ of degree~$17$, and by764computing the image of the modular symbol $XY\{0,\infty\}$ under765the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$766and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue767characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are768both equal to769$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7770+ \cdots\in \FF_{43}[[q]].$$771772There is no form in the Eisenstein subspaces of773$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with774$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so775$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is776prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a777level~$1$ form of weight~$4$. Thus we have checked the hypotheses778of Theorem~\ref{local}, so if $r$ is the dimension of779$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of780$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.781782Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,783since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup784of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion785subgroup of $\Sha$. Admitting these assumptions, we have786constructed the $\qq$-torsion in $\Sha$ predicted by the787Bloch-Kato conjecture.788789For particular examples of elliptic curves one can often find and790write down rational points predicted by the Birch and791Swinnerton-Dyer conjecture. It would be nice if likewise one could792explicitly produce algebraic cycles predicted by the793Beilinson-Bloch conjecture in the above examples. Since794$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary7950.3.2 of \cite{Z}), so ought to be trivial in796$\CH_0^{k/2}(M_g)\otimes\QQ$.797798\subsection{Some remarks on how the computation was performed}799We give a brief summary of how the computation was performed. The800algorithms that we used were implemented by the second author, and801most are a standard part of MAGMA (see \cite{magma}).802803Let~$g$,~$f$, and~$q$ be some data from a line of804Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We805verified the existence of a congruence modulo~$q$, that806$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq8070$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does808not arise from any $S_k(\Gamma_0(N/p))$, as follows:809810To prove there is a congruence, we showed that the corresponding811{\em integral} spaces of modular symbols satisfy an appropriate812congruence, which forces the existence of a congruence on the813level of Fourier expansions. We showed that $\rho_{g,\qq}$ is814irreducible by computing a set that contains all possible residue815characteristics of congruences between~$g$ and any Eisenstein816series of level dividing~$N$, where by congruence, we mean a817congruence for all Fourier coefficients of index~$n$ with818$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any819form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by820listing a basis of such~$h$ and finding the possible congruences,821where again we disregard the Fourier coefficients of index not822coprime to~$N$.823824To verify that $L(g,\frac{k}{2})=0$, we computed the image of the825modular symbol ${\mathbf826e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map827with the same kernel as the period mapping, and found that the828image was~$0$. The period mapping sends the modular829symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,830so that ${\mathbf e}$ maps to~$0$ implies that831$L(g,\frac{k}{2})=0$. In a similar way, we verified that832$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)833=(-1)^{k/2} g$ which, because of the functional equation, implies834that $L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of835independent interest because it includes examples of modular forms836of weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by837the functional equation. We found no such examples of weights838$\geq 8$.839840For the two examples \nf{581k4E} and \nf{684k4K}, the square of a841prime appears in the $q$-column, meaning $f$ and $g$ are congruent842$\bmod{\,\qq^2}$ for some $\qq\mid q$. In these cases, a843modification of Theorem \ref{local} will show that the844$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has845$O_E/\qq^2$-rank at least $r$.846847\subsection{Other examples}848We have some other examples where a congruence between forms can849be shown, but the levels differ. However, Remark 5.3 does not850apply, so that one of the forms could have an odd functional851equation, and the other could have an even functional equation.852For instance, we have a 13-congruence between $g=\nf{81k4A}$ and853$f=\nf{567k4L}$; here $L(\nf{567k4L},2)\neq 0$, while854$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation.855Here $f$ obviously fails the condition about not being congruent856to a form of lower level, so in Lemma 4.3 it is possible that857$\ord_{\qq}(c_7(2))>0$. In fact this does happen. Because858$V'_{\qq}$ (attached to g of level $81$) is unramified at $p=7$,859$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is860two-dimensional. As in (2) of the proof of Theorem \ref{local},861one of the eigenvalues of $\Frob_p^{-1}$ acting on this862two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where863$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that864$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that865$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of866$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.867Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in868our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is869non-trivial when $w_p=-1$, so (2) of the proof of Theorem870\ref{local} does not work. This is just as well, since had it871worked we would have expected872$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation873shows not to be the case.874875Here is an example where the divisibility between the levels is876the other way round: a 7-congruence between $g=\nf{122k6A}$ and877$f=\nf{61k6B}$. In this case both $L$-functions have even878functional equation, and we have $L(\nf{122k6A},3)=0$. In the879proof of Theorem 6.1, we find a problem with the local condition880at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to881$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its882kernel is at most one-dimensional, so we still get the883$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having884$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get885(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these886elements of $\Sha$ are {\it invisible} at level 61. When the887levels are different we are no longer able to apply Theorem 2.1 of888\cite{FJ}. However, we still have the congruences of integral889modular symbols required to make the proof of Proposition890\ref{div} go through. Indeed, as noted above, the congruences of891modular forms were found by producing congruences of modular892symbols. Despite these congruences of modular symbols, Remark 5.3893does not apply, since there is no reason to suppose that894$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.895896Finally, there are two examples where we have a form $g$ with even897functional equation such that $L(g,k/2)=0$, and a congruent form898$f$ which has odd functional equation; these are a 23-congruence899between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence900between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If901$\ord_{s=2}L(f,s)=1$, it ought to be the case that902$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and903similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately904the appropriate modification of Theorem \ref{local} does not905necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.906It only tells us that the $\qq$-torsion subgroup of907$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It908could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$909appears in the conjectural formula for the first derivative of the910complex $L$ function, evaluated at $s=k/2$, but in combination911with a regulator that we have no way of calculating.912913Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions914associated with $f$ and $g$ by the construction of Mazur, Tate and915Teitelbaum \cite{MTT}, each divided by a suitable canonical916period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not917quite clear what to make of this. This divisibility may be proved918as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit919times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably920normalised) are congruent $\bmod{\,\qq}$, as a result of the921congruence between the modular symbols out of which they are922constructed. Integrating an appropriate function against these923measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$924to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,925since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case926where the signs in the functional equations of $L(g,s)$ and927$L_q(g,s)$ are the same, positive in this instance. (According to928the proposition in Section 18 of \cite{MTT}, the signs differ929precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)930931932933\subsection{Excluded data}934We also found some examples for which the conditions of Theorem \ref{local}935were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},936but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar937problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here938$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a9395-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5940is less than the weight 6.941942\begin{thebibliography}{AL}943\bibitem[AL]{AL} A. O. L. Atkin, J. 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Math. }{\bf 130 }(1997), 99--152.1037\end{thebibliography}103810391040\end{document}1041127k4A 43 127k4C 17 [43]1042159k4A 5,23 159k4E 8 [5]x[23]1043365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)1044369k4A 5,13 369k4J 9 [5]x[13]x[2]1045453k4A 5,17 453k4E 23 [5]x[17]1046453k4A 23 151k4A 30 Odd func eq for g1047465k4A 11 465k4H 7 [11]x[5]x[2]1048477k4A 73 477k4M 12 [73]x[2]1049567k4A 23 567k4I 8 [23]x[3]105081k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.1051581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)1052639k4A 7 639k4H 12 [7]1053657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)1054657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)1055681k4A 43 227k4A 23 Odd func eq for g1056681k4A 59 681k4D 30 [59]x[3]x[2]1057684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)105895k6A 31,59 95k6D 9 [31]x[59]1059116k6A 5 116k6D 6 [5]x[29]x[2]1060122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)1061122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)1062260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP10631064