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Author: William A. Stein
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\begin{document}
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\title[Shafarevich-Tate Groups of Modular Motives]{Constructing Elements in Shafarevich-Tate Groups of Modular Motives}
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\author{Neil Dummigan}
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\author{William Stein}
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\author{Mark Watkins}
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\date{August 24th, 2002}
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\subjclass{11F33, 11F67, 11G40.}
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\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
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Shafarevich-Tate group.}
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\address{University of Sheffield\\ Department of Pure
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Mathematics\\
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Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
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U.K.}
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\address{Harvard University\\Department of Mathematics\\
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One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
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\address{Penn State Mathematics Department\\
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University Park\\State College, PA 16802\\ U.S.A.}
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\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
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\email{watkins@math.psu.edu}
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\begin{abstract}
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We study Shafarevich-Tate groups of motives attached to modular forms
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on $\Gamma_0(N)$ of weight bigger than~$2$. We deduce a criterion for
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the existence of nontrivial elements of these Shafarevich-Tate groups,
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and give $16$ examples in which the Beilinson-Bloch conjecture implies
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the existence of such elements. We also use modular symbols and
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observations about Tamagawa numbers to compute nontrivial conjectural
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lower bounds on the orders of the Shafarevich-Tate groups of modular
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motives of low level and weight at most $12$. Our methods build upon
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Mazur's idea of visibility, but in the context of motives instead of
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abelian varieties.
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\end{abstract}
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\maketitle
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\section{Introduction}
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Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
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be the associated $L$-function. The conjecture of Birch and
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Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
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at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
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also gives an interpretation of the leading term in the Taylor
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expansion in terms of various quantities, including the order of
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the Shafarevich-Tate group of $E$.
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Cremona and Mazur \cite{CM} look, among all strong Weil elliptic
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curves over $\QQ$ of conductor $N\leq 5500$, at those with
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non-trivial Shafarevich-Tate group (according to the Birch and
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Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
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group has predicted elements of prime order $m$. In most cases
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they find another elliptic curve, often of the same conductor,
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whose $m$-torsion is Galois-isomorphic to that of the first one,
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and which has positive rank. The rational points on the second elliptic
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curve produce classes in the common $H^1(\QQ,E[m])$. They show
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\cite{CM2} that these lie in the Shafarevich-Tate group of the
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first curve, so rational points on one curve explain elements of
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the Shafarevich-Tate group of the other curve.
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The Bloch-Kato conjecture \cite{BK} is the generalisation to
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arbitrary motives of the leading term part of the Birch and
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Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
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\cite{B} generalises the part about the order of vanishing at the
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central point, identifying it with the rank of a certain Chow
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group.
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This paper is a partial generalisation of \cite{CM} and \cite{AS}
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from abelian varieties over $\QQ$ associated to modular forms of
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weight~$2$ to the motives attached to modular forms of higher weight.
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It also does for congruences between modular forms of equal weight
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what \cite{Du2} did for congruences between modular forms of different
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weights.
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We consider the situation where two newforms~$f$ and~$g$, both of
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even weight $k>2$ and level~$N$, are congruent modulo a maximal ideal
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$\qq$ of odd residue characteristic, and $L(g,k/2)=0$ but
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$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
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to order at least $2$ at $s=k/2$. In Section~\ref{sec:examples},
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we give sixteen
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examples (all with $k=4$ and $k=6$), and in each $\qq$ divides the
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numerator of the algebraic number $L(f,k/2)/\vol_{\infty}$, where
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$\vol_{\infty}$ is a certain canonical period.
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In fact, we show how this divisibility may be deduced from the
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vanishing of $L(g,k/2)$ using recent work of Vatsal \cite{V}. The
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point is, the congruence between$f$ and~$g$ leads to a congruence
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between suitable ``algebraic parts'' of the special values
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$L(f,k/2)$ and $L(g,k/2)$. In slightly more detail, a multiplicity
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one result of Faltings and Jordan shows that the congruence of
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Fourier expansions leads to a congruence of certain associated
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cohomology classes. These are then identified with the modular
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symbols which give rise to the algebraic parts of special values.
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If $L(g,k/2)$ vanishes then the congruence implies that
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$L(f,k/2)/\vol_{\infty}$ must be divisible by $\qq$.
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The Bloch-Kato conjecture sometimes implies that the Shafarevich-Tate
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group attached to~$f$ has nonzero $\qq$-torsion. Under certain
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hypotheses and assumptions, the most substantial of which is the
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Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to the
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existence of algebraic cycles, we are able to construct some of the
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predicted elements of~$\Sha$ using the Galois-theoretic
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interpretation of the congruences to transfer elements from a Selmer
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group for~$g$ to a Selmer group for~$f$.
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One might say that algebraic cycles for one motive explain elements
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of~$\Sha$ for the other.
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%In proving the local
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%conditions at primes dividing the level, and also in examining the
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%local Tamagawa factors at these primes, we make use of a higher weight
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%level-lowering result due to Jordan and Livn\'e \cite{JL}.
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We also compute data which, assuming the Bloch-Kato
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conjecture, provides lower bounds for the orders of numerous
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Shafarevich-Tate group (see Section~\ref{sec:invis}).
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%Our data is consistent
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%with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the
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%congruence modulus is necessarily a perfect square (assuming that~$\Sha$
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%is finite).
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\section{Motives and Galois representations}
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Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
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$\Gamma_0(N)$, with coefficients in an algebraic number field~$E$,
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which is necessarily totally real. Let~$\lambda$ be any finite prime
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of~$E$, and let~$\ell$ denote its residue characteristic. A theorem
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of Deligne \cite{De1} implies the existence of a two-dimensional
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vector space $V_{\lambda}$ over $E_{\lambda}$, and a continuous
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representation
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$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
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such that
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\begin{enumerate}
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\item $\rho_{\lambda}$ is unramified at~$p$ for all primes~$p$ not dividing
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$lN$, and
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\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
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characteristic polynomial of $\Frob_p^{-1}$ acting on
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$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
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\end{enumerate}
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Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
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the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
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There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
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both $2$-dimensional $E$-vector spaces. For details of the
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construction see \cite{Sc}. The de Rham realisation has a Hodge
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filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
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F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
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cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
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cohomology.
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For each prime $\lambda$, there is a natural isomorphism
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$V_B\otimes E_{\lambda}\simeq V_{\lambda}$. We may choose a
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
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Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
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There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
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which amounts to multiplying the action of $\Frob_p$ by $p^j$.
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Following \cite{BK} (Section 3), for $p\neq l$ (including
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$p=\infty$) let
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$$
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H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).
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$$
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The subscript~$f$ stands for ``finite
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part'', $D_p$ is a decomposition subgroup at a prime above~$p$,
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$I_p$ is the inertia subgroup, and the cohomology is for
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continuous cocycles and coboundaries. For $p=l$ let
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$$
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H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow
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H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))
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$$
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(see Section 1 of
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\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
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$B_{\dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
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elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
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in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes~$p$.
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There is a natural exact sequence
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$$
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\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.
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$$
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Let
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$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
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Define the $\lambda$-Selmer group
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$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
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$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
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$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes~$p$. Note that the
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condition at $p=\infty$ is superfluous unless $l=2$. Define the
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Shafarevich-Tate group
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$$
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\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/
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\pi_*H^1_f(\QQ,V_{\lambda}(j)).
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$$
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Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
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prime ideal~$\lambda$ is the length of the $\lambda$-component of
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$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
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and write~$\Sha$ for~$\Sha(j)$.
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Define the group of global torsion points
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$$
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\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).
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$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
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which the exponent of any prime ideal~$\lambda$ is the length of
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the $\lambda$-component of $\Gamma_{\QQ}$.
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\section{Canonical periods}
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We assume from now on for convenience that $N\geq 3$. We need to
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choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
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and de Rham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
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in a way such that $T_B$ and $T_{\dR}\otimes_{O_E}O_E[1/Nk!]$
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agree with (respectively) the $O_E$-lattice $\mathfrak{M}_{f,B}$
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and the $O_E[1/Nk!]$-lattice $\mathfrak{M}_{f,\dR}$ defined in
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\cite{DFG} using cohomology, with non-constant coefficients, of
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modular curves. (In \cite{DFG}, see especially Sections 2.2 and
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5.4, and the paragraph preceding Lemma 2.3.)
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For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
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module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
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$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. Then the $O_{\lambda}$ module
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$T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable.
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Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
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generalised elliptic curves with full level-$N$ structure. Let
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$\mathfrak{E}$ be the universal generalised elliptic curve over
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$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
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of $\mathfrak{E}$ over $M(N)$. (The motive $M_f$ is constructed
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using a projector on the cohomology of a desingularisation of
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$\mathfrak{E}^{k-2}$. Realising $M(N)(\CC)$ as the disjoint union
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of $\phi(N)$ copies of the quotient
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$\Gamma(N)\backslash\mathfrak{H}^*$ (where $\mathfrak{H}^*$ is the
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completed upper half plane), and letting $\tau$ be a variable on
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$\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is isomorphic to
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the elliptic curve with period lattice generated by $1$ and
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$\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a variable on
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the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the fibre product.
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Then $2\pi i f(\tau)\,d\tau\wedge dz_1\wedge\ldots\wedge dz_{k-2}$
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is a well-defined differential form on (a desingularisation of)
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$\mathfrak{E}^{k-2}$ and naturally represents a generating element
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of $F^{k-1}T_{\dR}$. (At least we can make our choices locally at
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primes dividing $Nk!$ so that this is the case.) We shall call
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this element $e(f)$.
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Under the de Rham isomorphism between $V_{\dR}\otimes\CC$ and
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$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
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a natural action of complex conjugation on $V_B$, breaking it up
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into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
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Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
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to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
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$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
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are rank one $O_E$-modules, but not necessarily free, since the
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class number of $O_E$ may be greater than one. Choose nonzero
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elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
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the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
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$\Omega_f^{\pm}$ by $\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$.
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\section{The Bloch-Kato conjecture}\label{sec:bkconj}
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Let $L(f,s)$ be the $L$-function attached to~$f$. For
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$\Re(s)>\frac{k+1}{2}$ it is defined by the Dirichlet series with
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Euler product
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$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
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there is an analytic continuation given by an integral, as
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described in the next section. Suppose that $L(f,k/2)\neq 0$. The
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Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
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following equality of fractional ideals of~$E$:
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$$
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\frac{L(f,k/2)}{\vol_{\infty}}=
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\left(\prod_pc_p(k/2)\right)
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\frac{\#\Sha}{\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
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$$
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(Strictly speaking, the conjecture in \cite{BK}
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is only given for $E=\QQ$.) Here, $\pm$
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represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
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to $(2\pi i)^{k/2}$ multiplied by the determinant of the
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isomorphism $V_B^{\pm}\otimes\CC\simeq
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(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
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lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
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$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
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\begin{align*}
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\length&\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
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\ord_{\lambda}(P_p(p^{-j}))\\
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=&\length\>\> \left(H^0(\QQ_p,A_{\lambda}(j))/H^0\left(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
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\end{align*}
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We omit the definition of $\ord_{\lambda}(c_p(j))$ for
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$\lambda\mid p$, which requires one to assume Fontaine's de Rham
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conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
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$T_{\dR}$ and $T_B$, locally at~$\lambda$. (We shall mainly be
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concerned with the $q$-part of the Bloch-Kato conjecture, where~$q$
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is a prime of good reduction. For such primes, the de Rham
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conjecture follows from Theorem 5.6 of \cite{Fa1}.)
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\begin{lem}\label{vol}
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$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
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$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
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\end{lem}
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\begin{proof}
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$\vol_{\infty}$ is also equal to the determinant of the period map
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from $F^{k/2}V_{\dR}\otimes\CC$ to $V_B^{\pm}\otimes\CC$, with
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respect to lattices dual to those we used above in the definition
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of $\vol_{\infty}$ (c.f. the last paragraph of 1.7 of \cite{De2}).
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We are using here natural pairings. Recall that the index of
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$O_E\delta_f^{\pm}$ in $T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then
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the proof is completed by noting that, locally away from primes
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dividing $Nk!$, the index of $T_{\dR}$ in its dual is equal to the
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index of $T_B$ in its dual, both being equal to the ideal
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denoted~$\eta$ in \cite{DFG2}.
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\end{proof}
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\begin{lem} Let $p\nmid N$ be a prime and~$j$ an integer.
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Then the fractional ideal $c_p(j)$ is supported at most on
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divisors of~$p$.
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\end{lem}
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\begin{proof}
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As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
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$\ord_{\lambda}(c_p(j))$ is the length of the finite
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$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
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where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
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trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
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torsion free.
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\end{proof}
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\begin{lem}\label{local1}
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Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
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is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
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$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
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that $\,p\not\equiv -1\pmod{q}$. Suppose also that~$f$ is not congruent
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modulo $\qq$ to any newform of weight~$k$, trivial character and
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level dividing $N/p$. Then any integer~$j$,
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$\ord_{\qq}(c_p(j))=0$.
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\end{lem}
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\begin{proof}
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It suffices to show that
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$$
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\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
427
$$
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since this ensures that
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$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
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$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
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the dimensions differ then, given that $f$ is not congruent modulo
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$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
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\cite{L} shows that we are in the situation covered by one of the
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three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv
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-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
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unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
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result that $N$ is the prime-to-$q$ part of the conductor of
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$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
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the condition $q>k$) implies the existence of a newform of weight
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$k$, trivial character and level dividing $N/p$, congruent to~$g$
441
modulo $\qq$. This contradicts our hypotheses.
442
\end{proof}
443
444
\begin{remar}
445
For an example of what can be done when~$f$ is congruent to
446
a form of lower level, see the first example in Section~\ref{sec:other_ex}
447
below.
448
\end{remar}
449
450
\begin{lem}\label{at q}
451
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
452
$\ord_{\qq}(c_q)=0$.
453
\end{lem}
454
\begin{proof}
455
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
456
application, at the end of Section~2.2, of the results of
457
\cite{Fa1}) that $T_{\qq}$ is the
458
$O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the filtered
459
module $T_{\dR}\otimes O_{\qq}$ by the functor they call
460
$\mathbb{V}$. (This property is part of the definition of an
461
$S$-integral premotivic structure given in Section~1.2 of
462
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
463
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
464
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
465
2(h) of \cite{Fa1}.)
466
\end{proof}
467
468
\begin{lem}
469
If $A[\lambda]$ is an
470
irreducible representation of $\Gal(\Qbar/\QQ)$,
471
then
472
$$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0.$$
473
\end{lem}
474
This follows trivially from the definition.
475
476
Putting together the above lemmas we arrive at the following:
477
\begin{prop}\label{sha}
478
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
479
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
480
Assume the same hypotheses as in Lemma \ref{local1},
481
for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
482
are as in the previous section. If
483
$$
484
\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0
485
$$
486
(with numerator nonzero) then the Bloch-Kato conjecture
487
predicts that
488
$$
489
\ord_{\qq}(\#\Sha)>0.
490
$$
491
\end{prop}
492
493
\section{Congruences of special values}
494
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
495
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
496
large enough to contain all the coefficients $a_n$ and $b_n$.
497
Suppose that $\qq\mid q$ is a prime of~$E$ such that $f\equiv
498
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$, and assume
499
that $q\nmid N\phi(N)k!$.
500
Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
501
$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
502
$T_B^{\pm}$ locally at $\qq$.
503
Make two further assumptions:
504
$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
505
506
\begin{prop} \label{div}
507
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
508
\end{prop}
509
\begin{proof} This is based on some of the ideas used in Section 1 of
510
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
511
which presumably should refer to ``Condition 2''. Since
512
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
513
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
514
1=(-1)^{(k/2)-1}$. It is well known, and easy to prove, that
515
$$
516
\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).
517
$$
518
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
519
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
520
where the integral is taken along the positive imaginary axis,
521
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
522
Thus we are reduced
523
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
524
525
Let $\mathcal{D}_0$ be the group of divisors of degree zero
526
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
527
integer $r\geq 0$, let $P_r(R)$ be the additive group of
528
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
529
groups have a natural action of $\Gamma_1(N)$. Let
530
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
531
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
532
533
Via the isomorphism (8) in Section~1.5 of \cite{V}, combined with
534
the argument in 1.7 of \cite{V}, the cohomology class
535
$\omega_f^{\pm}$ corresponds to a modular symbol $\Phi_f^{\pm}\in
536
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
537
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
538
Inverting $N\phi(N)$ takes into account the fact that we are now
539
dealing with cohomology over $X_1(N)$ rather than $M(N)$. It
540
follows from the last line of Section~4.2 of \cite{St} that, up to
541
some small factorials which do not matter locally at $\qq$,
542
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
543
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
544
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
545
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
546
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
547
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
548
show is divisible by $\qq$.
549
Similarly
550
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
551
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
552
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
553
Therefore it would suffice to show that, for some $\mu\in O_E$,
554
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
555
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
556
that, for some $\mu\in O_E$, the element
557
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
558
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
559
non-constant coefficients. This would be the case if
560
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
561
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
562
a consequence of Theorem 2.1(1) of \cite{FJ}.
563
\end{proof}
564
\begin{remar}\label{sign}
565
The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
566
equal. They are determined by the eigenvalue of the involution $W_N$,
567
which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
568
$b_N$ are each $N^{k/2-1}$ times this sign and~$\qq$ has residue
569
characteristic coprime to $2N$. The common sign in the functional
570
equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
571
$W_N$ acting on~$f$ and~$g$.
572
\end{remar}
573
574
This is analogous to the remark at the end of Section~3 of \cite{CM},
575
which shows that if~$\qq$ has odd residue characteristic and
576
$L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then $L(g,s)$ must vanish to order
577
at least two at $s=k/2$. Note that Maeda's conjecture
578
implies that there are no examples of~$g$ of
579
level one with positive sign in their functional equation such that
580
$L(g,k/2)=0$ (see \cite{CF}).
581
582
\section{Constructing elements of the Shafarevich-Tate group}
583
Let~$f$ and~$g$ be as in the first paragraph of the previous
584
section. For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
585
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
586
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
587
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
588
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
589
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
590
591
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
592
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
593
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
594
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
595
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
596
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
597
rational equivalence classes of null-homologous,
598
algebraic cycles of codimension $k/2$
599
on the motive $M_g$. (This generalises the part
600
of the Birch--Swinnerton-Dyer conjecture which says that for an
601
elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
602
$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
603
604
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
605
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
606
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
607
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
608
get (assuming also the Beilinson-Bloch conjecture) a subspace of
609
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
610
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
611
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
612
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
613
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
614
Sections~1 and~6.5 of \cite{Fo2}.
615
616
Similarly, if $L(f,k/2)\neq 0$ then we expect that
617
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
618
coincides with the $\qq$-part of $\Sha$.
619
\begin{thm}\label{local}
620
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
621
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
622
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
623
that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
624
newform of weight~$k$, trivial character and level dividing $N/p$.
625
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
626
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
627
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
628
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
629
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
630
\end{thm}
631
632
\begin{proof}
633
Take a nonzero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
634
continuity and rescaling we may assume that it lies in
635
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
636
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
637
$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
638
nonzero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq
639
H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
640
is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
641
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
642
we get a nonzero, $\qq$-torsion class $\gamma\in
643
H^1(\QQ,A_{\qq}(k/2))$.
644
645
Our aim is to show that $\res_p(\gamma)\in
646
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
647
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
648
649
\begin{enumerate}
650
\item {\bf $p\nmid qN$. }
651
652
Consider the $I_p$-cohomology of the short exact sequence
653
$$
654
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
655
$$
656
where~$\pi$ is multiplication by a uniformising element of
657
$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
658
$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
659
$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
660
remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
661
$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in
662
H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
663
$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
664
that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq
665
H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
666
to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
667
\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
668
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
669
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
670
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
671
672
\item {\bf $p\mid N$. }
673
674
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
675
It suffices to show that
676
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
677
since then the natural map from
678
$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
679
surjective; this may be done as in the proof of Lemma
680
\ref{local1}. It follows as above that the image of $c\in
681
H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
682
$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
683
inflation-restriction. The order of this group is the same as the
684
order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
685
trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
686
$V_{\qq}(k/2)$ is ramified at $p$, so $\dim
687
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
688
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
689
consider the case where this common dimension is $1$. The
690
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
691
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by~$\alpha$
692
on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
693
follows from Theor\'eme A of \cite{Ca1} that this is the same as
694
the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
695
Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
696
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
697
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
698
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
699
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
700
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
701
$\res_p(\gamma)=0$ and certainly lies in
702
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
703
704
\item {\bf $p=q$. }
705
706
Since $q\nmid N$ is a prime of good reduction for the motive
707
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
708
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
709
$V'_{\qq}$ have the same dimension, where
710
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
711
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
712
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
713
the $O_{\qq}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
714
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
715
may now prove, in the same manner as Proposition 9.2 of
716
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
717
\end{enumerate}
718
\end{proof}
719
720
Theorem~2.7 of \cite{AS} is concerned with verifying local
721
conditions in the case $k=2$, where~$f$ and~$g$ are associated
722
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
723
abelian varieties over number fields.) Our restriction outlawing
724
congruences modulo $\qq$ with cusp forms of lower level is
725
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
726
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
727
curve with $\ord_l(j(A))<0$, consideration of a Tate
728
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
729
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
730
at~$l$.)
731
732
In this paper we have encountered two technical problems which we
733
dealt with in quite similar ways:
734
\begin{enumerate}
735
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
736
\item proving local conditions at primes $p\mid N$, for an element
737
of $\qq$-torsion.
738
\end{enumerate}
739
If our only interest was in testing the Bloch-Kato conjecture at
740
$\qq$, we could have made these problems cancel out, as in Lemma
741
8.11 of \cite{DFG}, by weakening the local conditions. However, we
742
have chosen not to do so, since we are also interested in the
743
Shafarevich-Tate group, and since the hypotheses we had to assume
744
are not particularly strong. Note that, since $A[\qq]$ is
745
irreducible, the $\qq$-part of $\Sha$ does not depend on the
746
choice of $T_{\qq}$.
747
748
\section{Examples and Experiments}
749
\label{sec:examples}
750
This section contains tables and numerical examples that illustrate
751
the main themes of this paper. In Section~\ref{sec:vistable}, we
752
explain Table~\ref{tab:newforms}, which contains~$16$ examples of
753
pairs $f,g$ such that the Beilinson-Bloch conjecture and
754
Theorem~\ref{local} together imply the existence of nontrivial
755
elements of the
756
Shafarevich-Tate group of the motive attached to~$f$.
757
Section~\ref{sec:howdone} outlines the higher-weight modules symbols
758
computations that were used in making Table~\ref{tab:newforms}.
759
Section~\ref{sec:invis} discusses Table~\ref{tab:invisforms},
760
which summarizes the results
761
of an extensive computation of conjectural orders of Shafarevich-Tate
762
groups for modular motives of low level and weight.
763
Section~\ref{sec:other_ex} gives specific examples in which various
764
hypotheses fail.
765
766
\subsection{Visible $\Sha$ Table~\ref{tab:newforms}}\label{sec:vistable}
767
\begin{table}
768
\caption{\label{tab:newforms}Visible $\Sha$}\vspace{-3ex}
769
770
$$
771
\begin{array}{|c|c|c|c|c|}\hline
772
g & \deg(g) & f & \deg(f) & q's \\\hline
773
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
774
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
775
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
776
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
777
\vspace{-2ex} & & & & \\
778
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
779
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
780
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
781
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
782
\vspace{-2ex} & & & & \\
783
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
784
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
785
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
786
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
787
\vspace{-2ex} & & & & \\
788
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
789
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
790
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
791
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
792
\hline
793
\end{array}
794
$$
795
\end{table}
796
797
798
Table~\ref{tab:newforms} on page~\pageref{tab:newforms} lists
799
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
800
along with at least one prime~$q$ such that there is a prime
801
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
802
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
803
The notation is as follows.
804
The first column contains a label whose structure is
805
\begin{center}
806
{\bf [Level]k[Weight][GaloisOrbit]}
807
\end{center}
808
This label determines a newform $g=\sum a_n q^n$, up to Galois
809
conjugacy. For example, \nf{127k4C} denotes a newform in the third
810
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
811
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
812
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
813
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
814
being first in the event that the two absolute values are equal,
815
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
816
and so on. The second column contains the degree of the field
817
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
818
contain~$f$ and its degree, respectively. The fifth column
819
contains at least one prime~$q$ such that there is a prime
820
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
821
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
822
satisfied for~$f$,~$g$, and~$\qq$.
823
824
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
825
prime $q$ appears in the $q$-column, meaning $q^2$ divides the
826
order of the group $S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp})$, defined
827
at the end of 7.3 below.
828
829
830
We describe the first line of Table~\ref{tab:newforms}
831
in more detail. See the next section for further details
832
on how the computations were performed.
833
834
Using modular symbols, we find that there is a newform
835
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
836
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
837
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
838
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
839
coefficients generate a number field~$K$ of degree~$17$, and by
840
computing the image of the modular symbol $XY\{0,\infty\}$ under
841
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
842
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
843
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
844
both equal to
845
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
846
+ \cdots\in \FF_{43}[[q]].$$
847
848
There is no form in the Eisenstein subspaces of
849
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
850
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
851
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
852
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
853
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
854
of Theorem~\ref{local}, so if $r$ is the dimension of
855
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
856
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
857
858
Recall that since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that
859
$r\geq 2$. Then, since $L(f,k/2)\neq 0$, we expect that the
860
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to
861
the $\qq$-torsion subgroup of $\Sha$. Admitting these assumptions,
862
we have constructed the $\qq$-torsion in $\Sha$ predicted by the
863
Bloch-Kato conjecture.
864
865
For particular examples of elliptic curves one can often find and
866
write down rational points predicted by the Birch and
867
Swinnerton-Dyer conjecture. It would be nice if likewise one could
868
explicitly produce algebraic cycles predicted by the
869
Beilinson-Bloch conjecture in the above examples. Since
870
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
871
0.3.2 of \cite{Z}), so ought to be trivial in
872
$\CH_0^{k/2}(M_g)\otimes\QQ$.
873
874
\subsection{How the computation was performed}\label{sec:howdone}
875
We give a brief summary of how the computation was performed. The
876
algorithms that we used were implemented by the second author, and
877
most are a standard part of MAGMA (see \cite{magma}).
878
879
Let~$g$,~$f$, and~$q$ be some data from a line of
880
Table~\ref{tab:newforms} and let~$N$ denote the level of~$g$. We
881
verified the existence of a congruence modulo~$q$, that
882
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
883
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
884
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
885
886
To prove there is a congruence, we showed that the corresponding
887
{\em integral} spaces of modular symbols satisfy an appropriate
888
congruence, which forces the existence of a congruence on the
889
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
890
irreducible by computing a set that contains all possible residue
891
characteristics of congruences between~$g$ and any Eisenstein
892
series of level dividing~$N$, where by congruence, we mean a
893
congruence for all Fourier coefficients of index~$n$ with
894
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
895
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
896
listing a basis of such~$h$ and finding the possible congruences,
897
where again we disregard the Fourier coefficients of index not
898
coprime to~$N$.
899
900
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
901
modular symbol ${\mathbf
902
e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
903
with the same kernel as the period mapping, and found that the
904
image was~$0$. The period mapping sends the modular
905
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
906
so that ${\mathbf e}$ maps to~$0$ implies that
907
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
908
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
909
=(-1)^{k/2} g$ which, because of the functional equation, implies
910
that $L'(g,\frac{k}{2})=0$. Table~\ref{tab:newforms} is of
911
independent interest because it includes examples of modular forms
912
of even weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
913
the functional equation. We found no such examples of weights
914
$\geq 8$.
915
916
\subsection{Conjecturally nontrivial $\Sha$}\label{sec:invis}
917
In this section we apply some of the results of
918
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
919
of Shafarevich-Tate groups of many modular motives. The results of
920
this section suggest that~$\Sha$ of a modular motive is usually not
921
``visible at level~$N$'', i.e., explained by congruences at level~$N$,
922
which agrees with the observations of \cite{CM} and \cite{AS}. For
923
example, when $k>6$ we find many examples of conjecturally
924
nontrivial~$\Sha$ but no examples of nontrivial visible~$\Sha$.
925
926
For any newform~$f$, let $L(M_f,s) = \prod_{i=1}^{d} L(f^{(i)},s)$
927
where $f^{(i)}$ runs over the $\Gal(\Qbar/\QQ)$-conjugates of~$f$.
928
Let~$T$ be the complex torus
929
$\CC^d/\mathcal{L}$, where the lattice $\mathcal{L}$ is defined by
930
integrating integral cuspidal modular symbols against the conjugates
931
of~$f$. Let $\Omega_{M_f}$ denote the volume of the $-1$
932
eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for complex
933
conjugation on~$T$.
934
935
936
{\begin{table}
937
\vspace{-2ex}
938
\caption{\label{tab:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
939
\vspace{-4ex}
940
941
$$
942
\begin{array}{|c|c|c|c|}\hline
943
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
944
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
945
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
946
\nf{263k4B} & 39 & 41^{2} & 263 \\
947
\nf{269k4C} & 39 & 23^{2} & 269 \\
948
\nf{271k4B} & 39 & 29^{2} & 271 \\
949
\nf{281k4B} & 40 & 29^{2} & 281 \\
950
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
951
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
952
\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
953
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
954
\hline
955
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
956
\nf{101k6B} & 24 & 17^{2} & 101 \\
957
\nf{103k6B} & 24 & 23^{2} & 103 \\
958
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
959
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
960
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
961
\nf{157k6B} & 34 & 251^{2} & 157 \\
962
\nf{167k6B} & 40 & 41^{2} & 167 \\
963
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
964
\nf{173k6B} & 39 & 71^{2} & 173 \\
965
\nf{181k6B} & 40 & 107^{2} & 181 \\
966
\nf{191k6B} & 46 & 85091^{2} & 191 \\
967
\nf{193k6B} & 41 & 31^{2} & 193 \\
968
\nf{199k6B} & 46 & 200329^2 & 199 \\
969
\hline
970
\nf{47k8B} & 16 & 19^{2} & 47 \\
971
\nf{59k8B} & 20 & 29^{2} & 59 \\
972
\nf{67k8B} & 20 & 29^{2} & 67 \\
973
\nf{71k8B} & 24 & 379^{2} & 71 \\
974
\nf{73k8B} & 22 & 197^{2} & 73 \\
975
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
976
\nf{79k8B} & 25 & 307^{2} & 79 \\
977
\nf{83k8B} & 27 & 1019^{2} & 83 \\
978
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
979
\nf{89k8B} & 29 & 44491^{2} & 89 \\
980
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
981
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
982
\nf{103k8B} & 32 & 75367^{2} & 103 \\
983
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
984
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
985
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
986
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
987
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
988
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
989
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
990
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
991
\hline
992
\end{array}
993
$$
994
\end{table}
995
\begin{table}
996
$$
997
\begin{array}{|c|c|c|c|}\hline
998
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
999
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
1000
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
1001
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
1002
\nf{125k8D} & 16 & 179^{2} & 5 \\
1003
\nf{127k8B} & 39 & 59^{2} & 127 \\
1004
\nf{128k8F} & 4 & 11^{2} & 1 \\
1005
\nf{131k8B} & 43 & 241^{2} \!\cdot\! 817838201^{2}&131\\
1006
\nf{134k8C} & 11 & 61^{2} & 11, 17, 41, 67, 71, 421, 2356138931854759 \\
1007
\nf{137k8B} & 42 & 71^{2} \!\cdot\! 749093^{2} & 137 \\
1008
\nf{139k8B} & 43 & 47^{2} \!\cdot\! 89^{2} \!\cdot\! 1021^{2} & 139 \\
1009
\nf{141k8C} & 14 & 13^{2} & 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
1010
\nf{142k8B} & 10 & 11^{2} & 3, 53, 71, 56377, 1965431024315921873 \\
1011
\nf{143k8C} & 19 & 307^{2} & 3, 11, 13, 89, 199, 409, 178397, 639259, 17440535
1012
97287 \\
1013
\nf{143k8D} & 21 & 109^{2} & 3, 7, 11, 13, 61, 79, 103, 173, 241, 769, 36583
1014
\\
1015
\nf{145k8C} & 17 & 29587^{2} & 5, 11, 29, 107, 251623, 393577, 518737, 9837145
1016
699 \\
1017
\nf{146k8C} & 12 & 3691^{2} & 11, 73, 269, 503, 1673540153, 11374452082219 \\
1018
\nf{148k8B} & 11 & 19^{2} & 3, 37 \\
1019
\nf{149k8B} & 47 & 11^{4} \!\cdot\! 40996789^{2} & 149\\
1020
\hline
1021
1022
\nf{43k10B} & 17 & 449^{2} & 43 \\
1023
\nf{47k10B} & 20 & 2213^{2} & 47 \\
1024
\nf{53k10B} & 21 & 673^{2} & 53 \\
1025
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
1026
\nf{59k10B} & 25 & 37^{2} & 59 \\
1027
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
1028
\nf{64k10K} & 2 & 19^{2} & 3 \\
1029
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
1030
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
1031
\nf{71k10B} & 30 & 1103^{2} & 71 \\
1032
1033
\hline
1034
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
1035
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
1036
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
1037
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
1038
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
1039
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
1040
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
1041
\nf{49k12H} & 12 & 271^{2} & 7 \\
1042
\hline
1043
\end{array}
1044
$$
1045
\end{table}
1046
1047
The following conjecture is probably not difficult to prove, but we
1048
haven't given a proof, so we formally state it as a conjecture, then
1049
assume it.
1050
\begin{conj}\label{conj:lrat}
1051
If $p\nmid Nk!$ is a prime then the $p$-parts of
1052
$$
1053
\frac{L(M_f,k/2)}{\Omega_{M_f}}\qquad\text{and}\qquad
1054
\Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right)
1055
$$
1056
are equal, where $\vol_\infty$ is as in Section~\ref{sec:bkconj}.
1057
\end{conj}
1058
For the rest of this section, {\em we officially assume the Bloch-Kato
1059
conjecture and Conjecture~\ref{conj:lrat}.}
1060
1061
Let~$\mathcal{S}$ be the set of newforms with~level $N$ and weight~$k$
1062
satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1063
$k=8$ and $N\leq 133$, or $k=10$ and $N\leq 72$, or $k=12$ and $N\leq
1064
49$. Given $f\in \mathcal{S}$, let~$B$ be the lower bound on $\#\Sha$ defined
1065
as follows:
1066
\begin{enumerate}
1067
\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1068
If $L_1=0$ let $B=1$ and terminate.
1069
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
1070
\item Let $L_3$ be the part of $L_2$ that is coprime to
1071
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1072
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
1073
of any prime of
1074
congruence between~$f$ and a form of weight~$k$ and
1075
lower level. (By congruence here, we mean a congruence for coefficients
1076
$a_n$ with $n$ coprime to the level of~$f$.)
1077
\item Let $B$ be the part of $L_4$ coprime to the residue characteristic
1078
of any prime of congruence
1079
between~$f$ and an Eisenstein series. (This eliminates
1080
residue characteristics of reducible representations.)
1081
\end{enumerate}
1082
Proposition~\ref{sha} implies that if
1083
$\ord_p(B) > 0$, then $\ord_p(\#\Sha) > 0$.
1084
1085
We computed~$B$ for every newform in~$\mathcal{S}$. There are many
1086
examples in which $L_3$ is large, but~$B$ is not, and this is because
1087
of Tamagawa factors. For example, {\bf 39k4C} has $L_3=19$, but $B=1$
1088
because of a $19$-congruence with a form of level~$13$; in this case
1089
we must have $19\mid c_{13}(2)$, where $c_{13}(2)$ is as in
1090
Section~\ref{sec:bkconj}. See Section~\ref{sec:other_ex} for more
1091
details. Also note that in every example~$B$ is a perfect square,
1092
which is consistent with the fact \cite{Fl2} that the part of the
1093
order of $\Sha$ coprime to the modular degree is necessarily a perfect
1094
square. That our computed value of~$B$ should be a square is not a
1095
priori obvious.
1096
1097
For simplicity, we discard residue characteristics instead of primes
1098
of rings of integers, so our definition of~$B$ is overly conservative.
1099
For example,~$5$ occurs in row~$2$ of Table~\ref{tab:newforms} but not
1100
in Table~\ref{tab:invisforms}, because \nf{159k4E} is Eisenstein at
1101
some prime above~$5$, but the prime of congruences of
1102
characteristic~$5$ between \nf{159k4B} and \nf{159k4E} is not
1103
Eisenstein.
1104
1105
1106
The newforms for which $B>1$ are given in
1107
Table~\ref{tab:invisforms}. The second column of the table records
1108
the degree of the field generated by the Fourier coefficients
1109
of~$f$. The third contains~$B$. Let~$W$ be the intersection of
1110
the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
1111
$W^{\perp}$ the Petersson orthogonal complement of~$W$ in
1112
$S_k(\Gamma_0(N),\ZZ)$. Then the fourth column contains the odd
1113
prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^{\perp}))$, which
1114
are exactly the possible primes of congruence for~$f$. We place a
1115
$*$ next to the four entries of Table~\ref{tab:invisforms} that
1116
also occur in Table~\ref{tab:newforms}.
1117
1118
\subsection{Examples in which hypotheses fail}\label{sec:other_ex}
1119
We have some other examples where forms of
1120
different levels are congruent.
1121
However, Remark~\ref{sign} does not
1122
apply, so that one of the forms could have an odd functional
1123
equation, and the other could have an even functional equation.
1124
For instance, we have a $19$-congruence between the
1125
newforms $g=\nf{13k4A}$ and $f=\nf{39k4C}$ of Fourier
1126
coefficients coprime to $39$.
1127
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
1128
has {\it odd} functional equation.
1129
Here~$f$ fails the condition about not being congruent
1130
to a form of lower level, so in Lemma~\ref{local1} it is possible that
1131
$\ord_{\qq}(c_{19}(2))>0$. In fact this does happen. Because
1132
$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=19$,
1133
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1134
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1135
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1136
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1137
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1138
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1139
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1140
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1141
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1142
our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1143
non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1144
does not work. This is just as well, since had it
1145
worked we would have expected
1146
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1147
shows not to be the case.
1148
1149
In the following example, the divisibility between the levels is the
1150
other way round. There is a $7$-congruence between $g=\nf{122k6A}$
1151
and $f=\nf{61k6B}$, both $L$-functions have even functional equation,
1152
and $L(g,3)=0$. In the proof of Theorem~\ref{local},
1153
there is a problem with the local condition at $p=2$. The map from
1154
$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily
1155
injective, but its kernel is at most one dimensional, so we still get
1156
the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1157
$\FF_{\qq}$-rank at least~$1$ (assuming $r\geq 2$), and thus get
1158
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1159
elements of $\Sha$ are {\it invisible} at level 61. When the levels
1160
are different we are no longer able to apply Theorem 2.1 of
1161
\cite{FJ}. However, we still have the congruences of integral modular
1162
symbols required to make the proof of Proposition \ref{div} go
1163
through. Indeed, as noted above, the congruences of modular forms were
1164
found by producing congruences of modular symbols. Despite these
1165
congruences of modular symbols, Remark 5.3 does not apply, since there
1166
is no reason to suppose that $w_N=w_{N'}$, where $N$ and $N'$ are the
1167
distinct levels.
1168
1169
Finally, there are two examples where we have a form $g$ with even
1170
functional equation such that $L(g,k/2)=0$, and a congruent form
1171
$f$ which has odd functional equation; these are a 23-congruence
1172
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1173
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1174
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1175
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1176
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1177
the appropriate modification of Theorem \ref{local} does not
1178
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1179
It only tells us that the $\qq$-torsion subgroup of
1180
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1181
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1182
appears in the conjectural formula for the first derivative of the
1183
complex $L$ function, evaluated at $s=k/2$, but in combination
1184
with a regulator that we have no way of calculating.
1185
1186
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1187
associated with $f$ and $g$ by the construction of Mazur, Tate and
1188
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1189
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1190
quite clear what to make of this. This divisibility may be proved
1191
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1192
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1193
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1194
congruence between the modular symbols out of which they are
1195
constructed. Integrating an appropriate function against these
1196
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1197
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1198
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1199
where the signs in the functional equations of $L(g,s)$ and
1200
$L_q(g,s)$ are the same, positive in this instance. (According to
1201
the proposition in Section 18 of \cite{MTT}, the signs differ
1202
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1203
1204
We also found some examples for which the conditions of Theorem~\ref{local}
1205
were not met. For example, we have a $7$-congruence between
1206
\nf{639k4B} and \nf{639k4H}, but $w_{71}=-1$, so that $71\equiv
1207
-w_{71}\pmod{7}$. There is a similar problem with a $7$-congruence
1208
between \nf{260k6A} and \nf{260k6E} --- here $w_{13}=1$ so that
1209
$13\equiv -w_{13}\pmod{7}$. Finally, there is a $5$-congruence between
1210
\nf{116k6A} and \nf{116k6D}, but here the prime~$5$ is less than the
1211
weight~$6$.
1212
1213
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1214
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1300
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1302
1303
1304
\end{document}
1305
127k4A 43 127k4C 17 [43]
1306
159k4A 5,23 159k4E 8 [5]x[23]
1307
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1308
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1309
453k4A 5,17 453k4E 23 [5]x[17]
1310
453k4A 23 151k4A 30 Odd func eq for g
1311
465k4A 11 465k4H 7 [11]x[5]x[2]
1312
477k4A 73 477k4M 12 [73]x[2]
1313
567k4A 23 567k4I 8 [23]x[3]
1314
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1315
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1316
639k4A 7 639k4H 12 [7]
1317
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1318
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1319
681k4A 43 227k4A 23 Odd func eq for g
1320
681k4A 59 681k4D 30 [59]x[3]x[2]
1321
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1322
95k6A 31,59 95k6D 9 [31]x[59]
1323
116k6A 5 116k6D 6 [5]x[29]x[2]
1324
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1325
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1326
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
\nf{263k4B} & & 41^2 & \\
1340
\nf{269k4C} & & 23^2 & \\
1341
\nf{271k4B} & & 29^2 &\\
1342
\nf{281k4B} & & 29^2\\
1343
\hline
1344
\nf{101k6B} & & 17^2 & 101\\
1345
\nf{103k6B} & & 23^2\\
1346
\nf{111k6C} & & 11^2\\
1347
\nf{153k6G} & & 7^2\\
1348
\nf{157k6B} & & 252^2\\
1349
\nf{167k6B} & & 41^2\\
1350
\nf{172k6B} & & 7^2\\
1351
\nf{173k6B} & & 71^2\\
1352
\nf{181k6B} & & 107^2\\
1353
\hline
1354
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1355