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Author: William A. Stein
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\begin{document}
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\title{Constructing Elements in
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Shafarevich-Tate Groups of Modular Motive}
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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{May 29th, 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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\maketitle {\bf Not for distribution}
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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 [2000] 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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The present work may be considered as a partial generalisation of
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the work of Cremona and Mazur, from elliptic curves over $\QQ$
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(which are associated to modular forms of weight $2$) to the
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motives attached to modular forms of higher weight. (See \cite{AS}
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for a different generalisation, to modular abelian varieties of
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higher dimension.) It may also be regarded as doing, for
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congruences between modular forms of equal weight, what \cite{Du2}
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did for congruences between modular forms of different weights.
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We consider the situation where two newforms~$f$ and~$g$, both of
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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 $s=k/2$. We are able to find sixteen
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examples (all with $k=4$ and $k=6$), and in each $\qq$
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divides the numerator of the algebraic number
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$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
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canonical period. In fact, we show how this divisibility may be
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deduced from the vanishing of $L(g,k/2)$ using recent work of
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Vatsal \cite{V}. The point is the congruence between$f$ and~$g$
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leads to a congruence between suitable ``algebraic parts'' of the
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special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
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other is divisible by $\qq$. Under certain hypotheses, the
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Bloch-Kato conjecture then implies that the Shafarevich-Tate group
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attached to~$f$ has non-zero $\qq$-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
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the existence of algebraic cycles, we are able to construct 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
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Selmer group for~$g$ to a Selmer group for~$f$. In proving the
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local conditions at primes dividing the level, and also in
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examining the local Tamagawa factors at these primes, we make use
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of a higher weight level-lowering result due to Jordan and Livn\'e
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\cite{JL}.
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One might say that algebraic cycles for one motive explain
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elements of~$\Sha$ for the other. A main point of \cite{CM} was to
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observe the frequency with which those elements of~$\Sha$
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predicted to exist for one elliptic curve may be explained by
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finding a congruence with another elliptic curve containing points
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of infinite order. One shortcoming of our work, compared to the
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elliptic curve case, is that, due to difficulties with local
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factors in the Bloch-Kato conjecture, we are unable to compute the
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exact order of~$\Sha$ predicted by the Bloch-Kato conjecture. We
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have to start with modular forms between which there exists a
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congruence. However, Vatsal's work allows us to explain how the
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vanishing of one $L$-function leads, via the congruence, to the
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divisibility by~$\qq$ of (an algebraic part of) another,
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independent of observations of computational data. The
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computational data does however show that there exist examples to
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which our results apply. Moreover, it displays factors of $\qq^2$,
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whose existence we do not prove theoretically, but which are
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predicted by Bloch-Kato.
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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. A theorem of Deligne \cite{De1}
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implies the existence, for each (finite) prime~$\lambda$ of~$E$,
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of a two-dimensional vector space $V_{\lambda}$ over
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$E_{\lambda}$, and a continuous 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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$lN$ (where $\lambda \mid l$);
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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. There are natural isomorphisms $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. 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 deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
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in a way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and
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$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the
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$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and
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$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially
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Sections 2.2 and 5.4 of \cite{DFG}.)
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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 for $\lambda\nmid Nk!$, the
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$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
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since it comes from $\ell$-adic cohomology with $O_{\lambda}$
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coefficients. We may assume that $T_{\lambda}$ is
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$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting
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$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.
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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 $M(N)$. Realising $M(N)(\CC)$ as the
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disjoint union of $\phi(N)$ copies of the quotient
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$\Gamma(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a
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variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is
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isomorphic to the elliptic curve with period lattice generated by
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$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a
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variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the
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fibre product. Then $2\pi i f(\tau)\,d\tau\wedge
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dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential
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form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and
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naturally represents a generating element of $F^{k-1}T_{\dR}$. (At
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least, we can make our choices locally at primes dividing $Nk!$ so
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that this is the case.) We shall call this element $e(f)$.
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Under the deRham 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 non-zero
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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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$$
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{L(f,k/2)\over \vol_{\infty}}=
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{\left(\prod_pc_p(k/2)\right)\#\Sha\over
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\aaa^{\pm}(\#\Gamma_{\QQ})^2}.
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$$
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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
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$q$ 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
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of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to
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$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
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used above in the definition of $\vol_{\infty}$ (c.f. the last
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paragraph of 1.7 of \cite{De2}). We are using here the natural
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pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
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$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
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by noting that, locally away from primes dividing $Nk!$, the index
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of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
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dual, both being equal to the ideal 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 for~$j$ any integer,
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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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$$
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\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),
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$$
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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
425
the condition $q>k$) implies the existence of a newform of weight
426
$k$, trivial character and level dividing $N/p$, congruent to~$g$
427
modulo $\qq$. This contradicts our hypotheses.
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\end{proof}
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\begin{remar}
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For an example of what can be done when~$f$ {\em is } congruent to
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a form of lower level, see the first example in Section~\ref{sec:other_ex}
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below.
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\end{remar}
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\begin{lem}\label{at q}
437
If $\qq\mid q$ is a prime of~$E$ such that $q\nmid Nk!$, then
438
$\ord_{\qq}(c_q)=0$.
439
\end{lem}
440
\begin{proof}
441
It follows from Lemma~5.7 of \cite{DFG} (whose proof relies on an
442
application, at the end of Section~2.2, of the results of
443
\cite{Fa1}) that $T_{\qq}$ is
444
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
445
filtered module $T_{\dR}\otimes O_{\qq}$ by the functor they call
446
$\mathbb{V}$. (This property is part of the definition of an
447
$S$-integral premotivic structure given in Section~1.2 of
448
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
449
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
450
in Theorem~4.1 of \cite{BK} follows from the first paragraph of
451
2(h) of \cite{Fa1}.)
452
\end{proof}
453
454
\begin{lem}
455
If $A[\lambda]$ is an
456
irreducible representation of $\Gal(\Qbar/\QQ)$,
457
then
458
$\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$.
459
\end{lem}
460
This follows trivially from the definition.
461
462
Putting together the above lemmas we arrive at the following:
463
\begin{prop}\label{sha}
464
Let $q\nmid N$ be a prime satisfying $q>k$ and suppose that $A[\qq]$
465
is an irreducible representation of $\Gal(\Qbar/\QQ)$, where $\qq\mid q$.
466
Assume the same hypotheses as in Lemma \ref{local1},
467
for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
468
are as in the previous section. If
469
$$
470
\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0
471
$$
472
(with numerator non-zero) then the Bloch-Kato conjecture
473
predicts that
474
$$
475
\ord_{\qq}(\#\Sha)>0.
476
$$
477
\end{prop}
478
479
\section{Congruences of special values}
480
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
481
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
482
large enough to contain all the coefficients $a_n$ and $b_n$.
483
Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
484
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
485
that $q\nmid N\phi(N)k!$ It is easy to see that we may choose the
486
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
487
$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
488
$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
489
has been made.
490
491
We shall now make two further assumptions:
492
\begin{enumerate}
493
\item $L(f,k/2)\neq 0$;
494
\item $L(g,k/2)=0$.
495
\end{enumerate}
496
\begin{prop} \label{div}
497
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
498
\end{prop}
499
\begin{proof} This is based on some of the ideas used in Section 1 of
500
\cite{V}. Note the apparent typo in Theorem~1.13 of \cite{V},
501
which presumably should refer to ``Condition 2''. Since
502
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
503
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
504
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
505
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
506
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
507
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
508
where the integral is taken along the positive imaginary axis,
509
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
510
Thus we are reduced
511
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
512
513
Let $\mathcal{D}_0$ be the group of divisors of degree zero
514
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
515
integer $r\geq 0$, let $P_r(R)$ be the additive group of
516
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
517
groups have a natural action of $\Gamma_1(N)$. Let
518
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
519
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
520
521
Via the isomorphism (8) in Section~1.5 of \cite{V},
522
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
523
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
524
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
525
(See also Section~4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
526
into account the fact that we are now dealing with $X_1(N)$ rather
527
that $M(N)$. Up to some small factorials which do not matter
528
locally at $\qq$,
529
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
530
(k/2)-1\pmod{2}}^{k-2} r_f(j)X^jY^{k-2-j}.$$ Since
531
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
532
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
533
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
534
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
535
show is divisible by $\qq$.
536
Similarly
537
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
538
(k/2)-1\pmod{2}}^{k-2} r_g(j)X^jY^{k-2-j}.$$ The coefficient of
539
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
540
Therefore it would suffice to show that, for some $\mu\in O_E$,
541
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
542
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
543
that, for some $\mu\in O_E$, the element
544
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
545
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
546
non-constant coefficients. This would be the case if
547
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
548
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
549
a consequence of Theorem 2.1(1) of \cite{FJ}.
550
\end{proof}
551
\begin{remar}
552
By Proposition~\ref{sha} (assuming, for all $p\mid N$ the same
553
hypotheses as in Lemma~\ref{local1}, together with
554
$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
555
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
556
construction of the required elements of $\Sha$.
557
\end{remar}
558
\begin{remar}\label{sign}
559
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
560
have to be equal, since they are determined by the action of the
561
involution $W_N$ on the common subspace generated by the
562
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
563
Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the
564
common eigenvalue of $W_N$ acting on~$f$ and~$g$.
565
\end{remar}
566
This is analogous to the remark at the end of Section~3 of
567
\cite{CM}, which shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$
568
then $L(g,s)$ must vanish to order at least two, as in all the
569
examples below. It is worth pointing out that there are no
570
examples of $g$ of level one, and positive sign in the functional
571
equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that
572
all the normalised cuspidal eigenforms of weight~$k$ and level one
573
are Galois conjugate) is false. See \cite{CF}.
574
575
\section{Constructing elements of the Shafarevich-Tate group}
576
Let~$f$ and~$g$ be as in the first paragraph of the previous
577
section. For~$f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
578
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
579
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
580
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
581
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
582
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
583
584
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
585
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
586
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
587
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
588
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
589
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
590
rational equivalence classes of null-homologous, codimension $k/2$
591
algebraic cycles on the motive $M_g$. (This generalises the part
592
of the Birch--Swinnerton-Dyer conjecture which says that for an
593
elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
594
$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
595
596
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
597
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
598
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
599
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
600
get (assuming also the Beilinson-Bloch conjecture) a subspace of
601
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
602
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
603
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
604
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
605
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
606
Sections~1 and~6.5 of \cite{Fo2}.
607
608
Similarly, if $L(f,k/2)\neq 0$ then we expect that
609
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
610
coincides with the $\qq$-part of $\Sha$.
611
\begin{thm}\label{local}
612
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
613
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
614
$A[\qq]$ is an irreducible representation of $\Gal(\Qbar/\QQ)$ and
615
that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
616
newform of weight~$k$, trivial character and level dividing $N/p$.
617
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
618
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
619
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
620
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
621
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
622
\end{thm}
623
624
\begin{proof}
625
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
626
continuity and rescaling we may assume that it lies in
627
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
628
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
629
$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
630
non-zero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq
631
H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
632
is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
633
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
634
we get a non-zero, $\qq$-torsion class $\gamma\in
635
H^1(\QQ,A_{\qq}(k/2))$.
636
637
Our aim is to show that $\res_p(\gamma)\in
638
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
639
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
640
641
\begin{enumerate}
642
\item {\bf $p\nmid qN$. }
643
644
Consider the $I_p$-cohomology of the short exact sequence
645
$$
646
\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},
647
$$
648
where~$\pi$ is multiplication by a uniformising element of
649
$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
650
$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
651
$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
652
remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
653
$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in
654
H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
655
$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
656
that the restriction of~$c$ (to $H^1(I_p,A'[\qq](k/2))\simeq
657
H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
658
to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line~3 of p.~125 of
659
\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
660
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
661
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
662
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
663
664
\item {\bf $p\mid N$. }
665
666
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
667
It suffices to show that
668
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
669
since this would imply that the natural map from
670
$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
671
surjective, but this may be done as in the proof of Lemma
672
\ref{local1}. It follows as above that the image of $c\in
673
H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
674
$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
675
inflation-restriction. The order of this group is the same as the
676
order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
677
trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
678
$V_{\qq}(k/2)$ is ramified at $p$, so $\dim
679
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
680
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
681
consider the case where this common dimension is $1$. The
682
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
683
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by
684
$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
685
follows from Theor\'eme A of \cite{Ca1} that this is the same as
686
the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
687
Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
688
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
689
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
690
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
691
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
692
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
693
$\res_p(\gamma)=0$ and certainly lies in
694
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
695
696
\item {\bf $p=q$. }
697
698
Since $q\nmid N$ is a prime of good reduction for the motive
699
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
700
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
701
$V'_{\qq}$ have the same dimension, where
702
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
703
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
704
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
705
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
706
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
707
may now prove, in the same manner as Proposition 9.2 of
708
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
709
\end{enumerate}
710
\end{proof}
711
712
Theorem~2.7 of \cite{AS} is concerned with verifying local
713
conditions in the case $k=2$, where~$f$ and~$g$ are associated
714
with abelian varieties~$A$ and~$B$. (Their theorem also applies to
715
abelian varieties over number fields.) Our restriction outlawing
716
congruences modulo $\qq$ with cusp forms of lower level is
717
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
718
$c_{A,l}$ and $c_{B,l}$. (In the case where~$A$ is an elliptic
719
curve with $\ord_l(j(A))<0$, consideration of a Tate
720
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
721
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
722
at~$l$.)
723
724
In this paper we have encountered two technical problems which we
725
dealt with in quite similar ways:
726
\begin{enumerate}
727
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
728
\item proving local conditions at primes $p\mid N$, for an element
729
of $\qq$-torsion.
730
\end{enumerate}
731
If our only interest was in testing the Bloch-Kato conjecture at
732
$\qq$, we could have made these problems cancel out, as in Lemma
733
8.11 of \cite{DFG}, by weakening the local conditions. However, we
734
have chosen not to do so, since we are also interested in the
735
Shafarevich-Tate group, and since the hypotheses we had to assume
736
are not particularly strong.
737
738
\section{Examples and Experiments}
739
740
\subsection{Visible $\Sha$ Table~\ref{fig:newforms}}
741
\begin{figure}
742
\caption{\label{fig:newforms}Visible $\Sha$}
743
$$
744
\begin{array}{|c|c|c|c|c|}\hline
745
g & \deg(g) & f & \deg(f) & q's \\\hline
746
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
747
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
748
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
749
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
750
\vspace{-2ex} & & & & \\
751
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
752
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
753
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
754
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
755
\vspace{-2ex} & & & & \\
756
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
757
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
758
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
759
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
760
\vspace{-2ex} & & & & \\
761
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
762
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
763
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
764
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
765
\hline
766
\end{array}
767
$$
768
\end{figure}
769
770
771
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
772
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
773
along with at least one prime~$q$ such that there is a prime
774
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
775
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
776
The notation is as follows.
777
The first column contains a label whose structure is
778
\begin{center}
779
{\bf [Level]k[Weight][GaloisOrbit]}
780
\end{center}
781
This label determines a newform $g=\sum a_n q^n$, up to Galois
782
conjugacy. For example, \nf{127k4C} denotes a newform in the third
783
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
784
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
785
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
786
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
787
being first in the event that the two absolute values are equal,
788
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
789
and so on. The second column contains the degree of the field
790
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
791
contain~$f$ and its degree, respectively. The fifth column
792
contains at least one prime~$q$ such that there is a prime
793
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
794
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
795
satisfied for~$f$,~$g$, and~$\qq$.
796
797
798
We describe the first line of Table~\ref{fig:newforms}
799
in more detail. See the next section for further details
800
on how the computations were performed.
801
802
Using modular symbols, we find that there is a newform
803
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
804
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
805
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
806
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
807
coefficients generate a number field~$K$ of degree~$17$, and by
808
computing the image of the modular symbol $XY\{0,\infty\}$ under
809
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
810
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
811
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
812
both equal to
813
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
814
+ \cdots\in \FF_{43}[[q]].$$
815
816
There is no form in the Eisenstein subspaces of
817
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
818
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
819
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
820
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
821
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
822
of Theorem~\ref{local}, so if $r$ is the dimension of
823
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
824
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
825
826
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
827
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
828
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
829
subgroup of $\Sha$. Admitting these assumptions, we have
830
constructed the $\qq$-torsion in $\Sha$ predicted by the
831
Bloch-Kato conjecture.
832
833
For particular examples of elliptic curves one can often find and
834
write down rational points predicted by the Birch and
835
Swinnerton-Dyer conjecture. It would be nice if likewise one could
836
explicitly produce algebraic cycles predicted by the
837
Beilinson-Bloch conjecture in the above examples. Since
838
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
839
0.3.2 of \cite{Z}), so ought to be trivial in
840
$\CH_0^{k/2}(M_g)\otimes\QQ$.
841
842
\subsection{How the computation was performed}
843
We give a brief summary of how the computation was performed. The
844
algorithms that we used were implemented by the second author, and
845
most are a standard part of MAGMA (see \cite{magma}).
846
847
Let~$g$,~$f$, and~$q$ be some data from a line of
848
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
849
verified the existence of a congruence modulo~$q$, that
850
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
851
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
852
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
853
854
To prove there is a congruence, we showed that the corresponding
855
{\em integral} spaces of modular symbols satisfy an appropriate
856
congruence, which forces the existence of a congruence on the
857
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
858
irreducible by computing a set that contains all possible residue
859
characteristics of congruences between~$g$ and any Eisenstein
860
series of level dividing~$N$, where by congruence, we mean a
861
congruence for all Fourier coefficients of index~$n$ with
862
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
863
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
864
listing a basis of such~$h$ and finding the possible congruences,
865
where again we disregard the Fourier coefficients of index not
866
coprime to~$N$.
867
868
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
869
modular symbol ${\mathbf
870
e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
871
with the same kernel as the period mapping, and found that the
872
image was~$0$. The period mapping sends the modular
873
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
874
so that ${\mathbf e}$ maps to~$0$ implies that
875
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
876
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
877
=(-1)^{k/2} g$ which, because of the functional equation, implies
878
that $L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of
879
independent interest because it includes examples of modular forms
880
of weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
881
the functional equation. We found no such examples of weights
882
$\geq 8$.
883
884
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
885
prime appears in the $q$-column, meaning $f$ and $g$ are congruent
886
$\bmod{\,\qq^2}$ for some $\qq\mid q$. In these cases, a
887
modification of Theorem \ref{local} will show that the
888
$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
889
$O_E/\qq^2$-rank at least $r$.
890
891
\subsection{Conjecturally nontrivial $\Sha$}
892
In this section we apply some of the results of
893
Section~\ref{sec:bkconj} to compute lower bounds on conjectural orders
894
of Shafarevich-Tate group of many modular motives.
895
The results of this section suggest that~$\Sha$ of modular motives
896
is usually not explained by congruences at level~$N$, which agrees
897
with the observations of \cite{CM} and \cite{AS}.
898
For example, when $k>6$ we find many examples of nontrivial~$\Sha$
899
but no examples of nontrivial visible~$\Sha$.
900
901
For any newform~$f$, let $L(M_f,s) = \prod_{\sigma}L(f^\sigma,s)$
902
where $f^\sigma$ runs over the $\Gal(\Qbar/\QQ)$-conjugates of~$f$.
903
Let~$T$ be the complex torus $\CC^d/\mathcal{L}$, where the
904
lattice $\mathcal{L}$ is defined by integrating integral modular
905
symbols against the conjugates of~$f$. Let $\Omega_{M_f}$ denote the
906
volume of the $-1$ eigenspace $T^{-}=\{z \in T : \overline{z}=-z\}$ for complex conjugation on~$T$.
907
908
909
{\begin{figure}
910
\vspace{-2ex}
911
\caption{\label{fig:invisforms}Conjecturally nontrivial $\Sha$ (mostly invisible)}
912
\vspace{-2ex}
913
$$
914
\begin{array}{|c|c|c|c|}\hline
915
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
916
\nf{127k4C}* & 17 & 43^{2} & 43, 127 \\
917
\nf{159k4E}* & 8 & 23^{2} & 3, 5, 11, 23, 53, 13605689 \\
918
\nf{263k4B} & 39 & 41^{2} & 263 \\
919
\nf{269k4C} & 39 & 23^{2} & 269 \\
920
\nf{271k4B} & 39 & 29^{2} & 271 \\
921
\nf{281k4B} & 40 & 29^{2} & 281 \\
922
\nf{295k4C} & 16 & 7^{2} & 3, 5, 11, 59, 101, 659, 70791023 \\
923
\nf{299k4C} & 20 & 29^{2} & 13, 23, 103, 20063, 21961 \\
924
\nf{319k4C} & 19 & 17^{2} & 3, 11, 23, 29, 37, 3181, 434348087 \\
925
\nf{321k4C} & 16 & 13^{2} & 3, 5, 107, 157, 12782373452377 \\
926
\hline
927
\nf{95k6D}* & 9 & 31^{2} \!\cdot\! 59^{2} & 3, 5, 17, 19, 31, 59, 113, 26701 \\
928
\nf{101k6B} & 24 & 17^{2} & 101 \\
929
\nf{103k6B} & 24 & 23^{2} & 103 \\
930
\nf{111k6C} & 9 & 11^{2} & 3, 37, 2796169609 \\
931
\nf{122k6D}* & 6 & 73^{2} & 3, 5, 61, 73, 1303196179 \\
932
\nf{153k6G} & 5 & 7^{2} & 3, 17, 61, 227 \\
933
\nf{157k6B} & 34 & 251^{2} & 157 \\
934
\nf{167k6B} & 40 & 41^{2} & 167 \\
935
\nf{172k6B} & 9 & 7^{2} & 3, 11, 43, 787 \\
936
\nf{173k6B} & 39 & 71^{2} & 173 \\
937
\nf{181k6B} & 40 & 107^{2} & 181 \\
938
\nf{191k6B} & 46 & 85091^{2} & 191 \\
939
\nf{193k6B} & 41 & 31^{2} & 193 \\
940
\nf{199k6B} & 46 & 200329^2 & 199 \\
941
\hline
942
\nf{47k8B} & 16 & 19^{2} & 47 \\
943
\nf{59k8B} & 20 & 29^{2} & 59 \\
944
\nf{67k8B} & 20 & 29^{2} & 67 \\
945
\nf{71k8B} & 24 & 379^{2} & 71 \\
946
\nf{73k8B} & 22 & 197^{2} & 73 \\
947
\nf{74k8C} & 6 & 23^{2} & 37, 127, 821, 8327168869 \\
948
\nf{79k8B} & 25 & 307^{2} & 79 \\
949
\nf{83k8B} & 27 & 1019^{2} & 83 \\
950
\nf{87k8C} & 9 & 11^{2} & 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
951
\nf{89k8B} & 29 & 44491^{2} & 89 \\
952
\nf{97k8B} & 29 & 11^{2} \!\cdot\! 277^{2} & 97 \\
953
\nf{101k8B} & 33 & 19^{2} \!\cdot\! 11503^{2} & 101 \\
954
\nf{103k8B} & 32 & 75367^{2} & 103 \\
955
\nf{107k8B} & 34 & 17^{2} \!\cdot\! 491^{2} & 107 \\
956
\nf{109k8B} & 33 & 23^{2} \!\cdot\! 229^{2} & 109 \\
957
\nf{111k8C} & 12 & 127^{2} & 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
958
\nf{113k8B} & 35 & 67^{2} \!\cdot\! 641^{2} & 113 \\
959
\nf{115k8B} & 12 & 37^{2} & 3, 5, 19, 23, 572437, 5168196102449 \\
960
\nf{117k8I} & 8 & 19^{2} & 3, 13, 181 \\
961
\nf{118k8C} & 8 & 37^{2} & 5, 13, 17, 59, 163, 3923085859759909 \\
962
\nf{119k8C} & 16 & 1283^{2} & 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
963
\hline
964
\end{array}
965
$$
966
\end{figure}
967
\begin{figure}
968
$$
969
\begin{array}{|c|c|c|c|}\hline
970
f & \deg(f) & B\,\, (\text{$\Sha$ bound})& \text{all odd congruence primes}\\\hline
971
\nf{121k8F} & 6 & 71^{2} & 3, 11, 17, 41 \\
972
\nf{121k8G} & 12 & 13^{2} & 3, 11 \\
973
\nf{121k8H} & 12 & 19^{2} & 5, 11 \\
974
\nf{125k8D} & 16 & 179^{2} & 5 \\
975
\hline
976
977
\nf{43k10B} & 17 & 449^{2} & 43 \\
978
\nf{47k10B} & 20 & 2213^{2} & 47 \\
979
\nf{53k10B} & 21 & 673^{2} & 53 \\
980
\nf{55k10D} & 9 & 71^{2} & 3, 5, 11, 251, 317, 61339, 19869191 \\
981
\nf{59k10B} & 25 & 37^{2} & 59 \\
982
\nf{62k10E} & 7 & 23^{2} & 3, 31, 101, 523, 617, 41192083 \\
983
\nf{64k10K} & 2 & 19^{2} & 3 \\
984
\nf{67k10B} & 26 & 191^{2} \!\cdot\! 617^{2} & 67 \\
985
\nf{68k10B} & 7 & 83^{2} & 3, 7, 17, 8311 \\
986
\nf{71k10B} & 30 & 1103^{2} & 71 \\
987
988
\hline
989
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571 \\
990
\nf{31k12B} & 15 & 67^{2} \!\cdot\! 71^{2} & 31, 13488901 \\
991
\nf{35k12C} & 6 & 17^{2} & 5, 7, 23, 29, 107, 8609, 1307051 \\
992
\nf{39k12C} & 6 & 73^{2} & 3, 13, 1491079, 3719832979693 \\
993
\nf{41k12B} & 20 & 54347^{2} & 7, 41, 3271, 6277 \\
994
\nf{43k12B} & 20 & 212969^{2} & 43, 1669, 483167 \\
995
\nf{47k12B} & 23 & 24469^{2} & 17, 47, 59, 2789 \\
996
\nf{49k12H} & 12 & 271^{2} & 7 \\
997
\hline
998
\end{array}
999
$$
1000
\end{figure}
1001
1002
The following conjecture is probably not difficult to prove, but we
1003
haven't given a proof, so we formally state it as a conjecture, then
1004
assume it.
1005
\begin{conj}\label{conj:lrat}
1006
If $p\nmid Nk$ is a prime and
1007
$p\mid \frac{L(M_f,k/2)}{\Omega_{k/2}}$, then
1008
$p\mid \Norm\left(\frac{L(f,k/2)}{\vol_{\infty}}\right)$.
1009
\end{conj}
1010
For the rest of this section, {\em we officially assume the Bloch-Kato
1011
conjecture and Conjecture~\ref{conj:lrat}.}
1012
1013
Let~$S$ be the set of newforms with~level $N$ and weight~$k$
1014
satisfying either $k=4$ and $N\leq 321$, or $k=6$ and $N\leq 199$, or
1015
$k=8$ and $N\leq 125$, or $k=10$ and $N\leq 72$, or $k=12$ and
1016
$N\leq 49$. Compute a bound~$B$ on $\#\Sha$, for each
1017
newform $f\in S$, as follows:
1018
\begin{enumerate}
1019
\item Let $L_1$ be the rational number $L(M_f,k/2)/\Omega_{M_f}$.
1020
If $L_1=0$ let $B=1$ and terminate.
1021
\item Let $L_2$ be the part of $L_1$ that is coprime to $N\cdot k!$.
1022
\item Let $L_3$ be the part of $L_2$ that is coprime to
1023
$p+1$ for every prime~$p$ such that $p^2\mid N$.
1024
\item Let $L_4$ be the part of $L_3$ coprime to any prime of
1025
congruence between~$f$ and a form of weight~$k$ and
1026
lower level.
1027
\item Let $B$ be the part of $L_4$ coprime to any prime of congruence
1028
between~$f$ and an Eisenstein series. (This eliminates
1029
residue characteristic of reducible representations.)
1030
\end{enumerate}
1031
Proposition~\ref{sha} implies that if
1032
$\ord_p(B) > 0$, then $\ord_p(\#\Sha) > 0$.
1033
1034
We computed~$B$ for every newform in~$S$. Because of Tamagawa
1035
numbers, there are many examples in which $L_3$ is
1036
large, but~$B$ is not.
1037
For example, {\bf 39k4C} has $L_3=19$, but $B=1$ because
1038
of a $19$-congruence with a form of level~$13$;
1039
in this case we must have $19\mid c_{13}$.
1040
1041
1042
The newforms for which $B>1$ are given in Table~\ref{fig:invisforms}.
1043
The second column of the table records the degree of the field
1044
generated by the Fourier coefficients of~$f$. This third
1045
contains~$B$. Let~$W$ be the intersection of the span of all
1046
conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and $W^{\perp}$ the
1047
Petersson orthogonal complement of~$W$ in $S_k(\Gamma_0(N),\ZZ)$.
1048
Then the fourth column contains the odd prime divisors of
1049
$\#(S_k(\Gamma_0(N),\ZZ))/(W+W^{\perp})$, which are exactly the
1050
possible primes of congruence for~$f$.
1051
We place a $*$ next to the four entries of Table~\ref{fig:invisforms}
1052
that also occur in Table~\ref{fig:newforms}.
1053
1054
\subsection{Other examples}\label{sec:other_ex}
1055
We have some other examples where forms of
1056
different levels are congruent.
1057
However, Remark~\ref{sign} does not
1058
apply, so that one of the forms could have an odd functional
1059
equation, and the other could have an even functional equation.
1060
For instance, we have a $13$-congruence between $g=\nf{81k4A}$ and
1061
$f=\nf{567k4L}$; here $L(\nf{567k4L},2)\neq 0$, while
1062
$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation.
1063
Here $f$ obviously fails the condition about not being congruent
1064
to a form of lower level, so in Lemma~\ref{local1} it is possible that
1065
$\ord_{\qq}(c_7(2))>0$. In fact this does happen. Because
1066
$V'_{\qq}$ (attached to~$g$ of level $81$) is unramified at $p=7$,
1067
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
1068
two-dimensional. As in (2) of the proof of Theorem~\ref{local},
1069
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
1070
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
1071
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
1072
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
1073
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
1074
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
1075
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
1076
our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
1077
non-trivial when $w_p=-1$, so (2) of the proof of Theorem~\ref{local}
1078
does not work. This is just as well, since had it
1079
worked we would have expected
1080
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
1081
shows not to be the case.
1082
1083
Here is an example where the divisibility between the levels is
1084
the other way round: a $7$-congruence between $g=\nf{122k6A}$ and
1085
$f=\nf{61k6B}$. In this case both $L$-functions have even
1086
functional equation, and we have $L(\nf{122k6A},3)=0$. In the
1087
proof of Theorem~\ref{local}, we find a problem with the local condition
1088
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
1089
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
1090
kernel is at most one-dimensional, so we still get the
1091
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
1092
$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get
1093
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
1094
elements of $\Sha$ are {\it invisible} at level 61. When the
1095
levels are different we are no longer able to apply Theorem 2.1 of
1096
\cite{FJ}. However, we still have the congruences of integral
1097
modular symbols required to make the proof of Proposition
1098
\ref{div} go through. Indeed, as noted above, the congruences of
1099
modular forms were found by producing congruences of modular
1100
symbols. Despite these congruences of modular symbols, Remark 5.3
1101
does not apply, since there is no reason to suppose that
1102
$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.
1103
1104
Finally, there are two examples where we have a form $g$ with even
1105
functional equation such that $L(g,k/2)=0$, and a congruent form
1106
$f$ which has odd functional equation; these are a 23-congruence
1107
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
1108
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
1109
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
1110
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
1111
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
1112
the appropriate modification of Theorem \ref{local} does not
1113
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
1114
It only tells us that the $\qq$-torsion subgroup of
1115
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
1116
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
1117
appears in the conjectural formula for the first derivative of the
1118
complex $L$ function, evaluated at $s=k/2$, but in combination
1119
with a regulator that we have no way of calculating.
1120
1121
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
1122
associated with $f$ and $g$ by the construction of Mazur, Tate and
1123
Teitelbaum \cite{MTT}, each divided by a suitable canonical
1124
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
1125
quite clear what to make of this. This divisibility may be proved
1126
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
1127
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
1128
normalised) are congruent $\bmod{\,\qq}$, as a result of the
1129
congruence between the modular symbols out of which they are
1130
constructed. Integrating an appropriate function against these
1131
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
1132
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
1133
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
1134
where the signs in the functional equations of $L(g,s)$ and
1135
$L_q(g,s)$ are the same, positive in this instance. (According to
1136
the proposition in Section 18 of \cite{MTT}, the signs differ
1137
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
1138
1139
1140
1141
\subsection{Excluded data}
1142
We also found some examples for which the conditions of Theorem \ref{local}
1143
were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},
1144
but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar
1145
problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here
1146
$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a
1147
5-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5
1148
is less than the weight 6.
1149
1150
\begin{thebibliography}{AL}
1151
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
1152
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
1153
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
1154
Shafarevich-Tate groups of abelian varieties, preprint.
1155
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
1156
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
1157
\bibitem[BCP]{magma}
1158
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1246
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1248
\end{document}
1249
127k4A 43 127k4C 17 [43]
1250
159k4A 5,23 159k4E 8 [5]x[23]
1251
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1252
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1253
453k4A 5,17 453k4E 23 [5]x[17]
1254
453k4A 23 151k4A 30 Odd func eq for g
1255
465k4A 11 465k4H 7 [11]x[5]x[2]
1256
477k4A 73 477k4M 12 [73]x[2]
1257
567k4A 23 567k4I 8 [23]x[3]
1258
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1259
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1260
639k4A 7 639k4H 12 [7]
1261
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1262
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1263
681k4A 43 227k4A 23 Odd func eq for g
1264
681k4A 59 681k4D 30 [59]x[3]x[2]
1265
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1266
95k6A 31,59 95k6D 9 [31]x[59]
1267
116k6A 5 116k6D 6 [5]x[29]x[2]
1268
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1269
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1270
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
\nf{263k4B} & & 41^2 & \\
1284
\nf{269k4C} & & 23^2 & \\
1285
\nf{271k4B} & & 29^2 &\\
1286
\nf{281k4B} & & 29^2\\
1287
\hline
1288
\nf{101k6B} & & 17^2 & 101\\
1289
\nf{103k6B} & & 23^2\\
1290
\nf{111k6C} & & 11^2\\
1291
\nf{153k6G} & & 7^2\\
1292
\nf{157k6B} & & 252^2\\
1293
\nf{167k6B} & & 41^2\\
1294
\nf{172k6B} & & 7^2\\
1295
\nf{173k6B} & & 71^2\\
1296
\nf{181k6B} & & 107^2\\
1297
\hline
1298
\nf{19k12B} & 9 & 67^{2} & 5, 17, 19, 31, 571\\
1299
1300