CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v5.texOpen in CoCalc with one click!
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 motives\\
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{\sc (NOT FOR DISTRIBUTION!)}}
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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{March 8th, 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
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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.
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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 order $m$. In most cases they find
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another elliptic curve, often of the same conductor, whose
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$m$-torsion is Galois-isomorphic to that of the first one, and
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which has rank two. The rational points on the second elliptic
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curve produce classes in the common $H^1(\QQ,E[m])$. They expect
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that these lie in the Shafarevich-Tate group of the first curve,
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so rational points on one curve explain elements of the
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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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(with which are associated 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 some $\qq$, and
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$L(g,s)$ vanishes to order at least $2$ at $s=k/2$. We are able to
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find eleven examples (all with $k=4$ and $k=6$), and in each case
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$\qq$ appears in 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 predict the
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exact order of $\Sha$. We have to start with modular forms between
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which there exists a congruence. However, Vatsal's work allows us
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to explain how the vanishing of one $L$-function leads, via the
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congruence, to the divisibility by $\qq$ of (an algebraic part of)
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another, 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 cannot prove theoretically, but which are
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predicted by Bloch-Kato, given the existence of a non-degenerate
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alternating pairing on $\Sha$ \cite{Fl2}.
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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 $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset
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F^k=\{0\}$. 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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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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$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).$$ 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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$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow
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H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (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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$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
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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 \newline
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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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$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
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The length of its $\lambda$-component may be taken for the
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exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Sha(j)$. We shall only concern ourselves with the case
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$j=k/2$, and write $\Sha$ for $\Sha(j)$.
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Define the set of global points
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$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. The length of its $\lambda$-component may be taken
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for the exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Gamma_{\QQ}$.
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\section{Canonical periods}
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From now on we assume 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 any 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 $X_0(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
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elliptic curves with cyclic $N$-isogenies. Let $\mathfrak{E}$ be
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the universal elliptic curve over $X_0(N)$. Let
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$\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product of
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$\mathfrak{E}$ over $X_0(N)$. Realising $X_0(N)$ as the quotient
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$\Gamma_0(N)\backslash\mathfrak{H}^*$, 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 $\omega_f^{\pm}=2\pi i
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\Omega_f^{\pm}\delta_f^{\pm}$.
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\section{The Bloch-Kato conjecture}
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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
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$\sum_{n=1}^{\infty}a_nn^{-s}$, but there is an analytic
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continuation given by an integral, as described in the next
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section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture
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for the motive $M_f(k/2)$ predicts that
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$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\aaa^{\pm}\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
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Here, $\pm$ represents the parity of $(k/2)-1$, and
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$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}\Omega_{\pm}$. For
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$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
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$$\length H^1_f(\QQ_p,T_{\lambda}(j))_{\tors.}-\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j})).$$
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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}.) The above
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formula is to be interpreted as an equality of fractional ideals
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of $E$. (Strictly speaking, the conjecture in \cite{BK} is only
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given for $E=\QQ$.)
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\begin{lem} Let $p\nmid N$ be a prime, $j$ 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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$\qq\mid q$, and that, for all primes $p\mid N$, $\,p\neq \pm
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1\pmod{q}$. Suppose also that $f$ is not congruent modulo $\qq$ to
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any newform of weight~$k$, trivial character and level dividing
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$N/p$, with~$p$ any prime that exactly divides~$N$. Then for any
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$p\mid N$, and $j$ an integer, $\ord_{\qq}(c_p(j))=0$.
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\end{lem}
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\begin{proof} Bearing in mind that
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$$\ord_{\qq}((1-a_pp^{-j}+p^{k-1-2j}))=\#H^0(\QQ_p,V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}),$$
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and also the long exact sequence in $I_p$-cohomology arising from
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$$\begin{CD}0@>>>T_{\qq}(j)@>>>V_{\qq}(j)@>\pi>>A_{\qq}(j)@>>>0\end{CD},$$
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it suffices to show that
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$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$
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If the dimensions differ then, given that $f$ is not congruent
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modulo $\qq$ to a newform of level strictly dividing $N$, and
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since $p\neq \pm 1\pmod{q}$, Propositions 3.1 and 2.3 of \cite{L}
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tell us that $A[\qq](j)$ is unramified at $p$ and that
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$\ord_p(N)=1$. (Here we are using Carayol's result that $N$ is the
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prime-to-$q$ part of the conductor of $V_{\qq}$ \cite{Ca1}.) But
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then Theorem 1 of \cite{JL} (which uses the condition $q>k$)
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implies the existence of a newform of weight $k$, trivial
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character and level dividing $N/p$, congruent to $g$ modulo $\qq$.
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This contradicts our hypotheses.
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\end{proof}
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\begin{lem} Let $\qq\mid q$ be a prime of $E$ such that $q\nmid
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Nk!$ Then $\ord_{\qq}(c_q)=0$.
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\end{lem}
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\begin{proof} It follows from the isomorphism at the end of
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Section 2.2 of \cite{DFG} (an application of the results of
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\cite{Fa1}) that $T_{\qq}$ is the
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$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
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filtered module $T_{\dR}\otimes O_{\lambda}$ by the functor they
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call $\mathbb{V}$. Given this, the lemma follows from Theorem
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4.1(iii) of \cite{BK}.
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\end{proof}
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\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
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irreducible representation of $\Gal(\Qbar/\QQ)$.
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\end{lem}
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This follows trivially from the definition.
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Putting together the above lemmas we arrive at the following:
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\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime such that $q>k$.
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Let $\qq\mid q$ be a prime of $E$ such that $A[\qq]$ is an
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irreducible representation of $\Gal(\Qbar/\QQ)$. Choose $T_{\dR}$
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and $T_B$ which locally at $\qq$ are as in the previous section.
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If $$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with
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numerator non-zero) then the Bloch-Kato conjecture predicts that
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$\ord_{\qq}(\#\Sha)>0$.
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\end{prop}
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\section{Congruences of special values}
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Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
409
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
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large enough to contain all the coefficients $a_n$ and $b_n$.
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Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
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g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. It is
413
easy to see that we may choose the $\delta_f^{\pm}\in T_B^{\pm}$
414
in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e.
415
$\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Let us
416
suppose that such a choice has been made.
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We shall now make two further assumptions:
419
\begin{enumerate}
420
\item $L(f,k/2)\neq 0$;
421
\item $L(g,k/2)=0$.
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\end{enumerate}
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\begin{prop}
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With assumptions as above,
425
$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$.
426
\end{prop}
427
\begin{proof} This is based on some of the ideas used in Section 1 of
428
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
429
which presumably should refer to ``Condition 2''. Since
430
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
431
$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm
432
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
433
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
434
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
435
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
436
where the integral is taken along the positive imaginary axis,
437
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
438
Thus we are reduced
439
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
440
441
Let $\mathcal{D}_0$ be the group of divisors of degree zero
442
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
443
integer $r\geq 0$, let $P_r(R)$ be the additive group of
444
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
445
groups have a natural action of $\Gamma_0(N)$. Let
446
$S_{\Gamma_0(N)}(k,R):=\Hom_{\Gamma_0(N)}(\mathcal{D}_0,P_{k-2}(R))$
447
be the $R$-module of weight $k$ modular symbols for $\Gamma_0(N)$.
448
449
Via the isomorphism (8) in Section 1.5 of \cite{V},
450
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
451
S_{\Gamma_0(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
452
element $\Delta_f^{\pm}\in S_{\Gamma_0(N)}(k,O_E)$.
453
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
454
(k/2)-1\pmod{2}}^{k-2}r_f(j)X^jY^{k-2-j}.$$ Since
455
$\omega_f^{\pm}=\Omega_f^{\pm}\delta_f^{\pm}$, we see that
456
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
457
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
458
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
459
show is divisible by $\qq$.
460
Similarly
461
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
462
(k/2)-1\pmod{2}}^{k-2}r_g(j)X^jY^{k-2-j}.$$ The coefficient of
463
$X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
464
Therefore it would suffice to show that, for some $\mu\in O_E$,
465
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
466
$\qq$ in $S_{\Gamma_0(N)}(k,O_E)$. It suffices to show that, for
467
some $\mu\in O_E$, the element $\delta_f^{\pm}-\mu\delta_g^{\pm}$
468
is divisible by $\qq$, considered as an element of $\qq$-adic
469
cohomology with coefficients. This would be the case if
470
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
471
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
472
a consequence of Theorem 2.1(1) of \cite{FJ}.
473
\end{proof}
474
\begin{remar}
475
By Proposition \ref{sha}, the Bloch-Kato conjecture now predicts
476
that $\ord_{\qq}(\#\Sha)>0$. The next section provides a
477
conditional construction of the required elements of $\Sha$.
478
\end{remar}
479
\begin{remar}\label{sign}
480
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
481
have to be equal, since they are determined by the action of the
482
involution $W_N$ on the common subspace generated by the
483
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
484
\end{remar}
485
This is analogous to the remark at the end of Section 3 of
486
\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then
487
$L(g,s)$ must vanish to order at least two, as in all the examples
488
below. It is worth pointing out that there are no examples of $g$
489
of level one such that $L(g,k/2)=0$, unless Maeda's conjecture
490
(that all the normalised cuspidal eigenforms are Galois conjugate)
491
is false. See \cite{CF}.
492
493
\section{Constructing elements of the Shafarevich-Tate group}
494
For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
495
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
496
$A'_{\lambda}$ be the corresponding objects for $g$. Let
497
$A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$. Since
498
$a_p$ is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows
499
from the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
500
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
501
502
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
503
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
504
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
505
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
506
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
507
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of
508
$\QQ$-rational, null-homologous, codimension $k/2$ algebraic
509
cycles on the motive $M_g$, modulo rational equivalence. (This
510
generalises the part of the Birch-Swinnerton-Dyer conjecture which
511
says that for an elliptic curve $E/\QQ$, the order of vanishing of
512
$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group
513
$E(\QQ)$.)
514
515
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
516
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
517
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
518
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
519
get (assuming also the Beilinson-Bloch conjecture) a subspace of
520
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
521
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
522
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
523
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
524
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
525
Sections 1 and 6.5 of \cite{Fo2}.
526
527
Similarly, if $L(f,k/2)\neq 0$ then we expect that
528
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
529
coincides with the $\qq$-part of $\Sha$.
530
\begin{thm}\label{local}
531
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
532
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
533
$A[\qq]$ and $A'[\qq]$ are irreducible representations of
534
$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm
535
1\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
536
modulo $\qq$ to any newform of weight~$k$, trivial character and
537
level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.
538
Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
539
$\FF_{\qq}$-rank at least $r-1$.
540
\end{thm}
541
\begin{proof}
542
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
543
continuity and rescaling we may assume that it lies in
544
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
545
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
546
class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
547
irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
548
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
549
we get a non-zero, $\qq$-torsion class $\gamma\in
550
H^1(\QQ,A_{\qq}(k/2))$.
551
552
Our aim is to show that $\res_p(\gamma)\in
553
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
554
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
555
The case $p=q$ does not quite work, which is the reason for the
556
``$r-1$'' in the statement of the theorem.
557
558
\begin{enumerate}
559
\item {\bf $p\nmid qN$. }
560
561
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
562
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
563
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
564
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
565
follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
566
$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
567
of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
568
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
569
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
570
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
571
572
\item {\bf $p\mid N$. }
573
574
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
575
$\qq$-divisible. It suffices to show that
576
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
577
but this may be done as in the proof of Lemma\ref{local1}. It
578
follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
579
$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
580
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
581
order of this group is the same as the order of the group
582
$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
583
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
584
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
585
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
586
H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
587
this common dimension is $1$. The (motivic) Euler factor at $p$
588
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
589
as multiplication by $\alpha$ on the one-dimensional space
590
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
591
\cite{Ca1} that this is the same as the Euler factor at $p$ of
592
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
593
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
594
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
595
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
596
$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm
597
1\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
598
$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
599
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
600
601
\item {\bf $p=q$. }
602
603
Since $q\nmid N$ is a prime of good reduction for the motive
604
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
605
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
606
$V'_{\qq}$ have the same dimension, where
607
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
608
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
609
It follows from Theorem 4.1(ii) of \cite{BK} that
610
$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$
611
via their exponential map.
612
613
Since $p$ is a prime of good reduction, the de Rham conjecture is
614
a consequence of the crystalline conjecture, which follows from
615
Theorem 5.6 of \cite{Fa1}. Hence
616
$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$
617
Observe that the dimension of the left-hand-side is $1$. Hence
618
there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at
619
least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It
620
follows that our $r$-dimensional $\FF_{\qq}$-subspace of the
621
$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an
622
$(r-1)$-dimensional subspace landing in
623
$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).
624
625
\end{enumerate}
626
\end{proof}
627
628
Theorem 2.7 of \cite{AS} is concerned with verifying local
629
conditions in the case $k=2$, where $f$ and $g$ are associated
630
with abelian varieties $A$ and $B$. (Their theorem also applies to
631
abelian varieties over number fields.) Our restriction outlawing
632
congruences modulo $\qq$ with cusp forms of lower level is
633
analogous to theirs forbidding $q$ from dividing Tamagawa factors
634
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
635
curve with $\ord_l(j(A))<0$, consideration of a Tate
636
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
637
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
638
at $l$.)
639
640
In this paper we have encountered two technical problems which we
641
dealt with in quite similar ways:
642
\begin{enumerate}
643
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
644
\item proving local conditions at primes $p\mid N$, for an element
645
of $\qq$-torsion.
646
\end{enumerate}
647
If our only interest was in testing the Bloch-Kato conjecture at
648
$\qq$, we could have made these problems cancel out, as in Lemma
649
8.11 of \cite{DFG}, by weakening the local conditions. However, we
650
have chosen not to do so, since we are also interested in the
651
Shafarevich-Tate group, and since the hypotheses we had to assume
652
are not particularly strong.
653
654
\section{Eleven examples}
655
\newcommand{\nf}[1]{\mbox{\bf #1}}
656
\begin{figure}
657
\caption{\label{fig:newforms}Newforms Relevant to
658
Theorem~\ref{local}}
659
$$
660
\begin{array}{|ccccccc|}\hline
661
g & \deg(g) & f & \deg(f) & q's \\\hline
662
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
663
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
664
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
665
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
666
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
667
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
668
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
669
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
670
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\
671
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
672
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
673
\hline
674
\end{array}
675
$$
676
\end{figure}
677
678
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
679
eleven pairs of newforms~$f$ and~$g$ (of equal weights and levels)
680
along with at least one prime~$q$ such that there is a prime
681
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
682
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
683
\subsection{Notation}
684
Table~\ref{fig:newforms} is laid out as follows.
685
The first column contains a label whose structure is
686
\begin{center}
687
{\bf [Level]k[Weight][GaloisOrbit]}
688
\end{center}
689
This label determines a newform $g=\sum a_n q^n$, up to Galois
690
conjugacy. For example, \nf{127k4C} denotes a newform in the third
691
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
692
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
693
\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
694
the level, with positive trace being first in the event that the
695
two absolute values are equal, and the first Galois orbit is
696
denoted {\bf A}, the second {\bf B}, and so on. The second column
697
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
698
third and fourth columns contain~$f$ and its degree, respectively.
699
The fifth column contains at least one prime~$q$ such that there
700
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
701
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
702
satisfied for~$f$,~$g$, and~$\qq$.
703
704
\subsection{The first example in detail}
705
\newcommand{\fbar}{\overline{f}}
706
We describe the first line of Table~\ref{fig:newforms}
707
in more detail. See the next section for further details
708
on how the computations were performed.
709
710
Using modular symbols, we find that there is a newform
711
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
712
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
713
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
714
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
715
coefficients generate a number field~$K$ of degree~$17$, and by
716
computing the image of the modular symbol $XY\{0,\infty\}$ under
717
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
718
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
719
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
720
both equal to
721
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
722
723
There is no form in the Eisenstein subspaces of
724
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
725
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
726
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
727
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
728
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
729
of Theorem~\ref{local}, so if $r$ is the dimension of
730
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
731
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
732
733
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
734
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
735
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
736
subgroup of $\Sha$. Admitting these assumptions, we have
737
constructed the $\qq$-torsion in $\Sha$ predicted by the
738
Bloch-Kato conjecture.
739
740
For particular examples of elliptic curves one can often find and
741
write down rational points predicted by the Birch and
742
Swinnerton-Dyer conjecture. It would be nice if likewise one could
743
explicitly produce algebraic cycles predicted by the
744
Beilinson-Bloch conjecture in the above examples. Since
745
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
746
0.3.2 of \cite{Z}), so ought to be trivial in
747
$\CH_0^{k/2}(M_g)\otimes\QQ$.
748
749
\subsection{Some remarks on how the computation was performed}
750
We give a brief summary of how the computation was performed. The
751
algorithms that we used were implemented by the second author, and
752
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
753
754
Let~$g$,~$f$, and~$q$ be some data from a line of
755
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
756
verified the existence of a congruence modulo~$q$, that
757
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
758
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
759
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
760
761
To prove there is a congruence, we showed that the corresponding
762
{\em integral} spaces of modular symbols satisfy an appropriate
763
congruence, which forces the existence of a congruence on the
764
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
765
irreducible by computing a set that contains all possible residue
766
characteristics of congruences between~$g$ and any Eisenstein
767
series of level dividing~$N$, where by congruence, we mean a
768
congruence for all Fourier coefficients of index~$n$ with
769
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
770
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
771
listing a basis of such~$h$ and finding the possible congruences,
772
where again we disregard the Fourier coefficients of index not
773
coprime to~$N$.
774
775
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
776
modular symbol ${\mathbf
777
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
778
map with the same kernel as the period mapping, and found that the
779
image was~$0$. The period mapping sends the modular
780
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
781
so that ${\mathbf e}$ maps to~$0$ implies that
782
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
783
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$
784
which, because of the functional equation, implies that
785
$L'(g,\frac{k}{2})=0$.
786
787
In the course of our search, we found~$17$ plausible examples and
788
had to discard~$6$ of them because either the representation
789
$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv
790
\pm 1\pmod{q}$ for some $p\mid N$. Table~\ref{fig:newforms} is
791
of independent interest because it includes examples of modular
792
forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
793
that is not forced by the functional equation. We found no such
794
examples of weights $\geq 8$.
795
796
797
798
\begin{thebibliography}{AL}
799
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
800
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
801
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
802
Shafarevich-Tate groups of abelian varieties: evidence for the
803
Birch and Swinnerton-Dyer conjecture, in preparation.
804
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
805
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
806
\bibitem[BCP]{magma}
807
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
808
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
809
235--265, Computational algebra and number theory (London, 1993).
810
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
811
of motives, The Grothendieck Festschrift Volume I, 333--400,
812
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
813
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
814
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
815
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
816
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
817
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
818
Duke Math. J. }{\bf 59 }(1989), 785--801.
819
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
820
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
821
13--28.
822
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
823
$L_f(s)$ at the center of the critical strip, preprint.
824
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
825
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
826
{\bf 179, } 139--172, Springer, 1969.
827
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
828
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
829
part 2, 313--346.
830
\bibitem[DFG]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
831
of modular forms and the Tamagawa number conjecture, preprint.
832
{{\sf
833
http://www.andromeda.rutgers.edu/\~{\mbox{}}krm/liguo/lgpapers.html}}
834
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
835
Math. Ann. }{\bf 318 }(2000), 621--636.
836
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
837
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
838
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
839
Galois representations, {\em in }Algebraic analysis, geometry and
840
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
841
Press, Baltimore, 1989.
842
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
843
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
844
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
845
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
846
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
847
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
848
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
849
Boston Berlin, 1993.
850
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
851
repr\'esentations $p$-adiques du groupe de Galois d'un corps
852
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
853
}{\bf 115 }(1982), 529--577.
854
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
855
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
856
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
857
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
858
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
859
51--56.
860
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
861
representations coming from modular forms, {\em J. Number Theory
862
}{\bf 31 }(1989), 133--141.
863
864
\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
865
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
866
99--125.
867
\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
868
heights. The arithmetic and geometry of algebraic cycles (Banff,
869
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
870
Soc., Providence, RI, 2000.
871
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
872
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
873
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
874
congruences for coefficients of modular forms,{\em Modular
875
functions of one variable} III, Lect. Notes Math. {\bf 350, }
876
Springer, 1973.
877
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
878
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
879
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
880
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
881
\end{thebibliography}
882
883
884
\end{document}
885
\subsection{Plan for finishing the computation}
886
887
What I (William) have to do with the below stuff:\\
888
\begin{enumerate}
889
\item Fill in the missing blanks at these levels:
890
$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
891
\item Make a MAGMA program that has a table of forms to try.
892
\item First, unfortunately, for some reason the
893
labels are sometimes wrong, e.g., \nf{159k4A} should
894
have been \nf{159k4B}, so I have to check all the
895
rational forms to see which has $L(1)=0$.
896
\item Try each level using the MAGMA program (this will take a long time to run).
897
\item Finish the ``Table of examples'' above.
898
\end{enumerate}
899
900
901
\begin{verbatim}
902
903
904
(This stuff below is being integrated into the above, as I do
905
the required (rather time consuming) computations.)
906
**************************************************
907
FROM Mark Watkins:
908
909
Here's a list of stuff; the left-hand (sinister) column
910
contains forms with a double zero at the central point,
911
whilst the right-hand (dexter) column contains forms which
912
have a large square factor in LRatio at 2. The middle column
913
is the large prime factor in ModularDegree of the LHS,
914
and/or the lpf in the LRatio at 2 of the RHS.
915
916
It seems that 567k4L has invisible Sha possibly.
917
The ModularDegree for 639k4B has no large factors.
918
919
127k4A 43 127k4C
920
159k4A 23 159k4E
921
365k4A 29 365k4E
922
369k4A 13 369k4I
923
453k4A 17 453k4E
924
453k4A 23
925
465k4A 11 465k4H
926
477k4A 73 477k4L
927
567k4A 23 567k4G
928
13 567k4L
929
581k4A 19 581k4E
930
639k4B --
931
932
933
Forms with spurious zeros to do:
934
935
657k4A
936
681k4A
937
684k4B
938
95k6A 31,59
939
116k6A --
940
122k6A 73
941
260k6A
942
943
If we allow 5 and 7 to be small primes, then we get more
944
visibility info.
945
946
159k4A 5 159k4E
947
369k4A 5 369k4I
948
453k4A 5 453k4E
949
639k4B 7 639k4H
950
951
Maybe the general theorem does not apply to primes which are so small,
952
but we might be able to show that they are OK in these specific cases.
953
954
I checked the first three weight 6 examples with AbelianIntersection,
955
but actually computing the LRatio bogs down too much. Will need
956
new version.
957
958
95k6A 31 95k6D
959
95k6A 59 95k6D
960
116k6A 5 116k6D
961
122k6A 73 122k6D
962
963
Incidentally, the LRatio for 581k4E has a power of 19^4.
964
Perhaps not surprising, as the ModularDegree of 581k4A
965
has a factor of 19^2. But maybe it means a bigger Sha.
966
\end{verbatim}
967