CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v6.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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\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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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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$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).$$ 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 group of global torsion 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} Assume the same hypotheses as in Lemma \ref{local1}.
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Choose $T_{\dR}$ and $T_B$ which locally at $\qq$ are as in the
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previous section. If
401
$$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with numerator
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non-zero) then the Bloch-Kato conjecture predicts that
403
$$\ord_{\qq}(\#\Sha)>0.$$
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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
408
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 $f\equiv
411
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. It is
412
easy to see that we may choose the $\delta_f^{\pm}\in T_B^{\pm}$
413
in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e.
414
$\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Let us
415
suppose that such a choice has been made.
416
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We shall now make two further assumptions:
418
\begin{enumerate}
419
\item $L(f,k/2)\neq 0$;
420
\item $L(g,k/2)=0$.
421
\end{enumerate}
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\begin{prop}
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With assumptions as above,
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$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$.
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\end{prop}
426
\begin{proof} This is based on some of the ideas used in Section 1 of
427
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
428
which presumably should refer to ``Condition 2''. Since
429
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
430
$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm
431
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
432
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
433
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
434
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
435
where the integral is taken along the positive imaginary axis,
436
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
437
Thus we are reduced
438
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
439
440
Let $\mathcal{D}_0$ be the group of divisors of degree zero
441
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
442
integer $r\geq 0$, let $P_r(R)$ be the additive group of
443
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
444
groups have a natural action of $\Gamma_0(N)$. Let
445
$S_{\Gamma_0(N)}(k,R):=\Hom_{\Gamma_0(N)}(\mathcal{D}_0,P_{k-2}(R))$
446
be the $R$-module of weight $k$ modular symbols for $\Gamma_0(N)$.
447
448
Via the isomorphism (8) in Section 1.5 of \cite{V},
449
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
450
S_{\Gamma_0(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
451
element $\Delta_f^{\pm}\in S_{\Gamma_0(N)}(k,O_E)$.
452
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
453
(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since
454
$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that
455
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
456
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
457
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
458
show is divisible by $\qq$.
459
Similarly
460
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
461
(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient
462
of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
463
Therefore it would suffice to show that, for some $\mu\in O_E$,
464
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
465
$\qq$ in $S_{\Gamma_0(N)}(k,O_E)$. It suffices to show that, for
466
some $\mu\in O_E$, the element $\delta_f^{\pm}-\mu\delta_g^{\pm}$
467
is divisible by $\qq$, considered as an element of $\qq$-adic
468
cohomology with coefficients. This would be the case if
469
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
470
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
471
a consequence of Theorem 2.1(1) of \cite{FJ}.
472
\end{proof}
473
\begin{remar}
474
By Proposition \ref{sha} (and under the same hypotheses as in
475
Lemma \ref{local1}), the Bloch-Kato conjecture now predicts that
476
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
477
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 $g$. Since $a_p$
497
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
498
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
499
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
500
501
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
502
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
503
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
504
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
505
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
506
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of
507
$\QQ$-rational, null-homologous, codimension $k/2$ algebraic
508
cycles on the motive $M_g$, modulo rational equivalence. (This
509
generalises the part of the Birch-Swinnerton-Dyer conjecture which
510
says that for an elliptic curve $E/\QQ$, the order of vanishing of
511
$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group
512
$E(\QQ)$.)
513
514
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
515
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
516
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
517
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
518
get (assuming also the Beilinson-Bloch conjecture) a subspace of
519
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
520
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
521
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
522
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
523
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
524
Sections 1 and 6.5 of \cite{Fo2}.
525
526
Similarly, if $L(f,k/2)\neq 0$ then we expect that
527
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
528
coincides with the $\qq$-part of $\Sha$.
529
\begin{thm}\label{local}
530
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
531
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
532
$A[\qq]$ and $A'[\qq]$ are irreducible representations of
533
$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm
534
1\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
535
modulo $\qq$ to any newform of weight~$k$, trivial character and
536
level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.
537
Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
538
$\FF_{\qq}$-rank at least $r-1$.
539
\end{thm}
540
\begin{proof}
541
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
542
continuity and rescaling we may assume that it lies in
543
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
544
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
545
class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
546
irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
547
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
548
we get a non-zero, $\qq$-torsion class $\gamma\in
549
H^1(\QQ,A_{\qq}(k/2))$.
550
551
Our aim is to show that $\res_p(\gamma)\in
552
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
553
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
554
The case $p=q$ does not quite work, which is the reason for the
555
``$r-1$'' in the statement of the theorem.
556
557
\begin{enumerate}
558
\item {\bf $p\nmid qN$. }
559
560
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
561
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
562
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
563
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
564
follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
565
$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
566
of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
567
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
568
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
569
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
570
571
\item {\bf $p\mid N$. }
572
573
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
574
$\qq$-divisible. It suffices to show that
575
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
576
but this may be done as in the proof of Lemma \ref{local1}. It
577
follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
578
$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
579
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
580
order of this group is the same as the order of the group
581
$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
582
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
583
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
584
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
585
H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
586
this common dimension is $1$. The (motivic) Euler factor at $p$
587
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
588
as multiplication by $\alpha$ on the one-dimensional space
589
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
590
\cite{Ca1} that this is the same as the Euler factor at $p$ of
591
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
592
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
593
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
594
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
595
$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm
596
1\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
597
$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
598
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
599
600
\item {\bf $p=q$. }
601
602
Since $q\nmid N$ is a prime of good reduction for the motive
603
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
604
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
605
$V'_{\qq}$ have the same dimension, where
606
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
607
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
608
It follows from Theorem 4.1(ii) of \cite{BK} that
609
$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$
610
via their exponential map.
611
612
Since $p$ is a prime of good reduction, the de Rham conjecture is
613
a consequence of the crystalline conjecture, which follows from
614
Theorem 5.6 of \cite{Fa1}. Hence
615
$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$
616
Observe that the dimension of the left-hand-side is $1$. Hence
617
there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at
618
least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It
619
follows that our $r$-dimensional $\FF_{\qq}$-subspace of the
620
$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an
621
$(r-1)$-dimensional subspace landing in
622
$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).
623
624
\end{enumerate}
625
\end{proof}
626
627
Theorem 2.7 of \cite{AS} is concerned with verifying local
628
conditions in the case $k=2$, where $f$ and $g$ are associated
629
with abelian varieties $A$ and $B$. (Their theorem also applies to
630
abelian varieties over number fields.) Our restriction outlawing
631
congruences modulo $\qq$ with cusp forms of lower level is
632
analogous to theirs forbidding $q$ from dividing Tamagawa factors
633
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
634
curve with $\ord_l(j(A))<0$, consideration of a Tate
635
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
636
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
637
at $l$.)
638
639
In this paper we have encountered two technical problems which we
640
dealt with in quite similar ways:
641
\begin{enumerate}
642
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
643
\item proving local conditions at primes $p\mid N$, for an element
644
of $\qq$-torsion.
645
\end{enumerate}
646
If our only interest was in testing the Bloch-Kato conjecture at
647
$\qq$, we could have made these problems cancel out, as in Lemma
648
8.11 of \cite{DFG}, by weakening the local conditions. However, we
649
have chosen not to do so, since we are also interested in the
650
Shafarevich-Tate group, and since the hypotheses we had to assume
651
are not particularly strong.
652
653
\section{Eleven examples}
654
\newcommand{\nf}[1]{\mbox{\bf #1}}
655
\begin{figure}
656
\caption{\label{fig:newforms}Newforms Relevant to
657
Theorem~\ref{local}}
658
$$
659
\begin{array}{|ccccc|}\hline
660
g & \deg(g) & f & \deg(f) & q's \\\hline
661
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
662
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
663
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
664
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
665
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
666
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
667
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
668
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
669
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\
670
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
671
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
672
\hline
673
\end{array}
674
$$
675
\end{figure}
676
677
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
678
eleven pairs of newforms~$f$ and~$g$ (of equal weights and levels)
679
along with at least one prime~$q$ such that there is a prime
680
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
681
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
682
\subsection{Notation}
683
Table~\ref{fig:newforms} is laid out as follows.
684
The first column contains a label whose structure is
685
\begin{center}
686
{\bf [Level]k[Weight][GaloisOrbit]}
687
\end{center}
688
This label determines a newform $g=\sum a_n q^n$, up to Galois
689
conjugacy. For example, \nf{127k4C} denotes a newform in the third
690
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
691
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
692
\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
693
the level, with positive trace being first in the event that the
694
two absolute values are equal, and the first Galois orbit is
695
denoted {\bf A}, the second {\bf B}, and so on. The second column
696
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
697
third and fourth columns contain~$f$ and its degree, respectively.
698
The fifth column contains at least one prime~$q$ such that there
699
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
700
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
701
satisfied for~$f$,~$g$, and~$\qq$.
702
703
\subsection{The first example in detail}
704
\newcommand{\fbar}{\overline{f}}
705
We describe the first line of Table~\ref{fig:newforms}
706
in more detail. See the next section for further details
707
on how the computations were performed.
708
709
Using modular symbols, we find that there is a newform
710
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
711
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
712
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
713
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
714
coefficients generate a number field~$K$ of degree~$17$, and by
715
computing the image of the modular symbol $XY\{0,\infty\}$ under
716
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
717
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
718
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
719
both equal to
720
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
721
722
There is no form in the Eisenstein subspaces of
723
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
724
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
725
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
726
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
727
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
728
of Theorem~\ref{local}, so if $r$ is the dimension of
729
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
730
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
731
732
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
733
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
734
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
735
subgroup of $\Sha$. Admitting these assumptions, we have
736
constructed the $\qq$-torsion in $\Sha$ predicted by the
737
Bloch-Kato conjecture.
738
739
For particular examples of elliptic curves one can often find and
740
write down rational points predicted by the Birch and
741
Swinnerton-Dyer conjecture. It would be nice if likewise one could
742
explicitly produce algebraic cycles predicted by the
743
Beilinson-Bloch conjecture in the above examples. Since
744
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
745
0.3.2 of \cite{Z}), so ought to be trivial in
746
$\CH_0^{k/2}(M_g)\otimes\QQ$.
747
748
\subsection{Some remarks on how the computation was performed}
749
We give a brief summary of how the computation was performed. The
750
algorithms that we used were implemented by the second author, and
751
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
752
753
Let~$g$,~$f$, and~$q$ be some data from a line of
754
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
755
verified the existence of a congruence modulo~$q$, that
756
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
757
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
758
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
759
760
To prove there is a congruence, we showed that the corresponding
761
{\em integral} spaces of modular symbols satisfy an appropriate
762
congruence, which forces the existence of a congruence on the
763
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
764
irreducible by computing a set that contains all possible residue
765
characteristics of congruences between~$g$ and any Eisenstein
766
series of level dividing~$N$, where by congruence, we mean a
767
congruence for all Fourier coefficients of index~$n$ with
768
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
769
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
770
listing a basis of such~$h$ and finding the possible congruences,
771
where again we disregard the Fourier coefficients of index not
772
coprime to~$N$.
773
774
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
775
modular symbol ${\mathbf
776
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
777
map with the same kernel as the period mapping, and found that the
778
image was~$0$. The period mapping sends the modular
779
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
780
so that ${\mathbf e}$ maps to~$0$ implies that
781
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
782
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$
783
which, because of the functional equation, implies that
784
$L'(g,\frac{k}{2})=0$.
785
786
In the course of our search, we found~$17$ plausible examples and
787
had to discard~$6$ of them because either the representation
788
$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv
789
\pm 1\pmod{q}$ for some $p\mid N$. Table~\ref{fig:newforms} is
790
of independent interest because it includes examples of modular
791
forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
792
that is not forced by the functional equation. We found no such
793
examples of weights $\geq 8$.
794
795
796
797
\begin{thebibliography}{AL}
798
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
799
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
800
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
801
Shafarevich-Tate groups of abelian varieties: evidence for the
802
Birch and Swinnerton-Dyer conjecture, in preparation.
803
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
804
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
805
\bibitem[BCP]{magma}
806
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
807
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
808
235--265, Computational algebra and number theory (London, 1993).
809
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
810
of motives, The Grothendieck Festschrift Volume I, 333--400,
811
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
812
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
813
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
814
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
815
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
816
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
817
Duke Math. J. }{\bf 59 }(1989), 785--801.
818
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
819
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
820
13--28.
821
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
822
$L_f(s)$ at the center of the critical strip, preprint.
823
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
824
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
825
{\bf 179, } 139--172, Springer, 1969.
826
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
827
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
828
part 2, 313--346.
829
\bibitem[DFG]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
830
of modular forms and the Tamagawa number conjecture, preprint.
831
{{\sf
832
http://www.andromeda.rutgers.edu/\~{\mbox{}}krm/liguo/lgpapers.html}}
833
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
834
Math. Ann. }{\bf 318 }(2000), 621--636.
835
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
836
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
837
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
838
Galois representations, {\em in }Algebraic analysis, geometry and
839
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
840
Press, Baltimore, 1989.
841
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
842
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
843
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
844
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
845
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
846
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
847
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
848
Boston Berlin, 1993.
849
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
850
repr\'esentations $p$-adiques du groupe de Galois d'un corps
851
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
852
}{\bf 115 }(1982), 529--577.
853
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
854
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
855
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
856
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
857
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
858
51--56.
859
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
860
representations coming from modular forms, {\em J. Number Theory
861
}{\bf 31 }(1989), 133--141.
862
863
\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
864
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
865
99--125.
866
\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
867
heights. The arithmetic and geometry of algebraic cycles (Banff,
868
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
869
Soc., Providence, RI, 2000.
870
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
871
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
872
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
873
congruences for coefficients of modular forms,{\em Modular
874
functions of one variable} III, Lect. Notes Math. {\bf 350, }
875
Springer, 1973.
876
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
877
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
878
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
879
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
880
\end{thebibliography}
881
882
883
\end{document}
884
\subsection{Plan for finishing the computation}
885
886
What I (William) have to do with the below stuff:\\
887
\begin{enumerate}
888
\item Fill in the missing blanks at these levels:
889
$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
890
\item Make a MAGMA program that has a table of forms to try.
891
\item First, unfortunately, for some reason the
892
labels are sometimes wrong, e.g., \nf{159k4A} should
893
have been \nf{159k4B}, so I have to check all the
894
rational forms to see which has $L(1)=0$.
895
\item Try each level using the MAGMA program (this will take a long time to run).
896
\item Finish the ``Table of examples'' above.
897
\end{enumerate}
898
899
900
\begin{verbatim}
901
902
903
(This stuff below is being integrated into the above, as I do
904
the required (rather time consuming) computations.)
905
**************************************************
906
FROM Mark Watkins:
907
908
Here's a list of stuff; the left-hand (sinister) column
909
contains forms with a double zero at the central point,
910
whilst the right-hand (dexter) column contains forms which
911
have a large square factor in LRatio at 2. The middle column
912
is the large prime factor in ModularDegree of the LHS,
913
and/or the lpf in the LRatio at 2 of the RHS.
914
915
It seems that 567k4L has invisible Sha possibly.
916
The ModularDegree for 639k4B has no large factors.
917
918
127k4A 43 127k4C
919
159k4A 23 159k4E
920
365k4A 29 365k4E
921
369k4A 13 369k4I
922
453k4A 17 453k4E
923
453k4A 23
924
465k4A 11 465k4H
925
477k4A 73 477k4L
926
567k4A 23 567k4G
927
13 567k4L
928
581k4A 19 581k4E
929
639k4B --
930
931
932
Forms with spurious zeros to do:
933
934
657k4A
935
681k4A
936
684k4B
937
95k6A 31,59
938
116k6A --
939
122k6A 73
940
260k6A
941
942
If we allow 5 and 7 to be small primes, then we get more
943
visibility info.
944
945
159k4A 5 159k4E
946
369k4A 5 369k4I
947
453k4A 5 453k4E
948
639k4B 7 639k4H
949
950
Maybe the general theorem does not apply to primes which are so small,
951
but we might be able to show that they are OK in these specific cases.
952
953
I checked the first three weight 6 examples with AbelianIntersection,
954
but actually computing the LRatio bogs down too much. Will need
955
new version.
956
957
95k6A 31 95k6D
958
95k6A 59 95k6D
959
116k6A 5 116k6D
960
122k6A 73 122k6D
961
962
Incidentally, the LRatio for 581k4E has a power of 19^4.
963
Perhaps not surprising, as the ModularDegree of 581k4A
964
has a factor of 19^2. But maybe it means a bigger Sha.
965
\end{verbatim}
966