CoCalc Shared Fileswww / papers / motive_visibility / dsw3.texOpen in CoCalc with one click!
Author: William A. Stein
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\documentclass{amsart}
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\newtheorem{defi}[prop]{Definition}
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\newtheorem{conj}[prop]{Conjecture}
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\newtheorem{examp}[prop]{Example}
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\newtheorem{remar}[prop]{Remark}
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% ---- SHA ----
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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{May 29th, 2002}
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\subjclass{11F33, 11F67, 11G40.}
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\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
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Shafarevich-Tate group.}
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\address{University of Sheffield\\ Department of Pure
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Mathematics\\
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Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
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U.K.}
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\address{Harvard University\\Department of Mathematics\\
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One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
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\address{Penn State Mathematics Department\\
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University Park\\State College, PA 16802\\ U.S.A.}
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\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
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\email{watkins@math.psu.edu}
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\maketitle {\bf Not for distribution}
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\section{Introduction}
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Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
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be the associated $L$-function. The conjecture of Birch and
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Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
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at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
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also gives an interpretation of the leading term in the Taylor
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expansion in terms of various quantities, including the order of
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the Shafarevich-Tate group of $E$.
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Cremona and Mazur [2000] look, among all strong Weil elliptic
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curves over $\QQ$ of conductor $N\leq 5500$, at those with
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non-trivial Shafarevich-Tate group (according to the Birch and
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Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
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group has predicted elements of prime order $m$. In most cases
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they find another elliptic curve, often of the same conductor,
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whose $m$-torsion is Galois-isomorphic to that of the first one,
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and which has 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 show
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\cite{CM2} that these lie in the Shafarevich-Tate group of the
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first curve, so rational points on one curve explain elements of
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the Shafarevich-Tate group of the other curve.
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The Bloch-Kato conjecture \cite{BK} is the generalisation to
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arbitrary motives of the leading term part of the Birch and
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Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
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\cite{B} generalises the part about the order of vanishing at the
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central point, identifying it with the rank of a certain Chow
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group.
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The present work may be considered as a partial generalisation of
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the work of Cremona and Mazur, from elliptic curves over $\QQ$
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(which are associated to modular forms of weight $2$) to the
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motives attached to modular forms of higher weight. (See \cite{AS}
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for a different generalisation, to modular abelian varieties of
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higher dimension.) It may also be regarded as doing, for
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congruences between modular forms of equal weight, what \cite{Du2}
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did for congruences between modular forms of different weights.
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We consider the situation where two newforms $f$ and $g$, both of
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weight $k>2$ and level $N$, are congruent modulo a maximal ideal
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$\qq$ of odd residue characteristic, $L(g,k/2)=0$ but
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$L(f,k/2)\neq 0$. It turns out that this forces $L(g,s)$ to vanish
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to order at least $2$ at $s=k/2$. We are able to find sixteen
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examples (all with $k=4$ and $k=6$), and in each case $\qq$
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divides the numerator of the algebraic number
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$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
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canonical period. In fact, we show how this divisibility may be
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deduced from the vanishing of $L(g,k/2)$ using recent work of
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Vatsal \cite{V}. The point is, the congruence between $f$ and $g$
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leads to a congruence between suitable ``algebraic parts'' of the
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special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
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other is divisible by $\qq$. Under certain hypotheses, the
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Bloch-Kato conjecture then implies that the Shafarevich-Tate group
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attached to $f$ has non-zero $\qq$-torsion. Under certain
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hypotheses and assumptions, the most substantial of which is the
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Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
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the existence of algebraic cycles, we are able to construct the
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predicted elements of $\Sha$, using the Galois-theoretic
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interpretation of the congruences to transfer elements from a
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Selmer group for $g$ to a Selmer group for $f$. In proving the
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local conditions at primes dividing the level, and also in
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examining the local Tamagawa factors at these primes, we make use
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of a higher weight level-lowering result due to Jordan and Livn\'e
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\cite{JL}.
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One might say that algebraic cycles for one motive explain
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elements of $\Sha$ for the other. A main point of \cite{CM} was to
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observe the frequency with which those elements of $\Sha$
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predicted to exist for one elliptic curve may be explained by
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finding a congruence with another elliptic curve containing points
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of infinite order. One shortcoming of our work, compared to the
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elliptic curve case, is that, due to difficulties with local
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factors in the Bloch-Kato conjecture, we are unable to compute the
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exact order of $\Sha$ predicted by the Bloch-Kato conjecture. We
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have to start with modular forms between which there exists a
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congruence. However, Vatsal's work allows us to explain how the
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vanishing of one $L$-function leads, via the congruence, to the
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divisibility by $\qq$ of (an algebraic part of) another,
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independent of observations of computational data. The
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computational data does however show that there exist examples to
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which our results apply. Moreover, it displays factors of $\qq^2$,
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whose existence we do not prove theoretically, but which are
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predicted by Bloch-Kato.
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\section{Motives and Galois representations}
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Let $f=\sum a_nq^n$ be a newform of weight $k\geq 2$ for
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$\Gamma_0(N)$, with coefficients in an algebraic number field $E$,
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which is necessarily totally real. A theorem of Deligne \cite{De1}
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implies the existence, for each (finite) prime $\lambda$ of $E$,
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of a two-dimensional vector space $V_{\lambda}$ over
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$E_{\lambda}$, and a continuous representation
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$$\rho_{\lambda}:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_{\lambda}),$$
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such that
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\begin{enumerate}
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\item $\rho_{\lambda}$ is unramified at $p$ for all primes $p$ not dividing
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$lN$ (where $\lambda \mid l$);
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\item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then the
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characteristic polynomial of $\Frob_p^{-1}$ acting on
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$V_{\lambda}$ is $x^2-a_px+p^{k-1}$.
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\end{enumerate}
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Following Scholl \cite{Sc}, $V_{\lambda}$ may be constructed as
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the $\lambda$-adic realisation of a Grothendieck motive $M_f$.
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There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$,
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both $2$-dimensional $E$-vector spaces. For details of the
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construction see \cite{Sc}. The de Rham realisation has a Hodge
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filtration $V_{\dR}=F^0\supset F^1=\cdots =F^{k-1}\supset
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F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
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cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
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cohomology. There are natural isomorphisms $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. We may choose a
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
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Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
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There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
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which amounts to multiplying the action of $\Frob_p$ by $p^j$.
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Following \cite{BK} (Section 3), for $p\neq l$ (including
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$p=\infty$) let
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$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).$$ 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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Define an ideal $\#\Sha(j)$ of $O_E$, in which the exponent of any
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prime ideal $\lambda$ is the length of the $\lambda$-component of
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$\Sha(j)$. We shall only concern ourselves with the case $j=k/2$,
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and write $\Sha$ for $\Sha(j)$.
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Define the group of global torsion points
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$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. Define an ideal $\#\Gamma_{\QQ}$ of $O_E$, in
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which the exponent of any prime ideal $\lambda$ is the length of
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the $\lambda$-component of $\Gamma_{\QQ}$.
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\section{Canonical periods}
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We assume from now on for convenience that $N\geq 3$. We need to
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choose convenient $O_E$-lattices $T_B$ and $T_{\dR}$ in the Betti
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and deRham realisations $V_B$ and $V_{\dR}$ of $M_f$. We do this
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in a way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and
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$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the
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$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and
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$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially
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Sections 2.2 and 5.4 of \cite{DFG}.)
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For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
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module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
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$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the
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$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
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since it comes from $\ell$-adic cohomology with $O_{\lambda}$
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coefficients. We may assume that $T_{\lambda}$ is
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$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting
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$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.
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Let $M(N)$ be the modular curve over $\ZZ[1/N]$ parametrising
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generalised elliptic curves with full level-$N$ structure. Let
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$\mathfrak{E}$ be the universal generalised elliptic curve over
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$M(N)$. Let $\mathfrak{E}^{k-2}$ be the $(k-2)$-fold fibre product
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of $\mathfrak{E}$ over $M(N)$. Realising $M(N)(\CC)$ as the
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disjoint union of $\phi(N)$ copies of the quotient
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$\Gamma(N)\backslash\mathfrak{H}^*$, and letting $\tau$ be a
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variable on $\mathfrak{H}$, the fibre $\mathfrak{E}_{\tau}$ is
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isomorphic to the elliptic curve with period lattice generated by
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$1$ and $\tau$. Let $z_i\in\CC/\langle 1,\tau \rangle$ be a
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variable on the $i^{th}$ copy of $\mathfrak{E}_{\tau}$ in the
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fibre product. Then $2\pi i f(\tau)\,d\tau\wedge
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dz_1\wedge\ldots\wedge dz_{k-2}$ is a well-defined differential
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form on (a desingularisation of) $\mathfrak{E}^{k-2}$ and
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naturally represents a generating element of $F^{k-1}T_{\dR}$. (At
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least, we can make our choices locally at primes dividing $Nk!$ so
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that this is the case.) We shall call this element $e(f)$.
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Under the deRham isomorphism between $V_{\dR}\otimes\CC$ and
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$V_B\otimes\CC$, $e(f)$ maps to some element $\omega_f$. There is
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a natural action of complex conjugation on $V_B$, breaking it up
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into one-dimensional $E$-vector spaces $V_B^{+}$ and $V_B^{-}$.
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Let $\omega_f^+$ and $\omega_f^-$ be the projections of $\omega_f$
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to $V_B^+\otimes\CC$ and $V_B^-\otimes\CC$, respectively. Let
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$T_B^{\pm}$ be the intersections of $V_B^{\pm}$ with $T_B$. These
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are rank one $O_E$-modules, but not necessarily free, since the
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class number of $O_E$ may be greater than one. Choose non-zero
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elements $\delta_f^{\pm}$ of $T_B^{\pm}$ and let $\aaa^{\pm}$ be
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the ideals $[T_B^{\pm}:O_E\delta_f^{\pm}]$. Define complex numbers
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$\Omega_f^{\pm}$ by $\omega_f^{\pm}=2\pi i
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\Omega_f^{\pm}\delta_f^{\pm}$.
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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 with
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Euler product
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$\sum_{n=1}^{\infty}a_nn^{-s}=\prod_p(P_p(p^{-s}))^{-1}$, but
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there is an analytic continuation given by an integral, as
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described in the next section. Suppose that $L(f,k/2)\neq 0$. The
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Bloch-Kato conjecture for the motive $M_f(k/2)$ predicts the
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following equality of fractional ideals of $E$:
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$${L(f,k/2)\over \vol_{\infty}}=
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{\left(\prod_pc_p(k/2)\right)\#\Sha\over
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\aaa^{\pm}(\#\Gamma_{\QQ})^2}.$$ (Strictly speaking, the
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conjecture in \cite{BK} is only given for $E=\QQ$.) Here, $\pm$
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represents the parity of $(k/2)-1$, and $\vol_{\infty}$ is equal
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to $(2\pi i)^{k/2}$ multiplied by the determinant of the
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isomorphism $V_B^{\pm}\otimes\CC\simeq
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(V_{\dR}/F^{k/2})\otimes\CC$, calculated with respect to the
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lattices $O_E\delta_f^{\pm}$ and the image of $T_{\dR}$. For
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$l\neq p$, $\ord_{\lambda}(c_p(j))$ is defined to be
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$$\length\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
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\ord_{\lambda}(P_p(p^{-j}))$$
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$$=\length\>\> (H^0(\QQ_p,A_{\lambda}(j))/H^0(\QQ_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p})).$$
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We omit the definition of $\ord_{\lambda}(c_p(j))$ for
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$\lambda\mid p$, which requires one to assume Fontaine's de Rham
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conjecture (\cite{Fo}, Appendix A6), and depends on the choices of
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$T_{\dR}$ and $T_B$, locally at $\lambda$. (We shall mainly be
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concerned with the $q$-part of the Bloch-Kato conjecture, where
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$q$ is a prime of good reduction. For such primes, the de Rham
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conjecture follows from Theorem 5.6 of \cite{Fa1}.)
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\begin{lem}\label{vol}
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$\vol_{\infty}=c(2\pi i)^{k/2}\Omega_{\pm}$, with $c\in E$ and
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$\ord_{\lambda}(c)=0$ for $\lambda\nmid \aaa^{\pm}Nk!$.
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\end{lem}
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\begin{proof} $\vol_{\infty}$ is also equal to the determinant
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of the period map from $F^{k/2}V_{\dR}\otimes\CC$ to
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$V_B^{\pm}\otimes\CC$, with respect to lattices dual to those we
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used above in the definition of $\vol_{\infty}$ (c.f. the last
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paragraph of 1.7 of \cite{De2}). We are using here the natural
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pairings. Recall that the index of $O_E\delta_f^{\pm}$ in
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$T_B^{\pm}$ is the ideal $\aaa^{\pm}$. Then the proof is completed
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by noting that, locally away from primes dividing $Nk!$, the index
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of $T_{\dR}$ in its dual is equal to the index of $T_B$ in its
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dual, both being equal to the ideal denoted $\eta$ in \cite{DFG2}.
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\end{proof}
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\begin{lem} Let $p\nmid N$ be a prime, $j$ an integer.
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Then the fractional ideal $c_p(j)$ is supported at most on
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divisors of $p$.
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\end{lem}
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\begin{proof}
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As on p.~30 of \cite{Fl1}, for odd $l\neq p$,
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$\ord_{\lambda}(c_p(j))$ is the length of the finite
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$O_{\lambda}$-module $H^0(\QQ_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
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where $I_p$ is an inertia group at $p$. But $T_{\lambda}(j)$ is a
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trivial $I_p$-module, so $H^1(I_p,T_{\lambda}(j))$ is
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torsion-free.
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\end{proof}
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\begin{lem}\label{local1}
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Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that $A[\qq]$
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is an irreducible representation of $\Gal(\Qbar/\QQ)$, where
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$\qq\mid q$. Let $p\mid N$ be a prime, and if $p^2\mid N$ suppose
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that $\,p\not\equiv -1\pmod{q}$. Suppose also that $f$ is not congruent
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modulo $\qq$ to any newform of weight~$k$, trivial character and
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level dividing $N/p$. Then for $j$ any integer,
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$\ord_{\qq}(c_p(j))=0$.
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\end{lem}
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\begin{proof}
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It suffices to show that
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$$\dim_{O_E/\qq} H^0(I_p,A[\qq](j))=\dim_{E_{\qq}} H^0(I_p,V_{\qq}(j)),$$
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since this ensures that
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$H^0(I_p,A_{\qq}(j))=V_{\qq}^{I_p}/T_{\qq}^{I_p}$, hence that
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$H^0(\QQ_p,A_{\qq}(j))=H^0(\QQ_p,V_{\qq}^{I_p}/T_{\qq}^{I_p})$. If
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the dimensions differ then, given that $f$ is not congruent modulo
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$\qq$ to a newform of level dividing $N/p$, Proposition 2.2 of
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\cite{L} shows that we are in the situation covered by one of the
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three cases in Proposition 2.3 of \cite{L}. Since $p\not\equiv
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-1\pmod{q}$ if $p^2\mid N$, Case 3 is excluded, so $A[\qq](j)$ is
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unramified at $p$ and $\ord_p(N)=1$. (Here we are using Carayol's
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result that $N$ is the prime-to-$q$ part of the conductor of
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$V_{\qq}$ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
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the condition $q>k$) implies the existence of a newform of weight
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$k$, trivial character and level dividing $N/p$, congruent to $g$
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modulo $\qq$. This contradicts our hypotheses.
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\end{proof}
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\begin{remar}
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For an example of what can be done when $f$ {\em is } congruent to
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a form of lower level, see the first example in 7.4 below.
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\end{remar}
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\begin{lem}\label{at q} 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 Lemma 5.7 of \cite{DFG} (whose proof relies on an
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application, at the end of section 2.2, 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 $T_{\dR}\otimes O_{\qq}$ by the functor they call
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$\mathbb{V}$. (This property is part of the definition of an
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$S$-integral premotivic structure given in Section 1.2 of
421
\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
422
of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
423
in Theorem 4.1 of \cite{BK} follows from the first paragraph of
424
2(h) of \cite{Fa1}.)
425
\end{proof}
426
427
\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
428
irreducible representation of $\Gal(\Qbar/\QQ)$.
429
\end{lem}
430
This follows trivially from the definition.
431
432
Putting together the above lemmas we arrive at the following:
433
\begin{prop}\label{sha} Assume the same hypotheses as in Lemma \ref{local1},
434
for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
435
are as in the previous section. If
436
$$\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0$$ (with numerator
437
non-zero) then the Bloch-Kato conjecture predicts that
438
$$\ord_{\qq}(\#\Sha)>0.$$
439
\end{prop}
440
441
\section{Congruences of special values}
442
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
443
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
444
large enough to contain all the coefficients $a_n$ and $b_n$.
445
Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
446
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
447
that $q\nmid N\phi(N)k!$ It is easy to see that we may choose the
448
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
449
$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
450
$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
451
has been made.
452
453
We shall now make two further assumptions:
454
\begin{enumerate}
455
\item $L(f,k/2)\neq 0$;
456
\item $L(g,k/2)=0$.
457
\end{enumerate}
458
\begin{prop} \label{div}
459
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
460
\end{prop}
461
\begin{proof} This is based on some of the ideas used in Section 1 of
462
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
463
which presumably should refer to ``Condition 2''. Since
464
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
465
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
466
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
467
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
468
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
469
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
470
where the integral is taken along the positive imaginary axis,
471
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
472
Thus we are reduced
473
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
474
475
Let $\mathcal{D}_0$ be the group of divisors of degree zero
476
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
477
integer $r\geq 0$, let $P_r(R)$ be the additive group of
478
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
479
groups have a natural action of $\Gamma_1(N)$. Let
480
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
481
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
482
483
Via the isomorphism (8) in Section 1.5 of \cite{V},
484
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
485
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
486
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
487
(See also Section 4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
488
into account the fact that we are now dealing with $X_1(N)$ rather
489
that $M(N)$. Up to some small factorials which do not matter
490
locally at $\qq$,
491
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
492
(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since
493
$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that
494
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
495
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
496
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
497
show is divisible by $\qq$.
498
Similarly
499
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
500
(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient
501
of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
502
Therefore it would suffice to show that, for some $\mu\in O_E$,
503
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
504
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
505
that, for some $\mu\in O_E$, the element
506
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
507
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
508
non-constant coefficients. This would be the case if
509
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
510
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
511
a consequence of Theorem 2.1(1) of \cite{FJ}.
512
\end{proof}
513
\begin{remar}
514
By Proposition \ref{sha} (assuming, for all $p\mid N$ the same
515
hypotheses as in Lemma \ref{local1}, together with
516
$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
517
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
518
construction of the required elements of $\Sha$.
519
\end{remar}
520
\begin{remar}\label{sign}
521
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
522
have to be equal, since they are determined by the action of the
523
involution $W_N$ on the common subspace generated by the
524
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
525
Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the
526
common eigenvalue of $W_N$ acting on $f$ and $g$.
527
\end{remar}
528
This is analogous to the remark at the end of Section 3 of
529
\cite{CM}, which shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$
530
then $L(g,s)$ must vanish to order at least two, as in all the
531
examples below. It is worth pointing out that there are no
532
examples of $g$ of level one, and positive sign in the functional
533
equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that
534
all the normalised cuspidal eigenforms of weight $k$ and level one
535
are Galois conjugate) is false. See \cite{CF}.
536
537
\section{Constructing elements of the Shafarevich-Tate group}
538
Let $f$ and $g$ be as in the first paragraph of the previous
539
section. For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
540
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
541
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
542
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
543
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
544
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
545
546
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
547
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
548
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
549
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
550
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
551
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
552
rational equivalence classes of null-homologous, codimension $k/2$
553
algebraic cycles on the motive $M_g$. (This generalises the part
554
of the Birch--Swinnerton-Dyer conjecture which says that for an
555
elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
556
$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
557
558
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
559
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
560
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
561
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
562
get (assuming also the Beilinson-Bloch conjecture) a subspace of
563
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
564
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
565
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
566
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
567
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
568
Sections 1 and 6.5 of \cite{Fo2}.
569
570
Similarly, if $L(f,k/2)\neq 0$ then we expect that
571
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
572
coincides with the $\qq$-part of $\Sha$.
573
\begin{thm}\label{local}
574
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
575
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
576
$A[\qq]$ is an irreducible representations of $\Gal(\Qbar/\QQ)$
577
and that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
578
newform of weight~$k$, trivial character and level dividing $N/p$.
579
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
580
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
581
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
582
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
583
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
584
\end{thm}
585
\begin{proof}
586
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
587
continuity and rescaling we may assume that it lies in
588
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
589
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
590
$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
591
non-zero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq
592
H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
593
is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
594
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
595
we get a non-zero, $\qq$-torsion class $\gamma\in
596
H^1(\QQ,A_{\qq}(k/2))$.
597
598
Our aim is to show that $\res_p(\gamma)\in
599
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
600
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
601
602
\begin{enumerate}
603
\item {\bf $p\nmid qN$. }
604
605
Consider the $I_p$-cohomology of the short exact sequence
606
$$\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},$$
607
where $\pi$ is multiplication by a uniformising element of
608
$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
609
$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
610
$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
611
remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
612
$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in
613
H^1_f(\QQ,V'_{\qq}(k/2))$ that the image in
614
$H^1(I_p,A'_{\qq}(k/2))$ of the restriction of $c$ is zero, hence
615
that the restriction of $c$ (to $H^1(I_p,A'[\qq](k/2))\simeq
616
H^1(I_p,A[\qq](k/2))$) is zero. Hence the restriction of $\gamma$
617
to $H^1(I_p,A_{\qq}(k/2))$ is also zero. By line 3 of p.~125 of
618
\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
619
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
620
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
621
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
622
623
\item {\bf $p\mid N$. }
624
625
First we show that $H^0(I_p, A'_{\qq}(k/2))$ is $\qq$-divisible.
626
It suffices to show that
627
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
628
since this would imply that the natural map from
629
$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
630
surjective, but this may be done as in the proof of Lemma
631
\ref{local1}. It follows as above that the image of $c\in
632
H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
633
$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
634
inflation-restriction. The order of this group is the same as the
635
order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
636
trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
637
$V_{\qq}(k/2)$ is ramified at $p$, so $\dim
638
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
639
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
640
consider the case where this common dimension is $1$. The
641
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
642
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by
643
$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
644
follows from Theor\'eme A of \cite{Ca1} that this is the same as
645
the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
646
Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
647
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
648
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
649
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
650
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
651
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
652
$\res_p(\gamma)=0$ and certainly lies in
653
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
654
655
\item {\bf $p=q$. }
656
657
Since $q\nmid N$ is a prime of good reduction for the motive
658
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
659
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
660
$V'_{\qq}$ have the same dimension, where
661
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
662
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
663
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
664
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
665
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
666
may now prove, in the same manner as Proposition 9.2 of
667
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
668
\end{enumerate}
669
\end{proof}
670
671
Theorem 2.7 of \cite{AS} is concerned with verifying local
672
conditions in the case $k=2$, where $f$ and $g$ are associated
673
with abelian varieties $A$ and $B$. (Their theorem also applies to
674
abelian varieties over number fields.) Our restriction outlawing
675
congruences modulo $\qq$ with cusp forms of lower level is
676
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
677
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
678
curve with $\ord_l(j(A))<0$, consideration of a Tate
679
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
680
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
681
at $l$.)
682
683
In this paper we have encountered two technical problems which we
684
dealt with in quite similar ways:
685
\begin{enumerate}
686
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
687
\item proving local conditions at primes $p\mid N$, for an element
688
of $\qq$-torsion.
689
\end{enumerate}
690
If our only interest was in testing the Bloch-Kato conjecture at
691
$\qq$, we could have made these problems cancel out, as in Lemma
692
8.11 of \cite{DFG}, by weakening the local conditions. However, we
693
have chosen not to do so, since we are also interested in the
694
Shafarevich-Tate group, and since the hypotheses we had to assume
695
are not particularly strong.
696
697
\section{Sixteen examples}
698
\newcommand{\nf}[1]{\mbox{\bf #1}}
699
\begin{figure}
700
\caption{\label{fig:newforms}Newforms Relevant to
701
Theorem~\ref{local}}
702
$$
703
\begin{array}{|ccccc|}\hline
704
g & \deg(g) & f & \deg(f) & q's \\\hline
705
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
706
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
707
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
708
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
709
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
710
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
711
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
712
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
713
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
714
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
715
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
716
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
717
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
718
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
719
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
720
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
721
\hline
722
\end{array}
723
$$
724
\end{figure}
725
726
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
727
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
728
along with at least one prime~$q$ such that there is a prime
729
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
730
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
731
\subsection{Notation}
732
Table~\ref{fig:newforms} is laid out as follows.
733
The first column contains a label whose structure is
734
\begin{center}
735
{\bf [Level]k[Weight][GaloisOrbit]}
736
\end{center}
737
This label determines a newform $g=\sum a_n q^n$, up to Galois
738
conjugacy. For example, \nf{127k4C} denotes a newform in the third
739
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
740
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
741
\ldots)$, then by the sequence of absolute values $|\mbox{\rm
742
Tr}(a_p(g))|$ for~$p$ not dividing the level, with positive trace
743
being first in the event that the two absolute values are equal,
744
and the first Galois orbit is denoted {\bf A}, the second {\bf B},
745
and so on. The second column contains the degree of the field
746
$\QQ(\ldots, a_n, \ldots)$. The third and fourth columns
747
contain~$f$ and its degree, respectively. The fifth column
748
contains at least one prime~$q$ such that there is a prime
749
$\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that the
750
hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
751
satisfied for~$f$,~$g$, and~$\qq$.
752
753
\subsection{The first example in detail}
754
\newcommand{\fbar}{\overline{f}}
755
We describe the first line of Table~\ref{fig:newforms}
756
in more detail. See the next section for further details
757
on how the computations were performed.
758
759
Using modular symbols, we find that there is a newform
760
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
761
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
762
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
763
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
764
coefficients generate a number field~$K$ of degree~$17$, and by
765
computing the image of the modular symbol $XY\{0,\infty\}$ under
766
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
767
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
768
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
769
both equal to
770
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
771
+ \cdots\in \FF_{43}[[q]].$$
772
773
There is no form in the Eisenstein subspaces of
774
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
775
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
776
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
777
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
778
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
779
of Theorem~\ref{local}, so if $r$ is the dimension of
780
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
781
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
782
783
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
784
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
785
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
786
subgroup of $\Sha$. Admitting these assumptions, we have
787
constructed the $\qq$-torsion in $\Sha$ predicted by the
788
Bloch-Kato conjecture.
789
790
For particular examples of elliptic curves one can often find and
791
write down rational points predicted by the Birch and
792
Swinnerton-Dyer conjecture. It would be nice if likewise one could
793
explicitly produce algebraic cycles predicted by the
794
Beilinson-Bloch conjecture in the above examples. Since
795
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
796
0.3.2 of \cite{Z}), so ought to be trivial in
797
$\CH_0^{k/2}(M_g)\otimes\QQ$.
798
799
\subsection{Some remarks on how the computation was performed}
800
We give a brief summary of how the computation was performed. The
801
algorithms that we used were implemented by the second author, and
802
most are a standard part of MAGMA (see \cite{magma}).
803
804
Let~$g$,~$f$, and~$q$ be some data from a line of
805
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
806
verified the existence of a congruence modulo~$q$, that
807
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
808
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
809
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
810
811
To prove there is a congruence, we showed that the corresponding
812
{\em integral} spaces of modular symbols satisfy an appropriate
813
congruence, which forces the existence of a congruence on the
814
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
815
irreducible by computing a set that contains all possible residue
816
characteristics of congruences between~$g$ and any Eisenstein
817
series of level dividing~$N$, where by congruence, we mean a
818
congruence for all Fourier coefficients of index~$n$ with
819
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
820
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
821
listing a basis of such~$h$ and finding the possible congruences,
822
where again we disregard the Fourier coefficients of index not
823
coprime to~$N$.
824
825
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
826
modular symbol ${\mathbf
827
e}=X^{\frac{k}{2}-1}Y^{\frac{k}{2}-1}\{0,\infty\}$ under a map
828
with the same kernel as the period mapping, and found that the
829
image was~$0$. The period mapping sends the modular
830
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
831
so that ${\mathbf e}$ maps to~$0$ implies that
832
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
833
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
834
=(-1)^{k/2} g$ which, because of the functional equation, implies
835
that $L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of
836
independent interest because it includes examples of modular forms
837
of weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
838
the functional equation. We found no such examples of weights
839
$\geq 8$.
840
841
For the two examples \nf{581k4E} and \nf{684k4K}, the square of a
842
prime appears in the $q$-column, meaning $f$ and $g$ are congruent
843
$\bmod{\,\qq^2}$ for some $\qq\mid q$. In these cases, a
844
modification of Theorem \ref{local} will show that the
845
$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
846
$O_E/\qq^2$-rank at least $r$.
847
848
\subsection{Other examples}
849
We have some other examples where a congruence between forms can
850
be shown, but the levels differ. However, Remark 5.3 does not
851
apply, so that one of the forms could have an odd functional
852
equation, and the other could have an even functional equation.
853
For instance, we have a 13-congruence between $g=\nf{81k4A}$ and
854
$f=\nf{567k4L}$; here $L(\nf{567k4L},2)\neq 0$, while
855
$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation.
856
Here $f$ obviously fails the condition about not being congruent
857
to a form of lower level, so in Lemma 4.3 it is possible that
858
$\ord_{\qq}(c_7(2))>0$. In fact this does happen. Because
859
$V'_{\qq}$ (attached to g of level $81$) is unramified at $p=7$,
860
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
861
two-dimensional. As in (2) of the proof of Theorem \ref{local},
862
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
863
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
864
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
865
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
866
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
867
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
868
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
869
our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
870
non-trivial when $w_p=-1$, so (2) of the proof of Theorem
871
\ref{local} does not work. This is just as well, since had it
872
worked we would have expected
873
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
874
shows not to be the case.
875
876
Here is an example where the divisibility between the levels is
877
the other way round: a 7-congruence between $g=\nf{122k6A}$ and
878
$f=\nf{61k6B}$. In this case both $L$-functions have even
879
functional equation, and we have $L(\nf{122k6A},3)=0$. In the
880
proof of Theorem 6.1, we find a problem with the local condition
881
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
882
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
883
kernel is at most one-dimensional, so we still get the
884
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
885
$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get
886
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
887
elements of $\Sha$ are {\it invisible} at level 61. When the
888
levels are different we are no longer able to apply Theorem 2.1 of
889
\cite{FJ}. However, we still have the congruences of integral
890
modular symbols required to make the proof of Proposition
891
\ref{div} go through. Indeed, as noted above, the congruences of
892
modular forms were found by producing congruences of modular
893
symbols. Despite these congruences of modular symbols, Remark 5.3
894
does not apply, since there is no reason to suppose that
895
$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.
896
897
Finally, there are two examples where we have a form $g$ with even
898
functional equation such that $L(g,k/2)=0$, and a congruent form
899
$f$ which has odd functional equation; these are a 23-congruence
900
between $g=\nf{453k4A}$ and $f=\nf{151k4A}$, and a 43-congruence
901
between $g=\nf{681k4A}$ and $f=\nf{227k4A}$. If
902
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
903
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
904
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
905
the appropriate modification of Theorem \ref{local} does not
906
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
907
It only tells us that the $\qq$-torsion subgroup of
908
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
909
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
910
appears in the conjectural formula for the first derivative of the
911
complex $L$ function, evaluated at $s=k/2$, but in combination
912
with a regulator that we have no way of calculating.
913
914
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
915
associated with $f$ and $g$ by the construction of Mazur, Tate and
916
Teitelbaum \cite{MTT}, each divided by a suitable canonical
917
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
918
quite clear what to make of this. This divisibility may be proved
919
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
920
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
921
normalised) are congruent $\bmod{\,\qq}$, as a result of the
922
congruence between the modular symbols out of which they are
923
constructed. Integrating an appropriate function against these
924
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{\,\qq}$
925
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
926
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
927
where the signs in the functional equations of $L(g,s)$ and
928
$L_q(g,s)$ are the same, positive in this instance. (According to
929
the proposition in Section 18 of \cite{MTT}, the signs differ
930
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
931
932
933
934
\subsection{Excluded data}
935
We also found some examples for which the conditions of Theorem \ref{local}
936
were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},
937
but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar
938
problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here
939
$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a
940
5-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5
941
is less than the weight 6.
942
943
\begin{thebibliography}{AL}
944
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
945
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
946
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
947
Shafarevich-Tate groups of abelian varieties, preprint.
948
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
949
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
950
\bibitem[BCP]{magma}
951
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
952
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
953
235--265, Computational algebra and number theory (London, 1993).
954
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
955
of motives, The Grothendieck Festschrift Volume I, 333--400,
956
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
957
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
958
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
959
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
960
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
961
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
962
Duke Math. J. }{\bf 59 }(1989), 785--801.
963
\bibitem[CM1]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
964
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
965
13--28.
966
\bibitem[CM2]{CM2} J. E. Cremona, B. Mazur, Appendix to A. Agashe,
967
W. Stein, Visible evidence for the Birch and Swinnerton-Dyer
968
conjecture for modular abelian varieties of rank zero, preprint.
969
\bibitem[CF]{CF} B. Conrey, D. Farmer, On the non-vanishing of
970
$L_f(s)$ at the center of the critical strip, preprint.
971
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
972
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
973
{\bf 179, } 139--172, Springer, 1969.
974
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
975
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
976
part 2, 313--346.
977
\bibitem[DFG1]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
978
of modular forms and the Tamagawa number conjecture, preprint.
979
{{\sf
980
http://www.andromeda.rutgers.edu/\~{\mbox{}}liguo/lgpapers.html}}
981
\bibitem[DFG2]{DFG2} F. Diamond, M. Flach, L. Guo, The Bloch-Kato
982
conjecture for adjoint motives of modular forms, {\em Math. Res.
983
Lett. }{\bf 8 }(2001), 437--442.
984
\bibitem[Du1]{Du3} N. Dummigan, Symmetric square $L$-functions and
985
Shafarevich-Tate groups, {\em Experiment. Math. }{\bf 10 }(2001),
986
383--400.
987
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
988
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
989
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
990
Galois representations, {\em in }Algebraic analysis, geometry and
991
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
992
Press, Baltimore, 1989.
993
\bibitem[FJ]{FJ} G. Faltings, B. Jordan, Crystalline cohomology
994
and $\GL(2,\QQ)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
995
\bibitem[Fl1]{Fl2} M. Flach, A generalisation of the Cassels-Tate
996
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.
997
\bibitem[Fl2]{Fl1} M. Flach, On the degree of modular parametrisations,
998
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
999
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
1000
Boston Berlin, 1993.
1001
\bibitem[Fo1]{Fo} J.-M. Fontaine, Sur certains types de
1002
repr\'esentations $p$-adiques du groupe de Galois d'un corps
1003
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
1004
}{\bf 115 }(1982), 529--577.
1005
\bibitem[Fo2]{Fo2} J.-M. Fontaine, Valeurs sp\'eciales des
1006
fonctions $L$ des motifs, S\'eminaire Bourbaki, Vol. 1991/92. {\em
1007
Ast\'erisque }{\bf 206 }(1992), Exp. No. 751, 4, 205--249.
1008
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture ``epsilon''
1009
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
1010
51--56.
1011
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
1012
representations coming from modular forms, {\em J. Number Theory
1013
}{\bf 31 }(1989), 133--141.
1014
\bibitem[MTT]{MTT} B. Mazur, J. Tate, J. Teitelbaum, On $p$-adic
1015
analogues of the conjectures of Birch and Swinnerton-Dyer, {\em
1016
Invent. Math. }{\bf 84 }(1986), 1--48.
1017
\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
1018
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
1019
99--125.
1020
\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
1021
heights. The arithmetic and geometry of algebraic cycles (Banff,
1022
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
1023
Soc., Providence, RI, 2000.
1024
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
1025
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
1026
\bibitem[St]{St} G. Stevens, $\Lambda$-adic modular forms of
1027
half-integral weight and a $\Lambda$-adic Shintani lifting.
1028
Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math.,
1029
174, Amer. Math. Soc., Providence, RI, 1994.
1030
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
1031
congruences for coefficients of modular forms,{\em Modular
1032
functions of one variable} III, Lect. Notes Math. {\bf 350, }
1033
Springer, 1973.
1034
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
1035
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
1036
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
1037
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
1038
\end{thebibliography}
1039
1040
1041
\end{document}
1042
127k4A 43 127k4C 17 [43]
1043
159k4A 5,23 159k4E 8 [5]x[23]
1044
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1045
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1046
453k4A 5,17 453k4E 23 [5]x[17]
1047
453k4A 23 151k4A 30 Odd func eq for g
1048
465k4A 11 465k4H 7 [11]x[5]x[2]
1049
477k4A 73 477k4M 12 [73]x[2]
1050
567k4A 23 567k4I 8 [23]x[3]
1051
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1052
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1053
639k4A 7 639k4H 12 [7]
1054
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1055
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1056
681k4A 43 227k4A 23 Odd func eq for g
1057
681k4A 59 681k4D 30 [59]x[3]x[2]
1058
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1059
95k6A 31,59 95k6D 9 [31]x[59]
1060
116k6A 5 116k6D 6 [5]x[29]x[2]
1061
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1062
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1063
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1064