CoCalc Shared Fileswww / papers / motive_visibility / motive_visibility_v4.tex
Author: William A. Stein
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\begin{document}
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\title{Constructing elements in
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Shafarevich-Tate groups of modular motives\\
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{\sc (NOT FOR DISTRIBUTION!)}}
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\author{Neil Dummigan}
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\author{William Stein}
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\author{Mark Watkins}
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\date{September 12, 2001}
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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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proved 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.
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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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A theorem of Deligne \cite{De1} implies the existence, for each
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(finite) prime $\lambda$ of $E$, of a two-dimensional vector space
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$V_{\lambda}$ over $E_{\lambda}$, 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 $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset 212 F^k=\{0\}$. 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 $V_B\otimes 215 E_{\lambda}\simeq V_{\lambda}$. We may choose a
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
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There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
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which amounts to multiplying the action of $\Frob_p$ by $p^j$.
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Following \cite{BK} (Section 3), for $p\neq l$ (including
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$p=\infty$) let
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$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow 224 H^1(I_p,V_{\lambda}(j))).$$ 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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$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow 229 H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of
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\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
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$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
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elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
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in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.
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There is a natural exact sequence
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$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
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Let
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$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
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Define the $\lambda$-Selmer group \newline
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$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
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$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
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$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the
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condition at $p=\infty$ is superfluous unless $l=2$. Define the
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Shafarevich-Tate group
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$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
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The length of its $\lambda$-component may be taken for the
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exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Sha(j)$. We shall only concern ourselves with the case
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$j=k/2$, and write $\Sha$ for $\Sha(j)$.
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Define the set of global points
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$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. The length of its $\lambda$-component may be taken
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for the exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Gamma_{\QQ}$.
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\section{Vatsal's work on congruences between special values}
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\section{The Bloch-Kato conjecture}
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Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture for the
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motive $M_f(k/2)$ predicts that
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$${L(f,k/2)\over \vol_{\infty}}={\left(\prod_pc_p(k/2)\right)\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
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The local factors $\vol_{\infty}$ and $c_p(k/2)$ depend on the
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choice of a basis for $V_{\dR}$, but their product does not. 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). (We shall mainly be concerned
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with the $q$-part of the Bloch-Kato conjecture, where $q$ is a
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prime of good reduction. For such primes, the de Rham conjecture
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follows from Theorem 5.6 of \cite{Fa1}.) The above formula is to
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be interpreted as an equality of fractional ideals of $E$.
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(Strictly speaking, the conjecture in \cite{BK} is only given for
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$E=\QQ$.)
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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 $p\mid N$, $\,p\neq \pm 293 1\pmod{q}$. 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 $q\nmid 316 N$. Choosing bases for $V_{\dR}$ and $V_B$ which locally at $\qq$
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are as in the previous section, we have $\ord_{\qq}(c_q)=0$.
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\end{lem}
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\begin{proof}
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\end{proof}
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\begin{lem} $\ord_{\lambda}(\#\Gamma_{\QQ}(j))=0$ if $A[\lambda]$ is an
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irreducible representation of $\Gal(\Qbar/\QQ)$.
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\end{lem}
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This follows trivially from the definition.
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Putting together the above lemmas we arrive at the following:
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\begin{prop}\label{sha} Let $q\nmid N$ be an odd prime such that $q>k$.
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Let $\qq\mid q$ be a prime of $E$ such that $A[\qq]$ is an
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irreducible representation of $\Gal(\Qbar/\QQ)$. Choose bases for
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$V_{\dR}$ and $V_B$ which locally at $\qq$ are as in the previous
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section. If
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$$\ord_{\qq}(L(f,k/2)/\vol_{\infty}>0$$
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(with numerator non-zero) then the Bloch-Kato conjecture predicts
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that $\ord_{\qq}(\#\Sha)>0$.
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\end{prop}
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\section{Constructing elements of the Shafarevich-Tate group}
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Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
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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 $f\equiv 343 g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. For $f$
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we have defined $V_{\lambda}$, $T_{\lambda}$ and $A_{\lambda}$.
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Let $V'_{\lambda}$, $T'_{\lambda}$ and $A'_{\lambda}$ be the
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corresponding objects for $g$. Let $A[\lambda]$ denote the
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$\lambda$-torsion in $A_{\lambda}$. Since $a_p$ is the trace of
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$\Frob_p^{-1}$ on $V_{\lambda}$, it follows from the Cebotarev
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Density Theorem that $A[\qq]$ and $A'[\qq]$, if irreducible, are
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isomorphic as $\Gal(\Qbar/\QQ)$-modules.
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Suppose that $L(g,k/2)=0$. If the sign in the functional equation
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is positive, this implies that the order of vanishing at $s=k/2$
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is at least $2$. According to the Beilinson-Bloch conjecture
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\cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$ is the
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rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational,
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null-homologous, codimension $k/2$ algebraic cycles on the motive
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$M_g$, modulo rational equivalence. (This generalises the part of
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the Birch-Swinnerton-Dyer conjecture which says that for an
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elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
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$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
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Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
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to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
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subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
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If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
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get (assuming also the Beilinson-Bloch conjecture) a subspace of
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$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
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vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
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conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
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equal to the order of vanishing of $L(g,s)$ at $s=k/2$, but it
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seems a shame not to mention the cycles.
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Similarly, for $j\geq k/2$, the rank of $H^1_f(\QQ,V_{\qq}(j))$ is
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conjectured to be the order of vanishing of $L(f,s)$ at $s=k-j$.
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If $L(f,k/2)\neq 0$ then we expect that
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$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
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coincides with the $\qq$-part of $\Sha$.
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\begin{thm}\label{local}
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Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
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$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
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$A[\qq]$ and $A'[\qq]$ are irreducible representations of
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$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\neq \pm 384 1\pmod{q}$. Suppose also that neither $f$ nor $g$ is congruent
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modulo $\qq$ to any newform of weight~$k$, trivial character and
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level dividing $N/p$, with~$p$ any prime that exactly divides~$N$.
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Then the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
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$\FF_{\qq}$-rank at least $r-1$.
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\end{thm}
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\begin{proof}
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Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
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continuity and rescaling we may assume that it lies in
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$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq 394 H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
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class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
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irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
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$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
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we get a non-zero, $\qq$-torsion class $\gamma\in 399 H^1(\QQ,A_{\qq}(k/2))$.
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Our aim is to show that $\res_p(\gamma)\in 402 H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
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consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
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The case $p=q$ does not quite work, which is the reason for the
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$r-1$'' in the statement of the theorem.
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\begin{enumerate}
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\item {\bf $p\nmid qN$. }
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Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p, 411 A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
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$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
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$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
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follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
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$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
416
of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
417
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
418
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
419
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
420
421
\item {\bf $p\mid N$. }
422
423
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
424
$\qq$-divisible. It suffices to show that
425
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
426
but this may be done as in the proof of Lemma\ref{local1}. It
427
follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
428
$H^1(I_p,A[\qq](k/2))$ is zero. $\res_p(c)$ then comes from
429
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
430
order of this group is the same as the order of the group
431
$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
432
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
433
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
434
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim 435 H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
436
this common dimension is $1$. The (motivic) Euler factor at $p$
437
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
438
as multiplication by $\alpha$ on the one-dimensional space
439
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
440
\cite{Ca1} that this is the same as the Euler factor at $p$ of
441
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
442
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
443
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
444
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
445
$H^0(I_p,A[\qq](k/2))$ as $\pm p^{-1}$. Since $p\neq \pm 446 1\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence
447
$\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
448
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
449
450
\item {\bf $p=q$. }
451
452
Since $q\nmid N$ is a prime of good reduction for the motive
453
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
454
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
455
$V'_{\qq}$ have the same dimension, where
456
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q} 457 B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
458
It follows from Theorem 4.1(ii) of \cite{BK} that
459
$$D_{\dR}(V'_{\qq})/(F^{k/2}D_{\dR}(V'_{\qq}))\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)),$$
460
via their exponential map.
461
462
Since $p$ is a prime of good reduction, the de Rham conjecture is
463
a consequence of the crystalline conjecture, which follows from
464
Theorem 5.6 of \cite{Fa1}. Hence
465
$$(V'_{\dR}\otimes_E E_{\qq})/(F^{k/2}V'_{\dR}\otimes_E E_{\qq})\simeq H^1_f(\QQ_q,V'_{\qq}(k/2)).$$
466
Observe that the dimension of the left-hand-side is $1$. Hence
467
there is a subspace of $H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension at
468
least $r-1$, mapping to zero in $H^1(\QQ_q,V'_{\qq}(k/2))$. It
469
follows that our $r$-dimensional $\FF_{\qq}$-subspace of the
470
$\qq$-torsion in $H^1(\QQ,A_{\qq}(k/2))$ has at least an
471
$(r-1)$-dimensional subspace landing in
472
$H^1_f(\QQ_{\qq},A_{\qq}(k/2))$ (in fact, mapping to zero).
473
474
\end{enumerate}
475
\end{proof}
476
477
Theorem 2.7 of \cite{AS} is concerned with verifying local
478
conditions in the case $k=2$, where $f$ and $g$ are associated
479
with abelian varieties $A$ and $B$. (Their theorem also applies to
480
abelian varieties over number fields.) Our restriction outlawing
481
congruences modulo $\qq$ with cusp forms of lower level is
482
analogous to theirs forbidding $q$ from dividing Tamagawa factors
483
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
484
curve with $\ord_l(j(A))<0$, consideration of a Tate
485
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
486
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
487
at $l$.)
488
489
\section{Eleven examples}
490
\newcommand{\nf}[1]{\mbox{\bf #1}}
491
\begin{figure}
492
\caption{\label{fig:newforms}Newforms Relevant to
493
Theorem~\ref{local}}
494
$$495 \begin{array}{|ccccc|}\hline 496 g & \deg(g) & f & \deg(f) & q's \\\hline 497 \nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\ 498 \nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\ 499 \nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\ 500 \nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\ 501 \nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\ 502 \nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\ 503 \nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\ 504 \nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\ 505 \nf{581k4A} & 1 & \nf{581k4E} & 34 & 19 \\ 506 \nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\ 507 \nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\ 508 \hline 509 \end{array} 510$$
511
\end{figure}
512
513
\edit{Is it possible to stop this table appearing part way through
514
the proof of the previous theorem?} Table~\ref{fig:newforms} on
515
page~\pageref{fig:newforms} lists eleven pairs of newforms~$f$
516
and~$g$ (of equal weights and levels) along with at least one
517
prime~$q$ such that there is a prime $\qq\mid q$ with $f\equiv 518 g\pmod{\qq}$. In each case, $\ord_{s=k/2}L(g,k/2)\geq 2$ while
519
$L(f,k/2)\neq 0$.
520
\subsection{Notation}
521
Table~\ref{fig:newforms} is laid out as follows.
522
The first column contains a label whose structure is
523
\begin{center}
524
{\bf [Level]k[Weight][GaloisOrbit]}
525
\end{center}
526
This label determines a newform $g=\sum a_n q^n$, up to Galois
527
conjugacy. For example, \nf{127k4C} denotes a newform in the third
528
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
529
orbits are ordered first by the degree of $\QQ(\ldots, a_n, 530 \ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
531
the level, with positive trace being first in the event that the
532
two absolute values are equal, and the first Galois orbit is
533
denoted {\bf A}, the second {\bf B}, and so on. The second column
534
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
535
third and fourth columns contain~$f$ and its degree, respectively.
536
The fifth column contains at least one prime~$q$ such that there
537
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
538
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
539
satisfied for~$f$,~$g$, and~$\qq$.
540
541
\subsection{The first example in detail}
542
\newcommand{\fbar}{\overline{f}}
543
We describe the first line of Table~\ref{fig:newforms}
544
in more detail. See the next section for further details
545
on how the computations were performed.
546
547
Using modular symbols, we find that there is a newform
548
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots 549 \in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
550
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
551
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
552
coefficients generate a number field~$K$ of degree~$17$, and by
553
computing the image of the modular symbol $XY\{0,\infty\}$ under
554
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
555
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
556
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
557
both equal to
558
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7 + \cdots\in \FF_{43}[[q]].$$
559
560
There is no form in the Eisenstein subspaces of
561
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
562
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
563
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
564
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
565
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
566
of Theorem~\ref{local}, so if $r$ is the dimension of
567
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
568
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r-1$.
569
570
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
571
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
572
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
573
subgroup of $\Sha$. Admitting these assumptions, we have
574
constructed the $\qq$-torsion in $\Sha$ predicted by the
575
Bloch-Kato conjecture.
576
577
For particular examples of elliptic curves one can often find and
578
write down rational points predicted by the Birch and
579
Swinnerton-Dyer conjecture. It would be nice if likewise one could
580
explicitly produce algebraic cycles predicted by the
581
Beilinson-Bloch conjecture in the above examples. Since
582
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
583
0.3.2 of \cite{Z}), so ought to be trivial in
584
$\CH_0^{k/2}(M_g)\otimes\QQ$.
585
586
\subsection{Some remarks on how the computation was performed}
587
We give a brief summary of how the computation was performed. The
588
algorithms that we used were implemented by the second author, and
589
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
590
591
Let~$g$,~$f$, and~$q$ be some data from a line of
592
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
593
verified the existence of a congruence modulo~$q$, that
594
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq 595 0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
596
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
597
598
To prove there is a congruence, we showed that the corresponding
599
{\em integral} spaces of modular symbols satisfy an appropriate
600
congruence, which forces the existence of a congruence on the
601
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
602
irreducible by computing a set that contains all possible residue
603
characteristics of congruences between~$g$ and any Eisenstein
604
series of level dividing~$N$, where by congruence, we mean a
605
congruence for all Fourier coefficients of index~$n$ with
606
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
607
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
608
listing a basis of such~$h$ and finding the possible congruences,
609
where again we disregard the Fourier coefficients of index not
610
coprime to~$N$.
611
612
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
613
modular symbol ${\mathbf 614 e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
615
map with the same kernel as the period mapping, and found that the
616
image was~$0$. The period mapping sends the modular
617
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
618
so that ${\mathbf e}$ maps to~$0$ implies that
619
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
620
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$
621
which, because of the functional equation, implies that
622
$L'(g,\frac{k}{2})=0$.
623
624
In the course of our search, we found~$17$ plausible examples and
625
had to discard~$6$ of them because either the representation
626
$\rho_{f,\qq}$ arose from lower level, was reducible, or $p\equiv 627 \pm 1\pmod{q}$ for some $p\mid N$. Table~\ref{fig:newforms} is
628
of independent interest because it includes examples of modular
629
forms of prime level and weight $>2$ with a zero at $\frac{k}{2}$
630
that is not forced by the functional equation. We found no such
631
examples of weights $\geq 8$.
632
633
634
635
636
\section{Some Remarks on Visibility}
637
It would be nice to add a section on visibility here. I.e.,
638
systematically list examples where the rational part of the central
639
value has a big square factor, which {\em probably} should be
640
nontrivial $\Sha$, and say whether or not the theory of this paper
641
predicts that it is visible. Answer the question: Does it look like
642
all Sha be visible for motives?'' probably no''.
643
644
645
\begin{thebibliography}{AL}
646
\bibitem[AL]{AL} A. O. L. Atkin, J. Lehner, Hecke operators on
647
$\Gamma_0(m)$, {\em Math. Ann. }{\bf 185 }(1970), 135--160.
648
\bibitem[AS]{AS} A. Agashe, W. Stein, Visibility of
649
Shafarevich-Tate groups of abelian varieties: evidence for the
650
Birch and Swinnerton-Dyer conjecture, in preparation.
651
\bibitem[B]{B} S. Bloch, Algebraic cycles and values of $L$-functions,
652
{\em J. reine angew. Math. }{\bf 350 }(1984), 94--108.
653
\bibitem[BCP]{magma}
654
W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}.
655
{T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3-4,
656
235--265, Computational algebra and number theory (London, 1993).
657
\bibitem[BK]{BK} S. Bloch, K. Kato, L-functions and Tamagawa numbers
658
of motives, The Grothendieck Festschrift Volume I, 333--400,
659
Progress in Mathematics, 86, Birkh\"auser, Boston, 1990.
660
\bibitem[Ca1]{Ca1} H. Carayol, Sur les repr\'esentations $\ell$-adiques
661
associ\'ees aux formes modulaires de Hilbert, {\em Ann. Sci.
662
\'Ecole Norm. Sup. (4)}{\bf 19 }(1986), 409--468.
663
\bibitem[Ca2]{Ca2} H. Carayol, Sur les repr\'esentations
664
Galoisiennes modulo $\ell$ attach\'ees aux formes modulaires, {\em
665
Duke Math. J. }{\bf 59 }(1989), 785--801.
666
\bibitem[CM]{CM} J. E. Cremona, B. Mazur, Visualizing elements in the
667
Shafarevich-Tate group, {\em Experiment. Math. }{\bf 9 }(2000),
668
13--28.
669
\bibitem[De1]{De1} P. Deligne, Formes modulaires et repr\'esentations
670
$\ell$-adiques. S\'em. Bourbaki, \'exp. 355, Lect. Notes Math.
671
{\bf 179, } 139--172, Springer, 1969.
672
\bibitem[De2]{De2} P. Deligne, Valeurs de Fonctions $L$ et P\'eriodes
673
d'Int\'egrales, {\em AMS Proc. Symp. Pure Math.,} Vol. 33 (1979),
674
part 2, 313--346.
675
\bibitem[DFG]{DFG} F. Diamond, M. Flach, L. Guo, Adjoint motives
676
of modular forms and the Tamagawa number conjecture, preprint.
677
{{\sf
678
http://www.andromeda.rutgers.edu/\~{\mbox{}}krm/liguo/lgpapers.html}}
679
\bibitem[Du1]{Du1} N. Dummigan, Period ratios of modular forms, {\em
680
Math. Ann. }{\bf 318 }(2000), 621--636.
681
\bibitem[Du2]{Du2} N. Dummigan, Congruences of modular forms and
682
Selmer groups, {\em Math. Res. Lett. }{\bf 8 }(2001), 479--494.
683
\bibitem[Fa]{Fa1} G. Faltings, Crystalline cohomology and $p$-adic
684
Galois representations, {\em in }Algebraic analysis, geometry and
685
number theory (J. Igusa, ed.), 25--80, Johns Hopkins University
686
Press, Baltimore, 1989.
687
\bibitem[Fl2]{Fl2} M. Flach, A generalisation of the Cassels-Tate
688
pairing, {\em J. reine angew. Math. }{\bf 412 }(1990), 113--127.\edit{I
689
switched these two Flach references, because they were out of chrono
690
order. --was}
691
\bibitem[Fl1]{Fl1} M. Flach, On the degree of modular parametrisations,
692
S\'eminaire de Th\'eorie des Nombres, Paris 1991-92 (S. David,
693
ed.), 23--36, Progress in mathematics, 116, Birkh\"auser, Basel
694
Boston Berlin, 1993.
695
\bibitem[Fo]{Fo} J.-M. Fontaine, Sur certains types de
696
repr\'esentations $p$-adiques du groupe de Galois d'un corps
697
local, construction d'un anneau de Barsotti-Tate, {\em Ann. Math.
698
}{\bf 115 }(1982), 529--577.
699
\bibitem[JL]{JL} B. W. Jordan, R. Livn\'e, Conjecture epsilon''
700
for weight $k>2$, {\em Bull. Amer. Math. Soc. }{\bf 21 }(1989),
701
51--56.
702
\bibitem[L]{L} R. Livn\'e, On the conductors of mod $\ell$ Galois
703
representations coming from modular forms, {\em J. Number Theory
704
}{\bf 31 }(1989), 133--141.
705
706
\bibitem[Ne1]{Ne1} J. Nekov\'ar, Kolyvagin's method for Chow groups of
707
Kuga-Sato varieties, {\em Invent. Math. }{\bf 107 }(1992),
708
99--125.
709
\bibitem[Ne2]{Ne2} J. Nekov\'ar, $p$-adic Abel-Jacobi maps and $p$-adic
710
heights. The arithmetic and geometry of algebraic cycles (Banff,
711
AB, 1998), 367--379, CRM Proc. Lecture Notes, 24, Amer. Math.
712
Soc., Providence, RI, 2000.
713
\bibitem[Sc]{Sc} A. J. Scholl, Motives for modular forms,
714
{\em Invent. Math. }{\bf 100 }(1990), 419--430.
715
\bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $\ell$-adic representations and
716
congruences for coefficients of modular forms,{\em Modular
717
functions of one variable} III, Lect. Notes Math. {\bf 350, }
718
Springer, 1973.
719
\bibitem[V]{V} V. Vatsal, Canonical periods and congruence
720
formulae, {\em Duke Math. J. }{\bf 98 }(1999), 397--419.
721
\bibitem[Z]{Z} S. Zhang, Heights of Heegner cycles and derivatives of $L$-series,
722
{\em Invent. Math. }{\bf 130 }(1997), 99--152.
723
\end{thebibliography}
724
725
726
\end{document}
727
\subsection{Plan for finishing the computation}
728
729
What I (William) have to do with the below stuff:\\
730
\begin{enumerate}
731
\item Fill in the missing blanks at these levels:
732
$$453, 639, 657, 681, 684, 95, 116, 122, 260$$
733
\item Make a MAGMA program that has a table of forms to try.
734
\item First, unfortunately, for some reason the
735
labels are sometimes wrong, e.g., \nf{159k4A} should
736
have been \nf{159k4B}, so I have to check all the
737
rational forms to see which has $L(1)=0$.
738
\item Try each level using the MAGMA program (this will take a long time to run).
739
\item Finish the Table of examples'' above.
740
\end{enumerate}
741
742
743
\begin{verbatim}
744
745
746
(This stuff below is being integrated into the above, as I do
747
the required (rather time consuming) computations.)
748
**************************************************
749
FROM Mark Watkins:
750
751
Here's a list of stuff; the left-hand (sinister) column
752
contains forms with a double zero at the central point,
753
whilst the right-hand (dexter) column contains forms which
754
have a large square factor in LRatio at 2. The middle column
755
is the large prime factor in ModularDegree of the LHS,
756
and/or the lpf in the LRatio at 2 of the RHS.
757
758
It seems that 567k4L has invisible Sha possibly.
759
The ModularDegree for 639k4B has no large factors.
760
761
127k4A 43 127k4C
762
159k4A 23 159k4E
763
365k4A 29 365k4E
764
369k4A 13 369k4I
765
453k4A 17 453k4E
766
453k4A 23
767
465k4A 11 465k4H
768
477k4A 73 477k4L
769
567k4A 23 567k4G
770
13 567k4L
771
581k4A 19 581k4E
772
639k4B --
773
774
775
Forms with spurious zeros to do:
776
777
657k4A
778
681k4A
779
684k4B
780
95k6A 31,59
781
116k6A --
782
122k6A 73
783
260k6A
784
785
If we allow 5 and 7 to be small primes, then we get more
786
visibility info.
787
788
159k4A 5 159k4E
789
369k4A 5 369k4I
790
453k4A 5 453k4E
791
639k4B 7 639k4H
792
793
Maybe the general theorem does not apply to primes which are so small,
794
but we might be able to show that they are OK in these specific cases.
795
796
I checked the first three weight 6 examples with AbelianIntersection,
797
but actually computing the LRatio bogs down too much. Will need
798
new version.
799
800
95k6A 31 95k6D
801
95k6A 59 95k6D
802
116k6A 5 116k6D
803
122k6A 73 122k6D
804
805
Incidentally, the LRatio for 581k4E has a power of 19^4.
806
Perhaps not surprising, as the ModularDegree of 581k4A
807
has a factor of 19^2. But maybe it means a bigger Sha.
808
\end{verbatim}
809