CoCalc Shared Fileswww / papers / motive_visibility / dsw2.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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% ---- 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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(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 $\qq$,
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$L(g,k/2)=0$ but $L(f,k/2)\neq 0$. It turns out that this forces
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$L(g,s)$ to vanish to order at least $2$ at $s=k/2$. We are able
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to find sixteen examples (all with $k=4$ and $k=6$), and in each
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case $\qq$ appears in the numerator of the algebraic number
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$L(f,k/2)/\vol_{\infty}$, where $\vol_{\infty}$ is a certain
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canonical period. In fact, we show how this divisibility may be
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deduced from the vanishing of $L(g,k/2)$ using recent work of
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Vatsal \cite{V}. The point is, the congruence between $f$ and $g$
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leads to a congruence between suitable ``algebraic parts'' of the
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special values $L(f,k/2)$ and $L(g,k/2)$. If one vanishes then the
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other is divisible by $\qq$. Under certain hypotheses, the
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Bloch-Kato conjecture then implies that the Shafarevich-Tate group
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attached to $f$ has non-zero $\qq$-torsion. Under certain
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hypotheses and assumptions, the most substantial of which is the
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Beilinson-Bloch conjecture relating the vanishing of $L(g,k/2)$ to
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the existence of algebraic cycles, we are able to construct the
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predicted elements of $\Sha$, using the Galois-theoretic
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interpretation of the congruences to transfer elements from a
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Selmer group for $g$ to a Selmer group for $f$. In proving the
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local conditions at primes dividing the level, and also in
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examining the local Tamagawa factors at these primes, we make use
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of a higher weight level-lowering result due to Jordan and Livn\'e
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\cite{JL}.
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One might say that algebraic cycles for one motive explain
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elements of $\Sha$ for the other. A main point of \cite{CM} was to
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observe the frequency with which those elements of $\Sha$
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predicted to exist for one elliptic curve may be explained by
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finding a congruence with another elliptic curve containing points
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of infinite order. One shortcoming of our work, compared to the
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elliptic curve case, is that, due to difficulties with local
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factors in the Bloch-Kato conjecture, we are unable to 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 cannot 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 $V_{\dR}=F^0\supset F^1=\ldots =F^{k-1}\supset
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F^k=\{0\}$. The Betti realisation $V_B$ comes from singular
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cohomology, while $V_{\lambda}$ comes from \'etale $l$-adic
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cohomology. There are natural isomorphisms $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. We may choose a
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$\Gal(\Qbar/\QQ)$-stable $O_{\lambda}$-module $T_{\lambda}$ inside
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each $V_{\lambda}$. Define $A_{\lambda}=V_{\lambda}/T_{\lambda}$.
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Let $A[\lambda]$ denote the $\lambda$-torsion in $A_{\lambda}$.
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There is the Tate twist $V_{\lambda}(j)$ (for any integer $j$),
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which amounts to multiplying the action of $\Frob_p$ by $p^j$.
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Following \cite{BK} (Section 3), for $p\neq l$ (including
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$p=\infty$) let
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$$H^1_f(\QQ_p,V_{\lambda}(j))=\ker (H^1(D_p,V_{\lambda}(j))\rightarrow
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H^1(I_p,V_{\lambda}(j))).$$ The subscript $f$ stands for ``finite
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part''. $D_p$ is a decomposition subgroup at a prime above $p$,
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$I_p$ is the inertia subgroup, and the cohomology is for
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continuous cocycles and coboundaries. For $p=l$ let
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$$H^1_f(\QQ_l,V_{\lambda}(j))=\ker (H^1(D_l,V_{\lambda}(j))\rightarrow
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H^1(D_l,V_{\lambda}(j)\otimes B_{\cris}))$$ (see Section 1 of
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\cite{BK} for definitions of Fontaine's rings $B_{\cris}$ and
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$B_{dR}$). Let $H^1_f(\QQ,V_{\lambda}(j))$ be the subspace of
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elements of $H^1(\QQ,V_{\lambda}(j))$ whose local restrictions lie
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in $H^1_f(\QQ_p,V_{\lambda}(j))$ for all primes $p$.
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There is a natural exact sequence
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$$\begin{CD}0@>>>T_{\lambda}(j)@>>>V_{\lambda}(j)@>\pi>>A_{\lambda}(j)@>>>0\end{CD}.$$
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Let
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$H^1_f(\QQ_p,A_{\lambda}(j))=\pi_*H^1_f(\QQ_p,V_{\lambda}(j))$.
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Define the $\lambda$-Selmer group \newline
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$H^1_f(\QQ,A_{\lambda}(j))$ to be the subgroup of elements of
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$H^1(\QQ,A_{\lambda}(j))$ whose local restrictions lie in
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$H^1_f(\QQ_p,A_{\lambda}(j))$ for all primes $p$. Note that the
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condition at $p=\infty$ is superfluous unless $l=2$. Define the
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Shafarevich-Tate group
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$$\Sha(j)=\oplus_{\lambda}H^1_f(\QQ,A_{\lambda}(j))/\pi_*H^1_f(\QQ,V_{\lambda}(j)).$$
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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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$$=\length\>\> (H^0(\QQ_p,A_{\lambda}(j))/H^0(\QQ_p, V_{\qq}(j)^{I_p}/T_{\qq}(j)^{I_p}).$$
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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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$$\dim H^0(I_p,A[\qq](j))=\dim H^0(I_p,V_{\qq}(j)).$$
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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 dividing $N/p$, Proposition 2.2
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of \cite{L} shows that we are in the situation covered by one of
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the 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{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
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\cite{DFG}.) Given this, the lemma follows from Theorem 4.1(iii)
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of \cite{BK}. (That $\mathbb{V}$ is the same as the functor used
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in Theorem 4.1 of \cite{BK} follows from the first paragraph of
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2(h) of \cite{Fa1}.)
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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.
424
425
Putting together the above lemmas we arrive at the following:
426
\begin{prop}\label{sha} Assume the same hypotheses as in Lemma \ref{local1},
427
for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
428
are as in the previous section. If
429
$$\ord_{\qq}(L(f,k/2)\aaa^{\pm}/\vol_{\infty})>0$$ (with numerator
430
non-zero) then the Bloch-Kato conjecture predicts that
431
$$\ord_{\qq}(\#\Sha)>0.$$
432
\end{prop}
433
434
\section{Congruences of special values}
435
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
436
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
437
large enough to contain all the coefficients $a_n$ and $b_n$.
438
Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
439
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
440
that $q\nmid N\phi(N)k!$. It is easy to see that we may choose the
441
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
442
$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
443
$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
444
has been made.
445
446
We shall now make two further assumptions:
447
\begin{enumerate}
448
\item $L(f,k/2)\neq 0$;
449
\item $L(g,k/2)=0$.
450
\end{enumerate}
451
\begin{prop} \label{div}
452
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
453
\end{prop}
454
\begin{proof} This is based on some of the ideas used in Section 1 of
455
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
456
which presumably should refer to ``Condition 2''. Since
457
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
458
$\ord_{\qq}(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
459
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
460
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
461
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
462
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
463
where the integral is taken along the positive imaginary axis,
464
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
465
Thus we are reduced
466
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
467
468
Let $\mathcal{D}_0$ be the group of divisors of degree zero
469
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
470
integer $r\geq 0$, let $P_r(R)$ be the additive group of
471
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
472
groups have a natural action of $\Gamma_1(N)$. Let
473
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
474
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
475
476
Via the isomorphism (8) in Section 1.5 of \cite{V},
477
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
478
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
479
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
480
(See also Section 4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
481
into account the fact that we are now dealing with $X_1(N)$ rather
482
that $M(N)$. Up to some small factorials which do not matter
483
locally at $\qq$,
484
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
485
(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since
486
$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that
487
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
488
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
489
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
490
show is divisible by $\qq$.
491
Similarly
492
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
493
(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient
494
of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
495
Therefore it would suffice to show that, for some $\mu\in O_E$,
496
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
497
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
498
that, for some $\mu\in O_E$, the element
499
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
500
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
501
non-constant coefficients. This would be the case if
502
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
503
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
504
a consequence of Theorem 2.1(1) of \cite{FJ}.
505
\end{proof}
506
\begin{remar}
507
By Proposition \ref{sha} (assuming, for all $p\mid N$ the same
508
hypotheses as in Lemma \ref{local1}, together with
509
$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
510
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
511
construction of the required elements of $\Sha$.
512
\end{remar}
513
\begin{remar}\label{sign}
514
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
515
have to be equal, since they are determined by the action of the
516
involution $W_N$ on the common subspace generated by the
517
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
518
Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the
519
common eigenvalue of $W_N$ acting on $f$ and $g$.
520
\end{remar}
521
This is analogous to the remark at the end of Section 3 of
522
\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then
523
$L(g,s)$ must vanish to order at least two, as in all the examples
524
below. It is worth pointing out that there are no examples of $g$
525
of level one, and positive sign in the functional equation, such
526
that $L(g,k/2)=0$, unless Maeda's conjecture (that all the
527
normalised cuspidal eigenforms of weight $k$ and level one are
528
Galois conjugate) is false. See \cite{CF}.
529
530
\section{Constructing elements of the Shafarevich-Tate group}
531
Let $f$ and $g$ be as in the first paragraph of the previous
532
section. For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
533
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
534
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
535
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
536
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
537
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
538
539
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
540
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
541
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
542
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
543
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
544
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of $\QQ$-rational
545
rational equivalence classes of null-homologous, codimension $k/2$
546
algebraic cycles on the motive $M_g$. (This generalises the part
547
of the Birch--Swinnerton-Dyer conjecture which says that for an
548
elliptic curve $E/\QQ$, the order of vanishing of $L(E,s)$ at
549
$s=1$ is equal to the rank of the Mordell-Weil group $E(\QQ)$.)
550
551
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
552
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
553
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
554
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
555
get (assuming also the Beilinson-Bloch conjecture) a subspace of
556
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
557
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
558
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
559
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
560
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
561
Sections 1 and 6.5 of \cite{Fo2}.
562
563
Similarly, if $L(f,k/2)\neq 0$ then we expect that
564
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
565
coincides with the $\qq$-part of $\Sha$.
566
\begin{thm}\label{local}
567
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
568
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
569
$A[\qq]$ is an irreducible representations of $\Gal(\Qbar/\QQ)$
570
and that for no prime $p\mid N$ is $f$ congruent modulo $\qq$ to a
571
newform of weight~$k$, trivial character and level dividing $N/p$.
572
Suppose that, for all primes $p\mid N$, $\,p\not\equiv
573
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. (Here
574
$w_p$ is the common eigenvalue of the Atkin-Lehner involution
575
$W_p$ acting on $f$ and $g$.) Then the $\qq$-torsion subgroup of
576
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
577
\end{thm}
578
\begin{proof}
579
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
580
continuity and rescaling we may assume that it lies in
581
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
582
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction (note that
583
$T'_{\qq}(k/2)/\qq T'_{\qq}(k/2)\simeq A'[\qq](k/2)$) we get a
584
non-zero class $c\in H^1(\QQ,A'[\qq](k/2))\simeq
585
H^1(\QQ,A[\qq](k/2))$. By irreducibility, $H^0(\QQ,A[\qq](k/2))$
586
is trivial. It follows that $H^0(\QQ,A_{\qq}(k/2))$ is trivial, so
587
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
588
we get a non-zero, $\qq$-torsion class $\gamma\in
589
H^1(\QQ,A_{\qq}(k/2))$.
590
591
Our aim is to show that $\res_p(\gamma)\in
592
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
593
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
594
595
\begin{enumerate}
596
\item {\bf $p\nmid qN$. }
597
598
Consider the $I_p$-cohomology of the short exact sequence
599
$$\begin{CD}0@>>>A'[\qq](k/2)@>>>A'_{\qq}(k/2)@>\pi>>A'_{\qq}(k/2)@>>>0\end{CD},$$
600
where $\pi$ is multiplication by a uniformising element of
601
$O_{\qq}$. Since in this case $A'_{\qq}(k/2)$ is unramified at
602
$p$, $H^0(I_p, A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is
603
$\qq$-divisible. Therefore $H^1(I_p,A'[\qq](k/2))$ (which,
604
remember, is the same as $H^1(I_p,A[\qq](k/2))$) injects into
605
$H^1(I_p,A'_{\qq}(k/2))$. It follows from the fact that $d\in
606
H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of $\gamma$ in
607
$H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p.~125 of
608
\cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
609
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
610
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
611
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
612
613
\item {\bf $p\mid N$. }
614
615
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
616
$\qq$-divisible. It suffices to show that
617
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
618
since this would imply that the natural map from
619
$H^0(I_p,V'_{\qq}(k/2))$ to $H^0(I_p, A'_{\qq}(k/2))$ is
620
surjective, but this may be done as in the proof of Lemma
621
\ref{local1}. It follows as above that the image of $c\in
622
H^1(\QQ,A[\qq](k/2))$ in $H^1(I_p,A[\qq](k/2))$ is zero. Then
623
$\res_p(c)$ comes from $H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by
624
inflation-restriction. The order of this group is the same as the
625
order of the group $H^0(\QQ_p,A[\qq](k/2))$, which we claim is
626
trivial. By the work of Carayol \cite{Ca1}, $p\mid N$ implies that
627
$V_{\qq}(k/2)$ is ramified at $p$, so $\dim
628
H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As above, we see that $\dim
629
H^0(I_p,V_{\qq}(k/2))=\dim H^0(I_p,A[\qq](k/2))$, so we need only
630
consider the case where this common dimension is $1$. The
631
(motivic) Euler factor at $p$ for $M_f$ is $(1-\alpha
632
p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts as multiplication by
633
$\alpha$ on the one-dimensional space $H^0(I_p,V_{\qq}(k/2))$. It
634
follows from Theor\'eme A of \cite{Ca1} that this is the same as
635
the Euler factor at $p$ of $L(f,s)$. By the work of Atkin and
636
Lehner \cite{AL}, it then follows that $p^2\nmid N$ and
637
$\alpha=-w_pp^{(k/2)-1}$, where $w_p=\pm 1$ is such that
638
$W_pf=w_pf$. Twisting by $k/2$, $\Frob_p^{-1}$ acts on
639
$H^0(I_p,V_{\qq}(k/2))$ (hence also on $H^0(I_p,A[\qq](k/2))$ as
640
$-w_pp^{-1}$. Since $p\not\equiv -w_p\pmod{q}$, we see that
641
$H^0(\QQ_p,A[\qq](k/2))$ is trivial. Hence $\res_p(c)=0$ so
642
$\res_p(\gamma)=0$ and certainly lies in
643
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
644
645
\item {\bf $p=q$. }
646
647
Since $q\nmid N$ is a prime of good reduction for the motive
648
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
649
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
650
$V'_{\qq}$ have the same dimension, where
651
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
652
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
653
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
654
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
655
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
656
may now prove, in the same manner as Proposition 9.2 of
657
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
658
\end{enumerate}
659
\end{proof}
660
661
Theorem 2.7 of \cite{AS} is concerned with verifying local
662
conditions in the case $k=2$, where $f$ and $g$ are associated
663
with abelian varieties $A$ and $B$. (Their theorem also applies to
664
abelian varieties over number fields.) Our restriction outlawing
665
congruences modulo $\qq$ with cusp forms of lower level is
666
analogous to theirs forbidding~$q$ from dividing Tamagawa factors
667
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
668
curve with $\ord_l(j(A))<0$, consideration of a Tate
669
parametrisation shows that if $q\mid c_{A,l}$, i.e., if
670
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
671
at $l$.)
672
673
In this paper we have encountered two technical problems which we
674
dealt with in quite similar ways:
675
\begin{enumerate}
676
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
677
\item proving local conditions at primes $p\mid N$, for an element
678
of $\qq$-torsion.
679
\end{enumerate}
680
If our only interest was in testing the Bloch-Kato conjecture at
681
$\qq$, we could have made these problems cancel out, as in Lemma
682
8.11 of \cite{DFG}, by weakening the local conditions. However, we
683
have chosen not to do so, since we are also interested in the
684
Shafarevich-Tate group, and since the hypotheses we had to assume
685
are not particularly strong.
686
687
\section{Sixteen examples}
688
\newcommand{\nf}[1]{\mbox{\bf #1}}
689
\begin{figure}
690
\caption{\label{fig:newforms}Newforms Relevant to
691
Theorem~\ref{local}}
692
$$
693
\begin{array}{|ccccc|}\hline
694
g & \deg(g) & f & \deg(f) & q's \\\hline
695
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
696
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
697
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
698
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
699
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
700
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
701
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
702
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
703
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
704
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
705
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
706
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
707
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
708
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
709
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
710
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
711
\hline
712
\end{array}
713
$$
714
\end{figure}
715
716
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
717
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
718
along with at least one prime~$q$ such that there is a prime
719
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
720
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
721
\subsection{Notation}
722
Table~\ref{fig:newforms} is laid out as follows.
723
The first column contains a label whose structure is
724
\begin{center}
725
{\bf [Level]k[Weight][GaloisOrbit]}
726
\end{center}
727
This label determines a newform $g=\sum a_n q^n$, up to Galois
728
conjugacy. For example, \nf{127k4C} denotes a newform in the third
729
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
730
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
731
\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
732
the level, with positive trace being first in the event that the
733
two absolute values are equal, and the first Galois orbit is
734
denoted {\bf A}, the second {\bf B}, and so on. The second column
735
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
736
third and fourth columns contain~$f$ and its degree, respectively.
737
The fifth column contains at least one prime~$q$ such that there
738
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
739
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
740
satisfied for~$f$,~$g$, and~$\qq$.
741
742
\subsection{The first example in detail}
743
\newcommand{\fbar}{\overline{f}}
744
We describe the first line of Table~\ref{fig:newforms}
745
in more detail. See the next section for further details
746
on how the computations were performed.
747
748
Using modular symbols, we find that there is a newform
749
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
750
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
751
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
752
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
753
coefficients generate a number field~$K$ of degree~$17$, and by
754
computing the image of the modular symbol $XY\{0,\infty\}$ under
755
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
756
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
757
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
758
both equal to
759
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
760
+ \cdots\in \FF_{43}[[q]].$$
761
762
There is no form in the Eisenstein subspaces of
763
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
764
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
765
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
766
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
767
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
768
of Theorem~\ref{local}, so if $r$ is the dimension of
769
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
770
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
771
772
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
773
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
774
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
775
subgroup of $\Sha$. Admitting these assumptions, we have
776
constructed the $\qq$-torsion in $\Sha$ predicted by the
777
Bloch-Kato conjecture.
778
779
For particular examples of elliptic curves one can often find and
780
write down rational points predicted by the Birch and
781
Swinnerton-Dyer conjecture. It would be nice if likewise one could
782
explicitly produce algebraic cycles predicted by the
783
Beilinson-Bloch conjecture in the above examples. Since
784
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
785
0.3.2 of \cite{Z}), so ought to be trivial in
786
$\CH_0^{k/2}(M_g)\otimes\QQ$.
787
788
\subsection{Some remarks on how the computation was performed}
789
We give a brief summary of how the computation was performed. The
790
algorithms that we used were implemented by the second author, and
791
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
792
793
Let~$g$,~$f$, and~$q$ be some data from a line of
794
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
795
verified the existence of a congruence modulo~$q$, that
796
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
797
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
798
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
799
800
To prove there is a congruence, we showed that the corresponding
801
{\em integral} spaces of modular symbols satisfy an appropriate
802
congruence, which forces the existence of a congruence on the
803
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
804
irreducible by computing a set that contains all possible residue
805
characteristics of congruences between~$g$ and any Eisenstein
806
series of level dividing~$N$, where by congruence, we mean a
807
congruence for all Fourier coefficients of index~$n$ with
808
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
809
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
810
listing a basis of such~$h$ and finding the possible congruences,
811
where again we disregard the Fourier coefficients of index not
812
coprime to~$N$.
813
814
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
815
modular symbol ${\mathbf
816
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
817
map with the same kernel as the period mapping, and found that the
818
image was~$0$. The period mapping sends the modular
819
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
820
so that ${\mathbf e}$ maps to~$0$ implies that
821
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
822
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g)
823
=(-1)^{k/2} g$ which, because of the functional equation, implies
824
that $L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of
825
independent interest because it includes examples of modular forms
826
of weight $>2$ with a zero at $\frac{k}{2}$ that is not forced by
827
the functional equation. We found no such examples of weights
828
$\geq 8$.
829
830
For the two examples \nf{581k4} and \nf{684k4K}, the square of a
831
prime appears in the $q$-column, meaning $f$ and $g$ are congruent
832
$\bmod{\qq^2}$ for some $\qq\mid q$. In these cases, a
833
modification of Theorem \ref{local} will show that the
834
$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
835
$O_E/\qq^2$-rank at least $r$.
836
837
\subsection{Other examples}
838
We have some other examples where a congruence between forms can
839
be shown, but the levels differ. However, Remark 5.3 does not
840
apply, so that one of the forms could have an odd functional
841
equation, and the other could have an even functional equation.
842
For instance, we have a 13-congruence between $g=\nf{81k4A}$ and
843
$f=$\nf{567k4L}; here $L(\nf{567k4L},2)\neq 0$, while
844
$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation.
845
Here $f$ obviously fails the condition about not being congruent
846
to a form of lower level, so in Lemma 4.3 it is possible that
847
$\ord_{\qq}(c_7(2))>0$. In fact this does happen. Because
848
$V'_{\qq}$ (attached to g of level $81$) is unramified at $p=7$,
849
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
850
two-dimensional. As in (2) of the proof of Theorem \ref{local},
851
one of the eigenvalues of $\Frob_p^{-1}$ acting on this
852
two-dimensional space is $\alpha=-w_pp^{(k/2)-1}$, where
853
$W_pf=w_pf$. The other must be $\beta=-w_pp^{k/2}$, so that
854
$\alpha\beta=p^{k-1}$. Twisting by $k/2$, we see that
855
$\Frob_p^{-1}$ acts as $-w_p$ on the quotient of
856
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_{\qq}(k/2))$.
857
Hence $\ord_{\qq}(c_p(k/2))>0$ when $w_p=-1$, which is the case in
858
our example here with $p=7$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
859
non-trivial when $w_p=-1$, so (2) of the proof of Theorem
860
\ref{local} does not work. This is just as well, since had it
861
worked we would have expected
862
$\ord_{\qq}(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
863
shows not to be the case.
864
865
Here is an example where the divisibility between the levels is
866
the other way round: a 7-congruence between $g=$\nf{122k6A} and
867
$f=$\nf{61k6B}. In this case both $L$-functions have even
868
functional equation, and we have $L(\nf{122k6A},3)=0$. In the
869
proof of Theorem 6.1, we find a problem with the local condition
870
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
871
$H^1(I_2,A'_{\qq}(3))$ is not necessarily injective, but its
872
kernel is at most one-dimensional, so we still get the
873
$\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
874
$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get
875
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
876
elements of $\Sha$ are {\it invisible} at level 61. When the
877
levels are different we are no longer able to apply Theorem 2.1 of
878
\cite{FJ}. However, we still have the congruences of integral
879
modular symbols required to make the proof of Proposition
880
\ref{div} go through. Indeed, as noted above, the congruences of
881
modular forms were found by producing congruences of modular
882
symbols. Despite these congruences of modular symbols, Remark 5.3
883
does not apply, since there is no reason to suppose that
884
$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.
885
886
Finally, there are two examples where we have a form $g$ with even
887
functional equation such that $L(g,k/2)=0$, and a congruent form
888
$f$ which has odd functional equation; these are a 23-congruence
889
between $g=$\nf{453k4A} and $f=$\nf{151k4A}, and a 43-congruence
890
between $g=$\nf{681k4A} and $f=$\nf{227k4A}. If
891
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
892
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
893
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
894
the appropriate modification of Theorem \ref{local} does not
895
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
896
It only tells us that the $\qq$-torsion subgroup of
897
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
898
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
899
appears in the conjectural formula for the first derivative of the
900
complex $L$ function, evaluated at $s=k/2$, but in combination
901
with a regulator that we have no way of calculating.
902
903
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
904
associated with $f$ and $g$ by the construction of Mazur, Tate and
905
Teitelbaum \cite{MTT}, each divided by a suitable canonical
906
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
907
quite clear what to make of this. This divisibility may be proved
908
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
909
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
910
normalised) are congruent $\bmod{\,\qq}$, as a result of the
911
congruence between the modular symbols out of which they are
912
constructed. Integrating an appropriate function against these
913
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{ \qq}$ to
914
$L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$, since
915
$L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case where the
916
signs in the functional equations of $L(g,s)$ and $L_q(g,s)$ are
917
the same, positive in this instance. (According to the proposition
918
in Section 18 of \cite{MTT}, the signs differ precisely when
919
$L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
920
921
922
923
\subsection{Excluded data}
924
We also found some examples for which the conditions of Theorem \ref{local}
925
were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},
926
but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar
927
problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here
928
$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a
929
5-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5
930
is less than the weight 6.
931
932
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933
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1028
1029
1030
\end{document}
1031
127k4A 43 127k4C 17 [43]
1032
159k4A 5,23 159k4E 8 [5]x[23]
1033
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1034
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1035
453k4A 5,17 453k4E 23 [5]x[17]
1036
453k4A 23 151k4A 30 Odd func eq for g
1037
465k4A 11 465k4H 7 [11]x[5]x[2]
1038
477k4A 73 477k4M 12 [73]x[2]
1039
567k4A 23 567k4I 8 [23]x[3]
1040
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1041
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1042
639k4A 7 639k4H 12 [7]
1043
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1044
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1045
681k4A 43 227k4A 23 Odd func eq for g
1046
681k4A 59 681k4D 30 [59]x[3]x[2]
1047
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1048
95k6A 31,59 95k6D 9 [31]x[59]
1049
116k6A 5 116k6D 6 [5]x[29]x[2]
1050
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1051
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1052
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1053