CoCalc Shared Fileswww / papers / motive_visibility / dsw.texOpen in CoCalc with one click!
Author: William A. Stein
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\documentclass{amsart}
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\usepackage{amssymb}
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\usepackage{amsmath}
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\usepackage{amscd}
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\newcommand{\edit}[1]{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
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\newtheorem{prop}{Proposition}[section]
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\newtheorem{defi}[prop]{Definition}
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\newtheorem{conj}[prop]{Conjecture}
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\newtheorem{lem}[prop]{Lemma}
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\newtheorem{thm}[prop]{Theorem}
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\newtheorem{cor}[prop]{Corollary}
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\newtheorem{examp}[prop]{Example}
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\newtheorem{remar}[prop]{Remark}
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\def\id{\mathop{\mathrm{ id}}\nolimits}
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\renewcommand{\Im}{{\mathrm {Im}}}
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\newcommand{\tors}{{\mathrm {tors}}}
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\newcommand{\cris}{{\mathrm {cris}}}
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\newcommand{\length}{{\mathrm {length}}}
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\newcommand{\dR}{{\mathrm {dR}}}
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\newcommand{\lcm}{{\mathrm {lcm}}}
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\newcommand{\Frob}{{\mathrm {Frob}}}
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\def\rank{\mathop{\mathrm{ rank}}\nolimits}
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\newcommand{\Gal}{\mathrm {Gal}}
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\newcommand{\AAA}{{\mathbb A}}
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\newcommand{\CC}{{\mathbb C}}
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\newcommand{\RR}{{\mathbb R}}
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\newcommand{\QQ}{{\mathbb Q}}
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\newcommand{\ZZ}{{\mathbb Z}}
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\newcommand{\NN}{{\mathbb N}}
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\newcommand{\EE}{{\mathbb E}}
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\newcommand{\TT}{{\mathbb T}}
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\newcommand{\HHH}{{\mathbb H}}
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\newcommand{\pp}{{\mathfrak p}}
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\newcommand{\qq}{{\mathfrak q}}
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%\newcommand{\Sha}{\underline{III}}
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%\newcommand{\Sha}{\amalg\kern-0.575em\amalg}
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% ---- SHA ----
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\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
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\newcommand{\textcyr}[1]{%
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{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}%
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\selectfont #1}}
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\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}
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\newcommand{\aaa}{{\mathfrak a}}
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\newcommand{\CH}{\mathrm {CH}}
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\begin{document}
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\title{Constructing elements in
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Shafarevich-Tate groups of modular motives}
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\author{Neil Dummigan}
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\author{William Stein}
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\author{Mark Watkins}
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\date{March 8th, 2002}
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\subjclass{11F33, 11F67, 11G40.}
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\keywords{modular form, $L$-function, visibility, Bloch-Kato conjecture,
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Shafarevich-Tate group.}
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\address{University of Sheffield\\ Department of Pure
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Mathematics\\
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Hicks Building\\ Hounsfield Road\\ Sheffield, S3 7RH\\
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U.K.}
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\address{Harvard University\\Department of Mathematics\\
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One Oxford Street\\ Cambridge, MA 02138\\ U.S.A.}
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\address{Penn State Mathematics Department\\
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University Park\\State College, PA 16802\\ U.S.A.}
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\email{n.p.dummigan@shef.ac.uk} \email{was@math.harvard.edu}
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\email{watkins@math.psu.edu}
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\maketitle
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\section{Introduction}
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Let $E$ be an elliptic curve defined over $\QQ$ and let $L(E,s)$
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be the associated $L$-function. The conjecture of Birch and
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Swinnerton-Dyer predicts that the order of vanishing of $L(E,s)$
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at $s=1$ is the rank of the group $E(\QQ)$ of rational points, and
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also gives an interpretation of the leading term in the Taylor
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expansion in terms of various quantities, including the order of
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the Shafarevich-Tate group.
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Cremona and Mazur [2000] look, among all strong Weil elliptic
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curves over $\QQ$ of conductor $N\leq 5500$, at those with
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non-trivial Shafarevich-Tate group (according to the Birch and
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Swinnerton-Dyer conjecture). Suppose that the Shafarevich-Tate
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group has predicted elements of order $m$. In most cases they find
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another elliptic curve, often of the same conductor, whose
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$m$-torsion is Galois-isomorphic to that of the first one, and
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which has rank two. The rational points on the second elliptic
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curve produce classes in the common $H^1(\QQ,E[m])$. They expect
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that these lie in the Shafarevich-Tate group of the first curve,
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so rational points on one curve explain elements of the
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Shafarevich-Tate group of the other curve.
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The Bloch-Kato conjecture \cite{BK} is the generalisation to
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arbitrary motives of the leading term part of the Birch and
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Swinnerton-Dyer conjecture. The Beilinson-Bloch conjecture
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\cite{B} generalises the part about the order of vanishing at the
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central point, identifying it with the rank of a certain Chow
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group.
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The present work may be considered as a partial generalisation of
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the work of Cremona and Mazur, from elliptic curves over $\QQ$
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(with which are associated modular forms of weight $2$) to the
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motives attached to modular forms of higher weight. (See \cite{AS}
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for a different generalisation, to modular abelian varieties of
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higher dimension.) It may also be regarded as doing, for
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congruences between modular forms of equal weight, what \cite{Du2}
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did for congruences between modular forms of different weights.
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We consider the situation where two newforms $f$ and $g$, both of
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weight $k>2$ and level $N$, are congruent modulo some $\qq$,
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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 predict the
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exact order of $\Sha$. We have to start with modular forms between
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which there exists a congruence. However, Vatsal's work allows us
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to explain how the vanishing of one $L$-function leads, via the
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congruence, to the divisibility by $\qq$ of (an algebraic part of)
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another, independent of observations of computational data. The
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computational data does however show that there exist examples to
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which our results apply. Moreover, it displays factors of $\qq^2$
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\edit{the meaning of the original L-value data still needs sorting
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out} , whose existence we cannot prove theoretically, but which
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are 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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The length of its $\lambda$-component may be taken for the
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exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Sha(j)$. We shall only concern ourselves with the case
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$j=k/2$, and write $\Sha$ for $\Sha(j)$.
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Define the group of global torsion points
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$$\Gamma_{\QQ}=\oplus_{\lambda}H^0(\QQ,A_{\lambda}(k/2)).$$
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This is analogous to the group of rational torsion points on an
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elliptic curve. The length of its $\lambda$-component may be taken
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for the exponent of $\lambda$ in an ideal of $O_E$, which we call
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$\#\Gamma_{\QQ}$.
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\section{Canonical periods}
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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 any way such that $T_B\otimes_{O_E}O_E[1/Nk!]$ and
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$T_{\dR}\otimes_{O_E}O_E[1/Nk!]$ agree with the
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$O_E[1/Nk!]$-lattices $\mathfrak{M}_{f,B}$ and
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$\mathfrak{M}_{f,\dR}$ defined in \cite{DFG}. (See especially
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Sections 2.2 and 5.4 of \cite{DFG}.)
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For any finite prime $\lambda$ of $O_E$ define the $O_{\lambda}$
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module $T_{\lambda}$ inside $V_{\lambda}$ to be the image of
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$T_B\otimes O_{\lambda}$ under the natural isomorphism $V_B\otimes
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E_{\lambda}\simeq V_{\lambda}$. Then for $\lambda\nmid Nk!$, the
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$O_{\lambda}$ module $T_{\lambda}$ is $\Gal(\Qbar/\QQ)$-stable,
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since it comes from $\ell$-adic cohomology with $O_{\lambda}$
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coefficients. We may assume that $T_{\lambda}$ is
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$\Gal(\Qbar/\QQ)$-stable for all finite $\lambda$, by adjusting
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$T_B$ locally at primes $\lambda\mid Nk!$ if necessary.
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Let $M(N)$ 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 $M(N)$ as the disjoint
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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
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$\sum_{n=1}^{\infty}a_nn^{-s}$, but there is an analytic
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continuation given by an integral, as described in the next
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section. Suppose that $L(f,k/2)\neq 0$. The Bloch-Kato conjecture
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for the motive $M_f(k/2)$ predicts that
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$${L(f,k/2)\over \vol_{\infty}}=
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{\left(\prod_pc_p(k/2)\right)\aaa^{\pm}\#\Sha\over (\#\Gamma_{\QQ})^2}.$$
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Here, $\pm$ represents the parity of $(k/2)-1$, and
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$\vol_{\infty}$ is equal to $(2\pi i)^{k/2}$ multiplied by the
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determinant of the 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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$$\length\>\> H^1_f(\QQ_p,T_{\lambda}(j))_{\tors}-
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\ord_{\lambda}((1-a_pp^{-j}+p^{k-1-2j}))$$
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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
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$\ord_{\lambda}(c_p(j))$ for $\lambda\mid p$, which requires one
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to assume Fontaine's de Rham conjecture (\cite{Fo}, Appendix A6),
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and depends on the choices of $T_{\dR}$ and $T_B$, locally at
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$\lambda$. (We shall mainly be concerned with the $q$-part of the
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Bloch-Kato conjecture, where $q$ is a prime of good reduction. For
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such primes, the de Rham conjecture follows from Theorem 5.6 of
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\cite{Fa1}.) The above formula is to be interpreted as an equality
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of fractional ideals of $E$. (Strictly speaking, the conjecture in
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\cite{BK} is only given for $E=\QQ$.)
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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} $\vol_{\infty}$ may be calculated as 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 the isomorphism at the end of
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Section 2.2 of \cite{DFG} (an application of the results of
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\cite{Fa1}) that $T_{\qq}$ is the
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$O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
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filtered module $T_{\dR}\otimes O_{\lambda}$ by the functor they
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call $\mathbb{V}$. Given this, the lemma follows from Theorem
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4.1(iii) of \cite{BK}. (That $\mathbb{V}$ is the same as the
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functor used in Theorem 4.1 of \cite{BK} follows from the first
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paragraph of 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
418
irreducible representation of $\Gal(\Qbar/\QQ)$.
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\end{lem}
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This follows trivially from the definition.
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Putting together the above lemmas we arrive at the following:
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\begin{prop}\label{sha} Assume the same hypotheses as in Lemma \ref{local1},
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for all $p\mid N$. Choose $T_{\dR}$ and $T_B$ which locally at $\qq$
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are as in the previous section. If
426
$$\ord_{\qq}(L(f,k/2)/\aaa^{\pm}\vol_{\infty})>0$$ (with numerator
427
non-zero) then the Bloch-Kato conjecture predicts that
428
$$\ord_{\qq}(\#\Sha)>0.$$
429
\end{prop}
430
431
\section{Congruences of special values}
432
Let $f=\sum a_nq^n$ and $g=\sum b_nq^n$ be newforms of equal
433
weight $k\geq 2$ for $\Gamma_0(N)$. Let $E$ be a number field
434
large enough to contain all the coefficients $a_n$ and $b_n$.
435
Suppose that $\qq\mid q$ is a prime of $E$ such that $f\equiv
436
g\pmod{\qq}$, i.e. $a_n\equiv b_n\pmod{\qq}$ for all $n$. Suppose
437
that $q\nmid N\phi(N)k!$. It is easy to see that we may choose the
438
$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
439
$\ord_{\qq}(\aaa^{\pm})=0$, i.e. $\delta_f^{\pm}$ generates
440
$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
441
has been made.
442
443
We shall now make two further assumptions:
444
\begin{enumerate}
445
\item $L(f,k/2)\neq 0$;
446
\item $L(g,k/2)=0$.
447
\end{enumerate}
448
\begin{prop} \label{div}
449
With assumptions as above, $\ord_{\qq}(L(f,k/2)/\vol_{\infty})>0$.
450
\end{prop}
451
\begin{proof} This is based on some of the ideas used in Section 1 of
452
\cite{V}. Note the apparent typo in Theorem 1.13 of \cite{V},
453
which presumably should refer to ``Condition 2''. Since
454
$\ord_{\qq}(\aaa^{\pm})=0$, we just need to show that
455
$\ord_{\qq}(L(f,k/2)/(2\pi i)^{k/2}\Omega_{\pm})>0$, where $\pm
456
1=(-1)^{(k/2)-1}$. It is well-known, and easy to prove, that
457
$$\int_0^{\infty}f(iy)y^{s-1}dy=(2\pi)^{-s}\Gamma(s)L(f,s).$$
458
Hence, if for $0\leq j\leq k-2$ we define the $j^{th}$ period
459
$$r_j(f)=\int_0^{i\infty}f(z)z^jdz,$$
460
where the integral is taken along the positive imaginary axis,
461
then $$r_j(f)=j!(-2\pi i)^{-(j+1)}L_f(j+1).$$
462
Thus we are reduced
463
to showing that $\ord_{\qq}(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
464
465
Let $\mathcal{D}_0$ be the group of divisors of degree zero
466
supported on $\mathbb{P}^1(\QQ)$. For a $\ZZ$-algebra $R$ and
467
integer $r\geq 0$, let $P_r(R)$ be the additive group of
468
homogeneous polynomials of degree $r$ in $R[X,Y]$. Both these
469
groups have a natural action of $\Gamma_1(N)$. Let
470
$S_{\Gamma_1(N)}(k,R):=\Hom_{\Gamma_1(N)}(\mathcal{D}_0,P_{k-2}(R))$
471
be the $R$-module of weight $k$ modular symbols for $\Gamma_1(N)$.
472
473
Via the isomorphism (8) in Section 1.5 of \cite{V},
474
$\omega_f^{\pm}$ corresponds to an element $\Phi_f^{\pm}\in
475
S_{\Gamma_1(N)}(k,\CC)$, and $\delta_f^{\pm}$ corresponds to an
476
element $\Delta_f^{\pm}\in S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$.
477
(See also Section 4.2 of \cite{St}.) Inverting $N\phi(N)$ takes
478
into account the fact that we are now dealing with $X_1(N)$ rather
479
that $M(N)$. Up to some small factorials which do not matter
480
locally at $\qq$,
481
$$\Phi_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
482
(k/2)-1\pmod{2}}^{k-2} 2\pi i r_f(j)X^jY^{k-2-j}.$$ Since
483
$\omega_f^{\pm}=2\pi i\Omega_f^{\pm}\delta_f^{\pm}$, we see that
484
$$\Delta_f^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
485
(k/2)-1\pmod{2}}^{k-2}(r_f(j)/\Omega_f^{\pm})X^jY^{k-2-j}.$$ The
486
coefficient of $X^{(k/2)-1}Y^{(k/2)-1}$ is what we would like to
487
show is divisible by $\qq$.
488
Similarly
489
$$\Phi_g^{\pm}([\infty]-[0])=\sum_{j=0,j\equiv
490
(k/2)-1\pmod{2}}^{k-2}2\pi i r_g(j)X^jY^{k-2-j}.$$ The coefficient
491
of $X^{(k/2)-1}Y^{(k/2)-1}$ in this is $0$, since $L(g,k/2)=0$.
492
Therefore it would suffice to show that, for some $\mu\in O_E$,
493
the element $\Delta_f^{\pm}-\mu\Delta_g^{\pm}$ is divisible by
494
$\qq$ in $S_{\Gamma_1(N)}(k,O_E[1/N\phi(N)])$. It suffices to show
495
that, for some $\mu\in O_E$, the element
496
$\delta_f^{\pm}-\mu\delta_g^{\pm}$ is divisible by $\qq$,
497
considered as an element of $\qq$-adic cohomology of $X_1(N)$ with
498
non-constant coefficients. This would be the case if
499
$\delta_f^{\pm}$ and $\delta_g^{\pm}$ generate the same
500
one-dimensional subspace upon reduction $\pmod{\qq}$. But this is
501
a consequence of Theorem 2.1(1) of \cite{FJ}.
502
\end{proof}
503
\begin{remar}
504
By Proposition \ref{sha} (assuming, for all $p\mid N$ the same
505
hypotheses as in Lemma \ref{local1}, together with
506
$q\nmid\phi(N)$), the Bloch-Kato conjecture now predicts that
507
$\ord_{\qq}(\#\Sha)>0$. The next section provides a conditional
508
construction of the required elements of $\Sha$.
509
\end{remar}
510
\begin{remar}\label{sign}
511
The signs in the functional equations of $L(f,s)$ and $L(g,s)$
512
have to be equal, since they are determined by the action of the
513
involution $W_N$ on the common subspace generated by the
514
reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
515
\end{remar}
516
This is analogous to the remark at the end of Section 3 of
517
\cite{CM}. It shows that if $L(f,k/2)\neq 0$ but $L(g,k/2)=0$ then
518
$L(g,s)$ must vanish to order at least two, as in all the examples
519
below. It is worth pointing out that there are no examples of $g$
520
of level one, and positive sign in the functional equation, such
521
that $L(g,k/2)=0$, unless Maeda's conjecture (that all the
522
normalised cuspidal eigenforms of weight $k$ and level one are
523
Galois conjugate) is false. See \cite{CF}.
524
525
\section{Constructing elements of the Shafarevich-Tate group}
526
For $f$ we have defined $V_{\lambda}$, $T_{\lambda}$ and
527
$A_{\lambda}$. Let $V'_{\lambda}$, $T'_{\lambda}$ and
528
$A'_{\lambda}$ be the corresponding objects for $g$. Since $a_p$
529
is the trace of $\Frob_p^{-1}$ on $V_{\lambda}$, it follows from
530
the Cebotarev Density Theorem that $A[\qq]$ and $A'[\qq]$, if
531
irreducible, are isomorphic as $\Gal(\Qbar/\QQ)$-modules.
532
533
Suppose that $L(g,k/2)=0$. If the sign in the functional equation
534
is positive (as it must be if $L(f,k/2)\neq 0$, see Remark
535
\ref{sign}), this implies that the order of vanishing of $L(g,s)$
536
at $s=k/2$ is at least $2$. According to the Beilinson-Bloch
537
conjecture \cite{B}, the order of vanishing of $L(g,s)$ at $s=k/2$
538
is the rank of the group $\CH_0^{k/2}(M_g)(\QQ)$ of
539
$\QQ$-rational, null-homologous, codimension $k/2$ algebraic
540
cycles on the motive $M_g$, modulo rational equivalence. (This
541
generalises the part of the Birch--Swinnerton-Dyer conjecture which
542
says that for an elliptic curve $E/\QQ$, the order of vanishing of
543
$L(E,s)$ at $s=1$ is equal to the rank of the Mordell-Weil group
544
$E(\QQ)$.)
545
546
Via the $\qq$-adic Abel-Jacobi map, $\CH_0^{k/2}(M_g)(\QQ)$ maps
547
to $H^1(\QQ,V'_{\qq}(k/2))$, and its image is contained in the
548
subspace $H^1_f(\QQ,V'_{\qq}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
549
If, as expected, the $\qq$-adic Abel-Jacobi map is injective, we
550
get (assuming also the Beilinson-Bloch conjecture) a subspace of
551
$H^1_f(\QQ,V'_{\qq}(k/2))$ of dimension equal to the order of
552
vanishing of $L(g,s)$ at $s=k/2$. In fact, one could simply
553
conjecture that the dimension of $H^1_f(\QQ,V'_{\qq}(k/2))$ is
554
equal to the order of vanishing of $L(g,s)$ at $s=k/2$. This would
555
follow from the ``conjectures'' $C_r(M)$ and $C^i_{\lambda}(M)$ in
556
Sections 1 and 6.5 of \cite{Fo2}.
557
558
Similarly, if $L(f,k/2)\neq 0$ then we expect that
559
$H^1_f(\QQ,V_{\qq}(k/2))=0$, so that $H^1_f(\QQ,A_{\qq}(k/2))$
560
coincides with the $\qq$-part of $\Sha$.
561
\begin{thm}\label{local}
562
Let $q\nmid N$ be a prime satisfying $q>k$. Suppose that
563
$H^1_f(\QQ,V'_{\qq}(k/2))$ has dimension $r>0$. Suppose that
564
$A[\qq]$ and $A'[\qq]$ are irreducible representations of
565
$\Gal(\Qbar/\QQ)$ and that, for all primes $p\mid N$, $\,p\not\equiv
566
-w_p\pmod{q}$, with $p\not\equiv -1\pmod{q}$ if $p^2\mid N$. Suppose
567
also that neither $f$ nor $g$ is congruent modulo $\qq$ to any
568
newform of weight~$k$, trivial character and level dividing $N/p$,
569
with~$p$ any prime that divides~$N$. Then the $\qq$-torsion
570
subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at
571
least $r$.
572
\end{thm}
573
\begin{proof}
574
Take a non-zero class $d\in H^1_f(\QQ,V'_{\qq}(k/2))$. By
575
continuity and rescaling we may assume that it lies in
576
$H^1_f(\QQ,T'_{\qq}(k/2))$ but not in $\qq
577
H^1_f(\QQ,T'_{\qq}(k/2))$. Then by reduction we get a non-zero
578
class $c\in H^1(\QQ,A'[\qq](k/2))\simeq H^1(\QQ,A[\qq](k/2))$. By
579
irreducibility, $H^0(\QQ,A[\qq](k/2))$ is trivial, so
580
$H^1(\QQ,A[\qq](k/2))$ injects into $H^1(\QQ,A_{\qq}(k/2))$, and
581
we get a non-zero, $\qq$-torsion class $\gamma\in
582
H^1(\QQ,A_{\qq}(k/2))$.
583
584
Our aim is to show that $\res_p(\gamma)\in
585
H^1_f(\QQ_p,A_{\qq}(k/2))$, for all (finite) primes $p$. We
586
consider separately the cases $p\nmid qN$, $p\mid N$ and $p=q$.
587
588
\begin{enumerate}
589
\item {\bf $p\nmid qN$. }
590
591
Since in this case $A'_{\qq}(k/2)$ is unramified at $p$, $H^0(I_p,
592
A'_{\qq}(k/2))=A'_{\qq}(k/2)$, which is $\qq$-divisible. Therefore
593
$H^1(I_p,A'[\qq](k/2))$ (which, remember, is the same as
594
$H^1(I_p,A[\qq](k/2))$) injects into $H^1(I_p,A'_{\qq}(k/2))$. It
595
follows from $d\in H^1_f(\QQ,V'_{\qq}(k/2))$ that the image of
596
$\gamma$ in $H^1(I_p,A_{\qq}(k/2))$ is zero. By line 3 of p. 125
597
of \cite{Fl2}, $H^1_f(\QQ_p,A_{\qq}(k/2))$ is equal to (not just
598
contained in) the kernel of the map from $H^1(\QQ_p,A_{\qq}(k/2))$
599
to $H^1(I_p,A_{\qq}(k/2))$, so we have shown that
600
$\res_p(\gamma)\in H^1_f(\QQ_p,A_{\qq}(k/2))$.
601
602
\item {\bf $p\mid N$. }
603
604
First we must show that $H^0(I_p, A'_{\qq}(k/2))$ is
605
$\qq$-divisible. It suffices to show that
606
$$\dim H^0(I_p,A'[\qq](k/2))=\dim H^0(I_p,V'_{\qq}(k/2)),$$
607
but this may be done as in the proof of Lemma \ref{local1}. It
608
follows as above that the image of $c\in H^1(\QQ,A[\qq](k/2))$ in
609
$H^1(I_p,A[\qq](k/2))$ is zero. Then $\res_p(c)$ comes from
610
$H^1(D_p/I_p,H^0(I_p,A[\qq](k/2)))$, by inflation-restriction. The
611
order of this group is the same as the order of the group
612
$H^0(\QQ_p,A[\qq](k/2))$, which we claim is trivial. By the work
613
of Carayol \cite{Ca1}, $p\mid N$ implies that $V_{\qq}(k/2)$ is
614
ramified at $p$, so $\dim H^0(I_p,V_{\qq}(k/2))=0$ or $1$. As
615
above, we see that $\dim H^0(I_p,V_{\qq}(k/2))=\dim
616
H^0(I_p,A[\qq](k/2))$, so we need only consider the case where
617
this common dimension is $1$. The (motivic) Euler factor at $p$
618
for $M_f$ is $(1-\alpha p^{-s})^{-1}$, where $\Frob_p^{-1}$ acts
619
as multiplication by $\alpha$ on the one-dimensional space
620
$H^0(I_p,V_{\qq}(k/2))$. It follows from Theor\'eme A of
621
\cite{Ca1} that this is the same as the Euler factor at $p$ of
622
$L(f,s)$. By the work of Atkin and Lehner \cite{AL}, it then
623
follows that $p^2\nmid N$ and $\alpha=-w_pp^{(k/2)-1}$, where
624
$w_p=\pm 1$ is such that $W_pf=w_pf$. Twisting by $k/2$,
625
$\Frob_p^{-1}$ acts on $H^0(I_p,V_{\qq}(k/2))$ (hence also on
626
$H^0(I_p,A[\qq](k/2))$ as $-w_pp^{-1}$. Since $p\not\equiv
627
-w_p\pmod{q}$, we see that $H^0(\QQ_p,A[\qq](k/2))$ is trivial.
628
Hence $\res_p(c)=0$ so $\res_p(\gamma)=0$ and certainly lies in
629
$H^1_f(\QQ_p,A_{\qq}(k/2))$.
630
631
\item {\bf $p=q$. }
632
633
Since $q\nmid N$ is a prime of good reduction for the motive
634
$M_g$, $\,V'_{\qq}$ is a crystalline representation of
635
$\Gal(\Qbar_q/\QQ_q)$, meaning $D_{\cris}(V'_{\qq})$ and
636
$V'_{\qq}$ have the same dimension, where
637
$D_{\cris}(V'_{\qq}):=H^0(\QQ_q,V'_{\qq}\otimes_{\QQ_q}
638
B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.)
639
As already noted in the proof of Lemma \ref{at q}, $T_{\qq}$ is
640
the $O_{\lambda}[\Gal(\Qbar_q/\QQ_q)]$-module associated to the
641
filtered module $T_{\dR}\otimes O_{\lambda}$. Since also $q>k$, we
642
may now prove, in the same manner as Proposition 9.2 of
643
\cite{Du3}, that $\res_q(\gamma)\in H^1_f(\QQ_q,A_{\qq}(k/2))$.
644
\end{enumerate}
645
\end{proof}
646
647
Theorem 2.7 of \cite{AS} is concerned with verifying local
648
conditions in the case $k=2$, where $f$ and $g$ are associated
649
with abelian varieties $A$ and $B$. (Their theorem also applies to
650
abelian varieties over number fields.) Our restriction outlawing
651
congruences modulo $\qq$ with cusp forms of lower level is
652
analogous to theirs forbidding $q$ from dividing Tamagawa factors
653
$c_{A,l}$ and $c_{B,l}$. (In the case where $A$ is an elliptic
654
curve with $\ord_l(j(A))<0$, consideration of a Tate
655
parametrisation shows that if $q\mid c_{A,l}$, i.e. if
656
$q\mid\ord_l(j(A))$, then it is possible that $A[q]$ is unramified
657
at $l$.)
658
659
In this paper we have encountered two technical problems which we
660
dealt with in quite similar ways:
661
\begin{enumerate}
662
\item dealing with the $\qq$-part of $c_p$ for $p\mid N$;
663
\item proving local conditions at primes $p\mid N$, for an element
664
of $\qq$-torsion.
665
\end{enumerate}
666
If our only interest was in testing the Bloch-Kato conjecture at
667
$\qq$, we could have made these problems cancel out, as in Lemma
668
8.11 of \cite{DFG}, by weakening the local conditions. However, we
669
have chosen not to do so, since we are also interested in the
670
Shafarevich-Tate group, and since the hypotheses we had to assume
671
are not particularly strong.
672
673
\section{Sixteen examples}
674
\newcommand{\nf}[1]{\mbox{\bf #1}}
675
\begin{figure}
676
\caption{\label{fig:newforms}Newforms Relevant to
677
Theorem~\ref{local}}
678
$$
679
\begin{array}{|ccccc|}\hline
680
g & \deg(g) & f & \deg(f) & q's \\\hline
681
\nf{127k4A} & 1 & \nf{127k4C} & 17 & 43 \\
682
\nf{159k4B} & 1 & \nf{159k4E} & 16 & 5, 23 \\
683
\nf{365k4A} & 1 & \nf{365k4E} & 18 & 29 \\
684
\nf{369k4B} & 1 & \nf{369k4I} & 9 & 13 \\
685
\nf{453k4A} & 1 & \nf{453k4E} & 23 & 17 \\
686
\nf{465k4B} & 1 & \nf{465k4I} & 7 & 11 \\
687
\nf{477k4B} & 1 & \nf{477k4L} & 12 & 73 \\
688
\nf{567k4B} & 1 & \nf{567k4H} & 8 & 23 \\
689
\nf{581k4A} & 1 & \nf{581k4E} & 34 & 19^2 \\
690
\nf{657k4A} & 1 & \nf{657k4C} & 7 & 5 \\
691
\nf{657k4A} & 1 & \nf{657k4G} & 12 & 5 \\
692
\nf{681k4A} & 1 & \nf{681k4D} & 30 & 59 \\
693
\nf{684k4C} & 1 & \nf{684k4K} & 4 & 7^2 \\
694
\nf{95k6A} & 1 & \nf{95k6D} & 9 & 31, 59 \\
695
\nf{122k6A} & 1 & \nf{122k6D} & 6 & 73 \\
696
\nf{260k6A} & 1 & \nf{260k6E} & 4 & 17 \\
697
\hline
698
\end{array}
699
$$
700
\end{figure}
701
702
Table~\ref{fig:newforms} on page~\pageref{fig:newforms} lists
703
sixteen pairs of newforms~$f$ and~$g$ (of equal weights and levels)
704
along with at least one prime~$q$ such that there is a prime
705
$\qq\mid q$ with $f\equiv g\pmod{\qq}$. In each case,
706
$\ord_{s=k/2}L(g,k/2)\geq 2$ while $L(f,k/2)\neq 0$.
707
\subsection{Notation}
708
Table~\ref{fig:newforms} is laid out as follows.
709
The first column contains a label whose structure is
710
\begin{center}
711
{\bf [Level]k[Weight][GaloisOrbit]}
712
\end{center}
713
This label determines a newform $g=\sum a_n q^n$, up to Galois
714
conjugacy. For example, \nf{127k4C} denotes a newform in the third
715
Galois orbit of newforms in $S_4(\Gamma_0(127))$. The Galois
716
orbits are ordered first by the degree of $\QQ(\ldots, a_n,
717
\ldots)$, then by $|\mbox{\rm Tr}(a_p(g))|$ for~$p$ not dividing
718
the level, with positive trace being first in the event that the
719
two absolute values are equal, and the first Galois orbit is
720
denoted {\bf A}, the second {\bf B}, and so on. The second column
721
contains the degree of the field $\QQ(\ldots, a_n, \ldots)$. The
722
third and fourth columns contain~$f$ and its degree, respectively.
723
The fifth column contains at least one prime~$q$ such that there
724
is a prime $\qq\mid q$ with $f\equiv g\pmod{\qq}$, and such that
725
the hypotheses of Theorem~\ref{local} (except possibly $r>0$) are
726
satisfied for~$f$,~$g$, and~$\qq$.
727
728
\subsection{The first example in detail}
729
\newcommand{\fbar}{\overline{f}}
730
We describe the first line of Table~\ref{fig:newforms}
731
in more detail. See the next section for further details
732
on how the computations were performed.
733
734
Using modular symbols, we find that there is a newform
735
$$g = q - q^2 - 8q^3 - 7q^4 - 15q^5 + 8q^6 - 25q^7 + \cdots
736
\in S_4(\Gamma_0(127))$$ with $L(g,2)=0$. Because $W_{127}(g)=g$,
737
the functional equation has sign~$+1$, so $L'(g,2)=0$ as well. We
738
also find a newform $f \in S_4(\Gamma_0(127))$ whose Fourier
739
coefficients generate a number field~$K$ of degree~$17$, and by
740
computing the image of the modular symbol $XY\{0,\infty\}$ under
741
the period mapping, we find that $L(f,2)\neq 0$. The newforms~$f$
742
and~$g$ are congruent modulo a prime $\qq$ of~$K$ of residue
743
characteristic~$43$. The mod~$\qq$ reductions of~$f$ and~$g$ are
744
both equal to
745
$$\fbar = q + 42q^2 + 35q^3 + 36q^4 + 28q^5 + 8q^6 + 18q^7
746
+ \cdots\in \FF_{43}[[q]].$$
747
748
There is no form in the Eisenstein subspaces of
749
$M_4(\Gamma_0(127))$ whose Fourier coefficients of index~$n$, with
750
$(n,127)=1$, are congruent modulo $43$ to those of $\fbar$, so
751
$\rho_{f,\qq}\approx\rho_{g,\qq}$ is irreducible. Since $127$ is
752
prime and $S_4(\SL_2(\ZZ))=0$,~$\fbar$ does not arise from a
753
level~$1$ form of weight~$4$. Thus we have checked the hypotheses
754
of Theorem~\ref{local}, so if $r$ is the dimension of
755
$H^1_f(\QQ,V'_{\qq}(k/2))$ then the $\qq$-torsion subgroup of
756
$H^1_f(\QQ,A_{\qq}(k/2))$ has $\FF_{\qq}$-rank at least $r$.
757
758
Since $\ord_{s=k/2}L(g,s)\geq 2$, we expect that $r\geq 2$. Then,
759
since $L(f,k/2)\neq 0$, we expect that the $\qq$-torsion subgroup
760
of $H^1_f(\QQ,A_{\qq}(k/2))$ is equal to the $\qq$-torsion
761
subgroup of $\Sha$. Admitting these assumptions, we have
762
constructed the $\qq$-torsion in $\Sha$ predicted by the
763
Bloch-Kato conjecture.
764
765
For particular examples of elliptic curves one can often find and
766
write down rational points predicted by the Birch and
767
Swinnerton-Dyer conjecture. It would be nice if likewise one could
768
explicitly produce algebraic cycles predicted by the
769
Beilinson-Bloch conjecture in the above examples. Since
770
$L'(g,k/2)=0$, Heegner cycles have height zero (see Corollary
771
0.3.2 of \cite{Z}), so ought to be trivial in
772
$\CH_0^{k/2}(M_g)\otimes\QQ$.
773
774
\subsection{Some remarks on how the computation was performed}
775
We give a brief summary of how the computation was performed. The
776
algorithms that we used were implemented by the second author, and
777
most are a standard part of the MAGMA V2.8 (see \cite{magma}).
778
779
Let~$g$,~$f$, and~$q$ be some data from a line of
780
Table~\ref{fig:newforms} and let~$N$ denote the level of~$g$. We
781
verified the existence of a congruence modulo~$q$, that
782
$L(g,\frac{k}{2})=L'(g,\frac{k}{2})=0$ and $L(f,\frac{k}{2})\neq
783
0$, and that $\rho_{f,\qq}=\rho_{g,\qq}$ is irreducible and does
784
not arise from any $S_k(\Gamma_0(N/p))$, as follows:
785
786
To prove there is a congruence, we showed that the corresponding
787
{\em integral} spaces of modular symbols satisfy an appropriate
788
congruence, which forces the existence of a congruence on the
789
level of Fourier expansions. We showed that $\rho_{g,\qq}$ is
790
irreducible by computing a set that contains all possible residue
791
characteristics of congruences between~$g$ and any Eisenstein
792
series of level dividing~$N$, where by congruence, we mean a
793
congruence for all Fourier coefficients of index~$n$ with
794
$(n,N)=1$. Similarly, we checked that~$g$ is not congruent to any
795
form~$h$ of level $N/p$ for any~$p$ that exactly divides~$N$ by
796
listing a basis of such~$h$ and finding the possible congruences,
797
where again we disregard the Fourier coefficients of index not
798
coprime to~$N$.
799
800
To verify that $L(g,\frac{k}{2})=0$, we computed the image of the
801
modular symbol ${\mathbf
802
e}=X^{\frac{k}{2}-1}Y^{k-2-(\frac{k}{2}-1)}\{0,\infty\}$ under a
803
map with the same kernel as the period mapping, and found that the
804
image was~$0$. The period mapping sends the modular
805
symbol~${\mathbf e}$ to a nonzero multiple of $L(g,\frac{k}{2})$,
806
so that ${\mathbf e}$ maps to~$0$ implies that
807
$L(g,\frac{k}{2})=0$. In a similar way, we verified that
808
$L(f,\frac{k}{2})\neq 0$. Next, we checked that $W_N(g) = g$
809
which, because of the functional equation, implies that
810
$L'(g,\frac{k}{2})=0$. Table~\ref{fig:newforms} is of
811
independent interest because it includes examples of modular
812
forms of weight $>2$ with a zero at $\frac{k}{2}$
813
that is not forced by the functional equation. We found no such
814
examples of weights $\geq 8$.
815
816
For the two examples \nf{581k4} and \nf{684k4K}, the square of a
817
prime appears in the $q$-column, meaning $f$ and $g$ are congruent
818
$\bmod{\qq^2}$ for some $\qq\mid q$. In these cases, a
819
modification of Theorem \ref{local} will show that the
820
$\qq^2$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(k/2))$ has
821
$O_E/\qq^2$-rank at least $r$.
822
823
\subsection{Other examples}
824
We have some other examples where a congruence between forms can
825
be shown, but the levels differ. However, Remark 5.3 does not
826
apply, so that one of the forms could have an odd functional
827
equation, and the other could have an even functional equation.
828
For instance, we have a 13-congruence between $g=$\nf{81k4A} and
829
$f=$\nf{567k4L}; here $L(\nf{567k4L},2)\neq 0$, while
830
$L(\nf{81k4},2)=0$ since it has {\it odd} functional equation. A
831
very minor modification of Theorem \ref{local} and its proof still
832
shows us, for this example, that the $\qq$-torsion subgroup of
833
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$
834
(assuming $r\geq 1$). Here $f$ obviously fails the condition about
835
not being congruent to a form of lower level, so in Lemma 4.3 it
836
is possible that $\ord_{\qq}(c_7(2))>0$. \edit{It may be possible
837
to rule this out in particular cases by showing that $\qq$ doesn't
838
divide the Euler factor of $f$ at $p$, evaluated at $s=k/2$. Does
839
that work here? }
840
841
In an example where the divisibility between the levels is the
842
other way round, a 7-congruence between $g=$\nf{122k6A} and
843
$f=$\nf{61k6B}, we have differing levels but both still have even
844
functional equation. In this case, we have that
845
$L(\nf{122k6A},3)=0$. In the proof of Theorem 6.1, we find a
846
problem with the local condition at $p=2$. The map from
847
$H^1(I_2,A'[\qq](3))$ to $H^1(I_2,A'_{\qq}(3))$ is not necessarily
848
injective, but its kernel is at most one-dimensional, so we still
849
get the $\qq$-torsion subgroup of $H^1_f(\QQ,A_{\qq}(2))$ having
850
$\FF_{\qq}$-rank at least $1$ (assuming $r\geq 2$), and thus get
851
(probable) elements of $\Sha$ for \nf{61k6B}. In particular, these
852
elements of $\Sha$ are {\it invisible} at level 61. When the
853
levels are different we are no longer able to apply Theorem 2.1 of
854
\cite{FJ}. However, we still have the congruences of integral
855
modular symbols required to make the proof of Proposition
856
\ref{div} go through. Indeed, as noted above, the congruences of
857
modular forms were found by producing congruences of modular
858
symbols. Despite these congruences of modular symbols, Remark 5.3
859
does not apply, since there is no reason to suppose that
860
$w_N=w_{N'}$, where $N$ and $N'$ are the distinct levels.
861
862
Finally, there are two examples where we have a form $g$ with even
863
functional equation such that $L(g,k/2)=0$, and a congruent form
864
$f$ which has odd functional equation; these are a 23-congruence
865
between $g=$\nf{453k4A} and $f=$\nf{151k4A}, and a 43-congruence
866
between $g=$\nf{681k4A} and $f=$\nf{227k4A}. If
867
$\ord_{s=2}L(f,s)=1$, it ought to be the case that
868
$\dim(H^1_f(\QQ,V_{\qq}(2)))=1$. If we assume this is so, and
869
similarly that $r=\ord_{s=2}(L(g,s))\geq 2$, then unfortunately
870
the appropriate modification of Theorem \ref{local} does not
871
necessarily provide us with non-trivial $\qq$-torsion in $\Sha$.
872
It only tells us that the $\qq$-torsion subgroup of
873
$H^1_f(\QQ,A_{\qq}(2))$ has $\FF_{\qq}$-rank at least $1$. It
874
could all be in the image of $H^1_f(\QQ,V_{\qq}(2))$. $\Sha$
875
appears in the conjectural formula for the first derivative of the
876
complex $L$ function, evaluated at $s=k/2$, but in combination
877
with a regulator that we have no way of calculating.
878
879
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
880
associated with $f$ and $g$ by the construction of Mazur, Tate and
881
Teitelbaum \cite{MTT}, each divided by a suitable canonical
882
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
883
quite clear what to make of this. This divisibility may be proved
884
as follows. The measures $d\mu_{f,\alpha}$ and (a $q$-adic unit
885
times) $d\mu_{g,\alpha'}$ in \cite{MTT} (again, suitably
886
normalised) are congruent $\bmod{ \qq}$, as a result of the
887
congruence between the modular symbols out of which they are
888
constructed. Integrating an appropriate function against these
889
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod{ \qq}$ to
890
$L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$, since
891
$L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case where the
892
signs in the functional equations of $L(g,s)$ and $L_q(g,s)$ are
893
the same, positive in this instance. (According to the proposition
894
in Section 18 of \cite{MTT}, the signs differ precisely when
895
$L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
896
897
898
899
\subsection{Excluded data}
900
We also found some examples for which the conditions of Theorem \ref{local}
901
were not met. We have a 7-congruence between \nf{639k4B} and \nf{639k4H},
902
but $w_{71}=-1$, so that $71\equiv -w_{71}\pmod{7}$. There is a similar
903
problem with a 7-congruence between \nf{260k6A} and \nf{260k6E} --- here
904
$w_{13}=1$ so that $13\equiv -w_{13}\pmod{7}$. Finally, there is a
905
5-congruence between \nf{116k6A} and \nf{116k6D}, but here the prime 5
906
is less than the weight 6.
907
908
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1004
1005
1006
\end{document}
1007
127k4A 43 127k4C 17 [43]
1008
159k4A 5,23 159k4E 8 [5]x[23]
1009
365k4B 29 365k4E 18 [29]x[5] (extra factor of 5 divides the level)
1010
369k4A 5,13 369k4J 9 [5]x[13]x[2]
1011
453k4A 5,17 453k4E 23 [5]x[17]
1012
453k4A 23 151k4A 30 Odd func eq for g
1013
465k4A 11 465k4H 7 [11]x[5]x[2]
1014
477k4A 73 477k4M 12 [73]x[2]
1015
567k4A 23 567k4I 8 [23]x[3]
1016
81k4A 13 567k4L 12 Odd func eq for f, Theorem 4.1 gives nothing.
1017
581k4A 19,19 581k4E 34 [19^2]x[4] (possible BIG Sha, as 19^2 divides MD)
1018
639k4A 7 639k4H 12 [7]
1019
657k4A 5 657k4C 7 [5]x[3]x[2] (see next note)
1020
657k4A 5 657k4G 12 [5]x[4] (does 657k4A make both these visible?)
1021
681k4A 43 227k4A 23 Odd func eq for g
1022
681k4A 59 681k4D 30 [59]x[3]x[2]
1023
684k4C 7,7 684k4K 4 [7^2]x[2] (see note to 581k4A)
1024
95k6A 31,59 95k6D 9 [31]x[59]
1025
116k6A 5 116k6D 6 [5]x[29]x[2]
1026
122k6A 7 61k6B 14 7^2 appears in L(61k6B,3)
1027
122k6A 73 122k6C 6 [73]x[3] (guess that 3 is a bad prime now)
1028
260k6A 7,17 260k6E 4 [7]x[17]x[4] <-- Did not compute MD or LROP
1029