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preceding: \begin<<838>>table<<838>>
\vspace<<839>>-2ex<<839>>
\caption<<1298>>\label<<840>>tab:invisforms<<840>>Conjecturally nontrivial $\Sha$ (mostly invisible)<<1298>>
\vspace<<841>>-4ex<<841>>
$$
\begin<<842>>array<<842>><<843>>|c|c|c|c|<<843>>\hline
f ;SPMamp; \deg(f) ;SPMamp; B\,\, (\text<<844>>$\Sha$ bound<<844>>);SPMamp; \text<<845>>all odd congruence primes<<845>>\\ \hline
\nf<<846>>127k4C<<846>>* ;SPMamp; 17 ;SPMamp; 43^<<847>>2<<847>> ;SPMamp; 43, 127 \\
\nf<<848>>159k4E<<848>>* ;SPMamp; 8 ;SPMamp; 23^<<849>>2<<849>> ;SPMamp; 3, 5, 11, 23, 53, 13605689 \\
\nf<<850>>263k4B<<850>> ;SPMamp; 39 ;SPMamp; 41^<<851>>2<<851>> ;SPMamp; 263 \\
\nf<<852>>269k4C<<852>> ;SPMamp; 39 ;SPMamp; 23^<<853>>2<<853>> ;SPMamp; 269 \\
\nf<<854>>271k4B<<854>> ;SPMamp; 39 ;SPMamp; 29^<<855>>2<<855>> ;SPMamp; 271 \\
\nf<<856>>281k4B<<856>> ;SPMamp; 40 ;SPMamp; 29^<<857>>2<<857>> ;SPMamp; 281 \\
\nf<<858>>295k4C<<858>> ;SPMamp; 16 ;SPMamp; 7^<<859>>2<<859>> ;SPMamp; 3, 5, 11, 59, 101, 659, 70791023 \\
\nf<<860>>299k4C<<860>> ;SPMamp; 20 ;SPMamp; 29^<<861>>2<<861>> ;SPMamp; 13, 23, 103, 20063, 21961 \\
\nf<<862>>321k4C<<862>> ;SPMamp; 16 ;SPMamp; 13^<<863>>2<<863>> ;SPMamp; 3, 5, 107, 157, 12782373452377 \\
\hline
\nf<<864>>95k6D<<864>>* ;SPMamp; 9 ;SPMamp; 31^<<865>>2<<865>> \!\cdot\! 59^<<866>>2<<866>> ;SPMamp; 3, 5, 17, 19, 31, 59, 113, 26701 \\
\nf<<867>>101k6B<<867>> ;SPMamp; 24 ;SPMamp; 17^<<868>>2<<868>> ;SPMamp; 101 \\
\nf<<869>>103k6B<<869>> ;SPMamp; 24 ;SPMamp; 23^<<870>>2<<870>> ;SPMamp; 103 \\
\nf<<871>>111k6C<<871>> ;SPMamp; 9 ;SPMamp; 11^<<872>>2<<872>> ;SPMamp; 3, 37, 2796169609 \\
\nf<<873>>122k6D<<873>>* ;SPMamp; 6 ;SPMamp; 73^<<874>>2<<874>> ;SPMamp; 3, 5, 61, 73, 1303196179 \\
\nf<<875>>153k6G<<875>> ;SPMamp; 5 ;SPMamp; 7^<<876>>2<<876>> ;SPMamp; 3, 17, 61, 227 \\
\nf<<877>>157k6B<<877>> ;SPMamp; 34 ;SPMamp; 251^<<878>>2<<878>> ;SPMamp; 157 \\
\nf<<879>>167k6B<<879>> ;SPMamp; 40 ;SPMamp; 41^<<880>>2<<880>> ;SPMamp; 167 \\
\nf<<881>>172k6B<<881>> ;SPMamp; 9 ;SPMamp; 7^<<882>>2<<882>> ;SPMamp; 3, 11, 43, 787 \\
\nf<<883>>173k6B<<883>> ;SPMamp; 39 ;SPMamp; 71^<<884>>2<<884>> ;SPMamp; 173 \\
\nf<<885>>181k6B<<885>> ;SPMamp; 40 ;SPMamp; 107^<<886>>2<<886>> ;SPMamp; 181 \\
\nf<<887>>191k6B<<887>> ;SPMamp; 46 ;SPMamp; 85091^<<888>>2<<888>> ;SPMamp; 191 \\
\nf<<889>>193k6B<<889>> ;SPMamp; 41 ;SPMamp; 31^<<890>>2<<890>> ;SPMamp; 193 \\
\nf<<891>>199k6B<<891>> ;SPMamp; 46 ;SPMamp; 200329^2 ;SPMamp; 199 \\
\hline
\nf<<892>>47k8B<<892>> ;SPMamp; 16 ;SPMamp; 19^<<893>>2<<893>> ;SPMamp; 47 \\
\nf<<894>>59k8B<<894>> ;SPMamp; 20 ;SPMamp; 29^<<895>>2<<895>> ;SPMamp; 59 \\
\nf<<896>>67k8B<<896>> ;SPMamp; 20 ;SPMamp; 29^<<897>>2<<897>> ;SPMamp; 67 \\
\nf<<898>>71k8B<<898>> ;SPMamp; 24 ;SPMamp; 379^<<899>>2<<899>> ;SPMamp; 71 \\
\nf<<900>>73k8B<<900>> ;SPMamp; 22 ;SPMamp; 197^<<901>>2<<901>> ;SPMamp; 73 \\
\nf<<902>>74k8C<<902>> ;SPMamp; 6 ;SPMamp; 23^<<903>>2<<903>> ;SPMamp; 37, 127, 821, 8327168869 \\
\nf<<904>>79k8B<<904>> ;SPMamp; 25 ;SPMamp; 307^<<905>>2<<905>> ;SPMamp; 79 \\
\nf<<906>>83k8B<<906>> ;SPMamp; 27 ;SPMamp; 1019^<<907>>2<<907>> ;SPMamp; 83 \\
\nf<<908>>87k8C<<908>> ;SPMamp; 9 ;SPMamp; 11^<<909>>2<<909>> ;SPMamp; 3, 5, 7, 29, 31, 59, 947, 22877, 3549902897 \\
\nf<<910>>89k8B<<910>> ;SPMamp; 29 ;SPMamp; 44491^<<911>>2<<911>> ;SPMamp; 89 \\
\nf<<912>>97k8B<<912>> ;SPMamp; 29 ;SPMamp; 11^<<913>>2<<913>> \!\cdot\! 277^<<914>>2<<914>> ;SPMamp; 97 \\
\nf<<915>>101k8B<<915>> ;SPMamp; 33 ;SPMamp; 19^<<916>>2<<916>> \!\cdot\! 11503^<<917>>2<<917>> ;SPMamp; 101 \\
\nf<<918>>103k8B<<918>> ;SPMamp; 32 ;SPMamp; 75367^<<919>>2<<919>> ;SPMamp; 103 \\
\nf<<920>>107k8B<<920>> ;SPMamp; 34 ;SPMamp; 17^<<921>>2<<921>> \!\cdot\! 491^<<922>>2<<922>> ;SPMamp; 107 \\
\nf<<923>>109k8B<<923>> ;SPMamp; 33 ;SPMamp; 23^<<924>>2<<924>> \!\cdot\! 229^<<925>>2<<925>> ;SPMamp; 109 \\
\nf<<926>>111k8C<<926>> ;SPMamp; 12 ;SPMamp; 127^<<927>>2<<927>> ;SPMamp; 3, 7, 11, 13, 17, 23, 37, 6451, 18583, 51162187 \\
\nf<<928>>113k8B<<928>> ;SPMamp; 35 ;SPMamp; 67^<<929>>2<<929>> \!\cdot\! 641^<<930>>2<<930>> ;SPMamp; 113 \\
\nf<<931>>115k8B<<931>> ;SPMamp; 12 ;SPMamp; 37^<<932>>2<<932>> ;SPMamp; 3, 5, 19, 23, 572437, 5168196102449 \\
\nf<<933>>117k8I<<933>> ;SPMamp; 8 ;SPMamp; 19^<<934>>2<<934>> ;SPMamp; 3, 13, 181 \\
\nf<<935>>118k8C<<935>> ;SPMamp; 8 ;SPMamp; 37^<<936>>2<<936>> ;SPMamp; 5, 13, 17, 59, 163, 3923085859759909 \\
\nf<<937>>119k8C<<937>> ;SPMamp; 16 ;SPMamp; 1283^<<938>>2<<938>> ;SPMamp; 3, 7, 13, 17, 109, 883, 5324191, 91528147213 \\
\hline
\end<<939>>array<<939>>
$$
\end<<940>>table<<940>>
\begin<<941>>table<<941>>
$$
\begin<<942>>array<<942>><<943>>|c|c|c|c|<<943>>\hline
f ;SPMamp; \deg(f) ;SPMamp; B\,\, (\text<<944>>$\Sha$ bound<<944>>);SPMamp; \text<<945>>all odd congruence primes<<945>>\\ \hline
\nf<<946>>121k8F<<946>> ;SPMamp; 6 ;SPMamp; 71^<<947>>2<<947>> ;SPMamp; 3, 11, 17, 41 \\
\nf<<948>>121k8G<<948>> ;SPMamp; 12 ;SPMamp; 13^<<949>>2<<949>> ;SPMamp; 3, 11 \\
\nf<<950>>121k8H<<950>> ;SPMamp; 12 ;SPMamp; 19^<<951>>2<<951>> ;SPMamp; 5, 11 \\
\nf<<952>>125k8D<<952>> ;SPMamp; 16 ;SPMamp; 179^<<953>>2<<953>> ;SPMamp; 5 \\
\nf<<954>>127k8B<<954>> ;SPMamp; 39 ;SPMamp; 59^<<955>>2<<955>> ;SPMamp; 127 \\
\nf<<956>>128k8F<<956>> ;SPMamp; 4 ;SPMamp; 11^<<957>>2<<957>> ;SPMamp; 1 \\
\nf<<958>>131k8B<<958>> ;SPMamp; 43 ;SPMamp; 241^<<959>>2<<959>> \!\cdot\! 817838201^<<960>>2<<960>>;SPMamp;131\\
\nf<<961>>134k8C<<961>> ;SPMamp; 11 ;SPMamp; 61^<<962>>2<<962>> ;SPMamp; 11, 17, 41, 67, 71, 421, 2356138931854759 \\
\nf<<963>>137k8B<<963>> ;SPMamp; 42 ;SPMamp; 71^<<964>>2<<964>> \!\cdot\! 749093^<<965>>2<<965>> ;SPMamp; 137 \\
\nf<<966>>139k8B<<966>> ;SPMamp; 43 ;SPMamp; 47^<<967>>2<<967>> \!\cdot\! 89^<<968>>2<<968>> \!\cdot\! 1021^<<969>>2<<969>> ;SPMamp; 139 \\
\nf<<970>>141k8C<<970>> ;SPMamp; 14 ;SPMamp; 13^<<971>>2<<971>> ;SPMamp; 3, 5, 7, 47, 4639, 43831013, 4047347102598757 \\
\nf<<972>>142k8B<<972>> ;SPMamp; 10 ;SPMamp; 11^<<973>>2<<973>> ;SPMamp; 3, 53, 71, 56377, 1965431024315921873 \\
\nf<<974>>143k8C<<974>> ;SPMamp; 19 ;SPMamp; 307^<<975>>2<<975>> ;SPMamp; 3, 11, 13, 89, 199, 409, 178397,
639259, 17440535
97287 \\
\nf<<976>>143k8D<<976>> ;SPMamp; 21 ;SPMamp; 109^<<977>>2<<977>> ;SPMamp; 3, 7, 11, 13, 61, 79, 103, 173, 241,
769, 36583
\\
\nf<<978>>145k8C<<978>> ;SPMamp; 17 ;SPMamp; 29587^<<979>>2<<979>> ;SPMamp; 5, 11, 29, 107, 251623, 393577,
518737, 9837145
699 \\
\nf<<980>>146k8C<<980>> ;SPMamp; 12 ;SPMamp; 3691^<<981>>2<<981>> ;SPMamp; 11, 73, 269, 503, 1673540153, 11374452082219 \\
\nf<<982>>148k8B<<982>> ;SPMamp; 11 ;SPMamp; 19^<<983>>2<<983>> ;SPMamp; 3, 37 \\
\nf<<984>>149k8B<<984>> ;SPMamp; 47 ;SPMamp; 11^<<985>>4<<985>> \!\cdot\! 40996789^<<986>>2<<986>> ;SPMamp; 149\\
\hline
\nf<<987>>43k10B<<987>> ;SPMamp; 17 ;SPMamp; 449^<<988>>2<<988>> ;SPMamp; 43 \\
\nf<<989>>47k10B<<989>> ;SPMamp; 20 ;SPMamp; 2213^<<990>>2<<990>> ;SPMamp; 47 \\
\nf<<991>>53k10B<<991>> ;SPMamp; 21 ;SPMamp; 673^<<992>>2<<992>> ;SPMamp; 53 \\
\nf<<993>>55k10D<<993>> ;SPMamp; 9 ;SPMamp; 71^<<994>>2<<994>> ;SPMamp; 3, 5, 11, 251, 317, 61339, 19869191 \\
\nf<<995>>59k10B<<995>> ;SPMamp; 25 ;SPMamp; 37^<<996>>2<<996>> ;SPMamp; 59 \\
\nf<<997>>62k10E<<997>> ;SPMamp; 7 ;SPMamp; 23^<<998>>2<<998>> ;SPMamp; 3, 31, 101, 523, 617, 41192083 \\
\nf<<999>>64k10K<<999>> ;SPMamp; 2 ;SPMamp; 19^<<1000>>2<<1000>> ;SPMamp; 3 \\
\nf<<1001>>67k10B<<1001>> ;SPMamp; 26 ;SPMamp; 191^<<1002>>2<<1002>> \!\cdot\! 617^<<1003>>2<<1003>> ;SPMamp; 67 \\
\nf<<1004>>68k10B<<1004>> ;SPMamp; 7 ;SPMamp; 83^<<1005>>2<<1005>> ;SPMamp; 3, 7, 17, 8311 \\
\nf<<1006>>71k10B<<1006>> ;SPMamp; 30 ;SPMamp; 1103^<<1007>>2<<1007>> ;SPMamp; 71 \\
\hline
\nf<<1008>>19k12B<<1008>> ;SPMamp; 9 ;SPMamp; 67^<<1009>>2<<1009>> ;SPMamp; 5, 17, 19, 31, 571 \\
\nf<<1010>>31k12B<<1010>> ;SPMamp; 15 ;SPMamp; 67^<<1011>>2<<1011>> \!\cdot\! 71^<<1012>>2<<1012>> ;SPMamp; 31, 13488901 \\
\nf<<1013>>35k12C<<1013>> ;SPMamp; 6 ;SPMamp; 17^<<1014>>2<<1014>> ;SPMamp; 5, 7, 23, 29, 107, 8609, 1307051 \\
\nf<<1015>>39k12C<<1015>> ;SPMamp; 6 ;SPMamp; 73^<<1016>>2<<1016>> ;SPMamp; 3, 13, 1491079, 3719832979693 \\
\nf<<1017>>41k12B<<1017>> ;SPMamp; 20 ;SPMamp; 54347^<<1018>>2<<1018>> ;SPMamp; 7, 41, 3271, 6277 \\
\nf<<1019>>43k12B<<1019>> ;SPMamp; 20 ;SPMamp; 212969^<<1020>>2<<1020>> ;SPMamp; 43, 1669, 483167 \\
\nf<<1021>>47k12B<<1021>> ;SPMamp; 23 ;SPMamp; 24469^<<1022>>2<<1022>> ;SPMamp; 17, 47, 59, 2789 \\
\nf<<1023>>49k12H<<1023>> ;SPMamp; 12 ;SPMamp; 271^<<1024>>2<<1024>> ;SPMamp; 7 \\
\hline
\end<<1025>>array<<1025>>
$$
\end<<1026>>table<<1026>>
\begin<<1027>>lem<<1027>>\label<<1028>>lem:lrat<<1028>>
If $p\nmid Nk!$ is such that $f$ is not congruent to any of its
Galois conjugates modulo a prime dividing $p$ then the $p$-parts
of
$$
\frac<<1029>>L(M_f/\QQ,k/2)<<1029>><<1299>>\Omega_<<1030>>M_f/\QQ<<1030>><<1299>>\qquad\text<<1031>>and<<1031>>\qquad
\Norm\left(\frac<<1032>>L(f,k/2)<<1032>><<1300>>\vol_<<1033>>\infty<<1033>><<1300>>\aaa^<<1034>>\pm<<1034>>\right)
$$
are equal, where $\vol_\infty$ is as in Section~\ref<<1035>>sec:bkconj<<1035>>.
\end<<1036>>lem<<1036>>
\begin<<1037>>proof<<1037>> (Sketch.) Let $H$ be the $\ZZ$-module of all
integral cuspidal modular symbols for $\Gamma_0(N)$. Let $I$ be
the image of $H$ under the projection into $H\otimes\QQ$
corresponding to $f$ and its Galois conjugates. $I$ is not
necessarily contained in $H$ since we will have inverted the
residue characteristics of any primes of congruence between $f$
and cuspforms for $\Gamma_0(N)$ which are not Galois conjugate to
$f$.
Now $\mathcal<<1038>>L<<1038>>$ is (up to divisors of $Nk!$) the lattice
obtained by pairing the cohomology modular symbols
$\Phi_<<1301>>f^<<1039>>(i)<<1039>><<1301>>^<<1040>>\pm<<1040>>$ (as in \S 5) with the homology modular
symbols in $H$, or equivalently in $I$. For $1\leq i\leq d$ let
$I_i$ be the $O_E$-module generated by the image of the projection
of $I$ into $I\otimes E$ corresponding to $f^<<1041>>(i)<<1041>>$. The finite
index of $I\otimes O_E$ in $\oplus_<<1042>>i=1<<1042>>^d I_i$ is divisible only
by primes of congruence between $f$ and its Galois conjugates. Up
to these primes, $\Omega_<<1043>>M_f/\QQ<<1043>>/(2\pi i)^<<1044>>((k/2)-1)d<<1044>>$ is then
a product of the $d$ complex numbers obtained by pairing
$\Phi_<<1302>>f^<<1045>>(i)<<1045>><<1302>>^<<1046>>\pm<<1046>>$ with $I_i$, for $1\leq i\leq d$. Bearing in
mind the last line of \S 3, and ignoring divisors of $\aaa^<<1047>>\pm<<1047>>$,
which are clearly of no importance, we see that these complex
numbers are the $\Omega^<<1048>>\pm<<1048>>_<<1303>>f^<<1049>>(i)<<1049>><<1303>>$, up to divisors of $Nk!$.
We have then a factorisation of the left hand side which shows it
to be equal to the right hand side, to the extent claimed by the
lemma.
\end<<1050>>proof<<1050>>
\begin<<1051>>remar<<1051>>
The newform $f=\nf<<1052>>319k4C<<1052>>$ is congruent to one of its Galois conjugates
modulo~$17$ and $17\mid \frac<<1053>>L(M_f/\QQ,k/2)<<1053>><<1304>>\Omega_<<1054>>M_f/\QQ<<1054>><<1304>>$ so the lemma
and our computations
say nothing about whether or not $17$ divides
$\Norm\left(\frac<<1055>>L(f,k/2)<<1055>><<1305>>\vol_<<1056>>\infty<<1056>><<1305>>\aaa^<<1057>>\pm<<1057>>\right)$.
\end<<1058>>remar<<1058>>
Let~$\mathcal<<1059>>S<<1059>>$ be the set of newforms with~level $N$ and
weight~$k$ satisfying either $k=4$ and $N\leq 321$, or $k=6$ and
$N\leq 199$, or $k=8$ and $N\leq 149$, or $k=10$ and $N\leq 72$,
or $k=12$ and $N\leq 49$. Given $f\in \mathcal<<1060>>S<<1060>>$, let~$B$ be
defined as follows:
\begin<<1061>>enumerate<<1061>>
\item Let $L_1$ be the numerator of the
rational number $L(M_f/\QQ,k/2)/\Omega_<<1062>>M_f/\QQ<<1062>>$.
If $L_1=0$ let $B=1$ and terminate.
\item Let $L_2$ be the part of $L_1$ that is coprime to $Nk!$.
\item Let $L_3$ be the part of $L_2$ that is coprime to
$p+1$ for every prime~$p$ such that $p^2\mid N$.
\item Let $L_4$ be the part of $L_3$ coprime to the residue characteristic
of any prime of
congruence between~$f$ and a form of weight~$k$ and
lower level. (By congruence here, we mean a congruence for coefficients
$a_n$ with $n$ coprime to the level of~$f$.)
\item Let $L_5$ be the part of $L_4$ coprime to the residue characteristic
of any prime of congruence
between~$f$ and an Eisenstein series. (This eliminates
residue characteristics of reducible representations.)
\item Let $B$ be the part of $L_5$ coprime to the residue characteristic
of any prime of congruence between $f$ and any one of its Galois
conjugates.
\end<<1063>>enumerate<<1063>>
Proposition~\ref<<1064>>sha<<1064>> and Lemma~\ref<<1065>>lem:lrat<<1065>> imply that if
$\ord_p(B)
;SPMgt; 0$ then, according
to the Bloch-Kato conjecture, $\ord_p(\#\Sha)=\ord_p(B) ;SPMgt; 0$. We
have left the congruence primes in $B$ in the starred examples
since the squares are still suggestive.
We computed~$B$ for every newform in~$\mathcal<<1066>>S<<1066>>$. There are
many examples in which $L_3$ is large, but~$B$ is not, and this is
because of Tamagawa factors. For example, <<1067>>\bf 39k4C<<1067>> has
$L_3=19$, but $B=1$ because of a $19$-congruence with a form of
level~$13$; in this case we must have $19\mid c_<<1068>>3<<1068>>(2)$, where
$c_<<1069>>3<<1069>>(2)$ is as in Section~\ref<<1070>>sec:bkconj<<1070>>. See
Section~\ref<<1071>>sec:other_ex<<1071>> for more details. Also note that in
every example~$B$ is a perfect square, which is as predicted by
the existence of Flach's generalised Cassels-Tate pairing
\cite<<1072>>Fl2<<1072>>. (Note that for $\lambda\mid l$, a non-congruence prime
for $f$, the lattice $T_<<1073>>\lambda<<1073>>$ is self-dual, so the pairing
shows that the order of the $\ell$-part of $\Sha$, if finite,
is a square.) That our computed value of~$B$ should be a square is
not <<1074>>\it a priori<<1074>> obvious.
For simplicity, we discard residue characteristics instead of primes
of rings of integers, so our definition of~$B$ is overly conservative.
For example,~$5$ occurs in row~$2$ of Table~\ref<<1075>>tab:newforms<<1075>> but not
in Table~\ref<<1076>>tab:invisforms<<1076>>, because \nf<<1077>>159k4E<<1077>> is Eisenstein at
some prime above~$5$, but the prime of congruences of
characteristic~$5$ between \nf<<1078>>159k4B<<1078>> and \nf<<1079>>159k4E<<1079>> is not
Eisenstein.
The newforms for which $B;SPMgt;1$ are given in
Table~\ref<<1080>>tab:invisforms<<1080>>. The second column of the table records
the degree of the field generated by the Fourier coefficients
of~$f$. The third contains~$B$. Let~$W$ be the intersection of
the span of all conjugates of~$f$ with $S_k(\Gamma_0(N),\ZZ)$ and
$W^<<1081>>\perp<<1081>>$ the Petersson orthogonal complement of~$W$ in
$S_k(\Gamma_0(N),\ZZ)$. The fourth column contains the odd
prime divisors of $\#(S_k(\Gamma_0(N),\ZZ)/(W+W^<<1082>>\perp<<1082>>))$, which
are exactly the possible primes of congruence for~$f$. We place a
$*$ next to the four entries of Table~\ref<<1083>>tab:invisforms<<1083>> that
also occur in Table~\ref<<1084>>tab:newforms<<1084>>.
\subsection<<1085>>Examples in which hypotheses fail<<1085>>\label<<1086>>sec:other_ex<<1086>>
We have some other examples where forms of
different levels are congruent.
However, Remark~\ref<<1087>>sign<<1087>> does not
apply, so that one of the forms could have an odd functional
equation, and the other could have an even functional equation.
For instance, we have a $19$-congruence between the
newforms $g=\nf<<1088>>13k4A<<1088>>$ and $f=\nf<<1089>>39k4C<<1089>>$ of Fourier
coefficients coprime to $39$.
Here $L(f,2)\neq 0$, while $L(g,2)=0$ since $L(g,s)$
has <<1090>>\it odd<<1090>> functional equation.
Here~$f$ fails the condition about not being congruent
to a form of lower level, so in Lemma~\ref<<1091>>local1<<1091>> it is possible that
$\ord_<<1092>>\qq<<1092>>(c_<<1093>>3<<1093>>(2));SPMgt;0$. In fact this does happen. Because
$V'_<<1094>>\qq<<1094>>$ (attached to~$g$ of level $13$) is unramified at $p=3$,
$H^0(I_p,A[\qq])$ (the same as $H^0(I_p,A'[\qq])$) is
two-dimensional. As in (2) of the proof of Theorem~\ref<<1095>>local<<1095>>,
one of the eigenvalues of $\Frob_p^<<1096>>-1<<1096>>$ acting on this
two-dimensional space is $\alpha=-w_pp^<<1097>>(k/2)-1<<1097>>$, where
$W_pf=w_pf$. The other must be $\beta=-w_pp^<<1098>>k/2<<1098>>$, so that
$\alpha\beta=p^<<1099>>k-1<<1099>>$. Twisting by $k/2$, we see that
$\Frob_p^<<1100>>-1<<1100>>$ acts as $-w_p$ on the quotient of
$H^0(I_p,A[\qq](k/2))$ by the image of $H^0(I_p,V_<<1101>>\qq<<1101>>(k/2))$.
Hence $\ord_<<1102>>\qq<<1102>>(c_p(k/2));SPMgt;0$ when $w_p=-1$, which is the case in
our example here with $p=3$. Likewise $H^0(\QQ_p,A[\qq](k/2))$ is
nontrivial when $w_p=-1$, so (2) of the proof of Theorem~\ref<<1103>>local<<1103>>
does not work. This is just as well, since had it
worked we would have expected
$\ord_<<1104>>\qq<<1104>>(L(f,k/2)/\vol_<<1105>>\infty<<1105>>)\geq 3$, which computation
shows not to be the case.
In the following example, the divisibility between the levels is
the other way round. There is a $7$-congruence between
$g=\nf<<1106>>122k6A<<1106>>$ and $f=\nf<<1107>>61k6B<<1107>>$, both $L$-functions have even
functional equation, and $L(g,3)=0$. In the proof of
Theorem~\ref<<1108>>local<<1108>>, there is a problem with the local condition
at $p=2$. The map from $H^1(I_2,A'[\qq](3))$ to
$H^1(I_2,A'_<<1109>>\qq<<1109>>(3))$ is not necessarily injective, but its
kernel is at most one dimensional, so we still get the
$\qq$-torsion subgroup of $H^1_f(\QQ,A_<<1110>>\qq<<1110>>(2))$ having
$\FF_<<1111>>\qq<<1111>>$-rank at least~$1$ (assuming $r\geq 2$), and thus get
elements of $\Sha$ for \nf<<1112>>61k6B<<1112>> (assuming all along the strong
Beilinson-Bloch conjecture). In particular, these elements of
$\Sha$ are <<1113>>\it invisible<<1113>> at level 61. When the levels are
different we are no longer able to apply Theorem 2.1 of \cite<<1114>>FJ<<1114>>.
However, we still have the congruences of integral modular symbols
required to make the proof of Proposition \ref<<1115>>div<<1115>> go through.
Indeed, as noted above, the congruences of modular forms were
found by producing congruences of modular symbols. Despite these
congruences of modular symbols, Remark~\ref<<1116>>sign<<1116>> does not apply,
since there is no reason to suppose that $w_N=w_<<1117>>N'<<1117>>$, where $N$
and $N'$ are the distinct levels.
Finally, there are two examples where we have a form $g$ with even
functional equation such that $L(g,k/2)=0$, and a congruent form
$f$ which has odd functional equation; these are a 23-congruence
between $g=\nf<<1118>>453k4A<<1118>>$ and $f=\nf<<1119>>151k4A<<1119>>$, and a 43-congruence
between $g=\nf<<1120>>681k4A<<1120>>$ and $f=\nf<<1121>>227k4A<<1121>>$. If
$\ord_<<1122>>s=2<<1122>>L(f,s)=1$, it ought to be the case that
$\dim(H^1_f(\QQ,V_<<1123>>\qq<<1123>>(2)))=1$. If we assume this is so, and
similarly that $r=\ord_<<1124>>s=2<<1124>>(L(g,s))\geq 2$, then unfortunately
the appropriate modification of Theorem \ref<<1125>>local<<1125>> (with strong
Beilinson-Bloch conjecture) does not necessarily provide us with
nontrivial $\qq$-torsion in $\Sha$. It only tells us that the
$\qq$-torsion subgroup of $H^1_f(\QQ,A_<<1126>>\qq<<1126>>(2))$ has
$\FF_<<1127>>\qq<<1127>>$-rank at least $1$. It could all be in the image of
$H^1_f(\QQ,V_<<1128>>\qq<<1128>>(2))$. $\Sha$ appears in the conjectural formula
for the first derivative of the complex $L$ function, evaluated at
$s=k/2$, but in combination with a regulator that we have no way
of calculating.
Let $L_q(f,s)$ and $L_q(g,s)$ be the $q$-adic $L$ functions
associated with $f$ and $g$ by the construction of Mazur, Tate and
Teitelbaum \cite<<1129>>MTT<<1129>>, each divided by a suitable canonical
period. We may show that $\qq\mid L_q'(f,k/2)$, though it is not
quite clear what to make of this. This divisibility may be proved
as follows. The measures $d\mu_<<1130>>f,\alpha<<1130>>$ and (a $q$-adic unit
times) $d\mu_<<1131>>g,\alpha'<<1131>>$ in \cite<<1132>>MTT<<1132>> (again, suitably
normalised) are congruent $\bmod<<1133>>\,\qq<<1133>>$, as a result of the
congruence between the modular symbols out of which they are
constructed. Integrating an appropriate function against these
measures, we find that $L_q'(f,k/2)$ is congruent $\bmod<<1134>>\,\qq<<1134>>$
to $L_q'(g,k/2)$. It remains to observe that $L_q'(g,k/2)=0$,
since $L(g,k/2)=0$ forces $L_q(g,k/2)=0$, but we are in a case
where the signs in the functional equations of $L(g,s)$ and
$L_q(g,s)$ are the same, positive in this instance. (According to
the proposition in Section 18 of \cite<<1135>>MTT<<1135>>, the signs differ
precisely when $L_q(g,s)$ has a ``trivial zero'' at $s=k/2$.)
We also found some examples for which the conditions of
Theorem~\ref<<1136>>local<<1136>> were not met. For example, we have a
$7$-congruence between \nf<<1137>>639k4B<<1137>> and \nf<<1138>>639k4H<<1138>>, but
$w_<<1139>>71<<1139>>=-1$, so that $71\equiv -w_<<1140>>71<<1140>>\pmod<<1141>>7<<1141>>$. There is a
similar problem with a $7$-congruence between \nf<<1142>>260k6A<<1142>> and
\nf<<1143>>260k6E<<1143>> --- here $w_<<1144>>13<<1144>>=1$ so that $13\equiv
-w_<<1145>>13<<1145>>\pmod<<1146>>7<<1146>>$. According to Propositions \ref<<1147>>div<<1147>> and
\ref<<1148>>sha<<1148>>, Bloch-Kato still predicts that the $\qq$-part of $\Sha$
is non-trivial in these examples. Finally, there is a
$5$-congruence between \nf<<1149>>116k6A<<1149>> and \nf<<1150>>116k6D<<1150>>, but here the
prime~$5$ is less than the weight~$6$ so Propositions \ref<<1151>>div<<1151>>
and \ref<<1152>>sha<<1152>> do not even apply.
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\end<<1243>>document<<1243>>
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@@
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*** using "q" as the argument instead; is this correct? ***
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@@@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})\rightarrow \Aut(V_{\lambda}),\end{displaymath}
*** using "Q" as the argument instead; is this correct? ***
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*** using "Q" as the argument instead; is this correct? ***
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*** using "Q" as the argument instead; is this correct? ***
@@
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
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*** using "Q" as the argument instead; is this correct? ***
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*** using "Q" as the argument instead; is this correct? ***
@
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*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
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*** no brace for \mathbb , before:
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*** no brace for \mathbb , before:
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@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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@@
*** no brace for \mathbb , before:
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*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
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@
*** no brace for \mathbb , before:
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*** using "Q" as the argument instead; is this correct? ***
@@@@@
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*** no brace for \mathbb , before:
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@@@@@
*** no brace for \mathbb , before:
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@@@@
*** no brace for \mathbb , before:
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@
*** no brace for \mathbb , before:
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*** no brace for \mathbb , before:
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@
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*** no brace for \mathfrak , before:
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@@@
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*** using "a" as the argument instead; is this correct? ***
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@@
*** no brace for \mathbb , before:
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*** using "C" as the argument instead; is this correct? ***
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*** using "C" as the argument instead; is this correct? ***
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*** using "Q" as the argument instead; is this correct? ***
@
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Q{_p,A_{\lambda}(j))/H^0\left({\mathbb Q}_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p, V_{\lambda}(j)^{I_p}/T_{\lambda}(j)^{I_p}\right)\right).
*** using "Q" as the argument instead; is this correct? ***
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a{^{\pm}Nk!$.\end{theorem_type}
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@
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@
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*** using "a" as the argument instead; is this correct? ***
@
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Q{_p,H^1(I_p,T_{\lambda}(j))_{\tors}),$
*** using "Q" as the argument instead; is this correct? ***
@@
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*** using "q" as the argument instead; is this correct? ***
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*** no brace for \mathbb , before:
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@
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@
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*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
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@
*** no brace for \mathfrak , before:
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*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
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*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
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*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
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@
*** no brace for \mathfrak , before:
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*** no brace for \mathfrak , before:
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@
*** no brace for \mathfrak , before:
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*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](j))\geq \dim_{E_{{\mathfrak q}}}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{{
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{ H^0(I_p,A[{\mathfrak q}](j))=\dim_{E_{{\mathfrak q}}} H^0(I_p,V_{{\mathfrak q}}(j)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](j))=\dim_{E_{{\mathfrak q}}} H^0(I_p,V_{{\mathfrak q}}(j)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{{ H^0(I_p,V_{{\mathfrak q}}(j)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))=
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)^{I_p}/T_{{\mathfrak q}}(j)^{I_p}$, hence that
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)^{I_p}$, hence that
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(j))=H^0({\mathbb Q}_p,V_{{\mathfrak q}}(j)^{I_p}/T_{{\mathfrak q}}(j)^{I_p})$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))=H^0({\mathbb Q}_p,V_{{\mathfrak q}}(j)^{I_p}/T_{{\mathfrak q}}(j)^{I_p})$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,V_{{\mathfrak q}}(j)^{I_p}/T_{{\mathfrak q}}(j)^{I_p})$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)^{I_p}/T_{{\mathfrak q}}(j)^{I_p})$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j)^{I_p})$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ to a newform of level dividing $N/p$, Proposition 2.2
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](j)$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ \cite{Ca1}.) But then Theorem 1 of \cite{JL} (which uses
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$. This contradicts our hypotheses.
*** using "q" as the argument instead; is this correct? ***
@@@
*** no brace for \mathfrak , before:
q{\mid q$\ is a prime of~$E$\ such that $q\nmid Nk!$, then
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(c_q)=0$.\end{theorem_type}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ is the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{[\mathrm {Gal}(\overline{\mathbb Q}_q/{\mathbb Q}_q)]$-module associated to the filtered
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q/{\mathbb Q}_q)]$-module associated to the filtered
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q)]$-module associated to the filtered
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ by the functor they call
*** using "q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$,
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$,
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{{)=0.\end{displaymath}\end{theorem_type}
*** using "Q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{]$\ is an irreducible representation of $\mathrm {Gal}(\overline{\mathbb Q}/{\mathbb Q})$,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$,
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$,
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{\mid q$. Assume the same hypotheses as in Lemma
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ are as in the previous section. If
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm}/\vol_{\infty}\neq 0$\ then the Bloch-Kato
*** using "a" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(\#{\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}})=\ord_{{\mathfrak q}}(L(f,k/2){\mathfrak a}^{\pm}/\vol_{\infty}).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(L(f,k/2){\mathfrak a}^{\pm}/\vol_{\infty}).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm}/\vol_{\infty}).
*** using "a" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{\mid q$\ is a prime of~$E$\ such that $f\equiv
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$, i.e. $a_n\equiv b_n\pmod{{\mathfrak q}}$\ for all $n$. Assume
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ for all $n$. Assume
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{]$\ is an irreducible representation of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$, and that $q\nmid N\phi(N)k!$. Choose
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$, and that $q\nmid N\phi(N)k!$. Choose
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{({\mathfrak a}^{\pm})=0$, i.e., $\delta_f^{\pm}$\ generates
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm})=0$, i.e., $\delta_f^{\pm}$\ generates
*** using "a" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$. Make two further assumptions:
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{{(L(f,k/2)/\vol_{\infty})>0$.\end{theorem_type}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{({\mathfrak a}^{\pm})=0$, we just need to show that
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm})=0$, we just need to show that
*** using "a" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(L(f,k/2)/((2\pi i)^{k/2}\Omega_{\pm}))>0$, where $\pm
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(r_{(k/2)-1}(f)/\Omega_{\pm})>0$.
*** using "q" as the argument instead; is this correct? ***
@@@
*** no brace for \mathbb , before:
Q{)$. For a ${\mathbb Z}$-algebra $R$\ and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{$-algebra $R$\ and
*** using "Z" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
C{)$, and $\delta_f^{\pm}$\ corresponds to an
*** using "C" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)$. We are
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ in $S_{\Gamma_1(N)}(k,O_{E,{\mathfrak q}})$. It suffices to show that,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)$. It suffices to show that,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-adic cohomology of $X_1(N)$\ with
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$. But this is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{]$).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$, because $a_N$\ and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ has residue
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$\ has odd residue characteristic and
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$\ be as in the first paragraph of the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ of an ``algebraic part'' of
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{]$\ and $A'[{\mathfrak q}]$, if
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{]$, if
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$-modules.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$-modules.
*** using "Q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{)$\ of ${\mathbb Q}$-rational rational equivalence
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{$-rational rational equivalence
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{$, the order of vanishing of $L(E,s)$\ at $s=1$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$.)
*** using "Q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$-adic Abel-Jacobi map, $\mathrm {CH}_0^{k/2}(M_g)({\mathbb Q})$\ maps
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$\ maps
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$, and its image is contained in the
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, and its image is contained in the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, by 3.1 and 3.2 of \cite{Ne2}.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-adic Abel-Jacobi map is injective, we
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$\ of dimension equal to the order of
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ of dimension equal to the order of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{,V_{{\mathfrak q}}(k/2))=0$, so that $H^1_f({\mathbb Q},A_{{\mathfrak q}}(k/2))$
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))=0$, so that $H^1_f({\mathbb Q},A_{{\mathfrak q}}(k/2))$
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-part of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$. Suppose that $A[{\mathfrak q}]$\ is an irreducible
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$. Suppose that $A[{\mathfrak q}]$\ is an irreducible
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{]$\ is an irreducible
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$\ and that for no prime $p\mid N$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$\ and that for no prime $p\mid N$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ to a newform of weight~$k$, trivial
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of $H^1_f({\mathbb Q},A_{{\mathfrak q}}(k/2))$\ has
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$\ has
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ has
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{{\mathfrak q}}$-rank at least $r$.\end{theorem_type}
*** using "F" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-rank at least $r$.\end{theorem_type}
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{,T'_{{\mathfrak q}}(k/2))$\ is~$r$. The natural map from
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is~$r$. The natural map from
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,T'_{{\mathfrak q}}(k/2))/{\mathfrak q}H^1_f({\mathbb Q},T'_{{\mathfrak q}}(k/2))$\ to
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))/{\mathfrak q}H^1_f({\mathbb Q},T'_{{\mathfrak q}}(k/2))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{H^1_f({\mathbb Q},T'_{{\mathfrak q}}(k/2))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,T'_{{\mathfrak q}}(k/2))$\ to
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A'[{\mathfrak q}](k/2))$\ is injective. Take a nonzero class $c$\ in
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is injective. Take a nonzero class $c$\ in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{{\mathfrak q}}$-rank $r$. Choose $d\in
*** using "F" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-rank $r$. Choose $d\in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,T'_{{\mathfrak q}}(k/2))$\ mapping to $c$. Consider the
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ mapping to $c$. Consider the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$-cohomology of the short exact sequence
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$-cohomology of the short exact sequence
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2)@>>>A'_{{\mathfrak q}}(k/2)@>\pi>>A'_{{\mathfrak q}}(k/2)@>>>0\end{CD},
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)@>\pi>>A'_{{\mathfrak q}}(k/2)@>>>0\end{CD},
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)@>>>0\end{CD},
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$. By irreducibility, $H^0({\mathbb Q},A[{\mathfrak q}](k/2))$\ is trivial.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A[{\mathfrak q}](k/2))$\ is trivial.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is trivial.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$\ is trivial, so
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is trivial, so
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A[{\mathfrak q}](k/2))$\ injects into $H^1({\mathbb Q},A_{{\mathfrak q}}(k/2))$, and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ injects into $H^1({\mathbb Q},A_{{\mathfrak q}}(k/2))$, and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$, and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion class $\gamma\in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(k/2))$, for all (finite) primes $p$. We
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, for all (finite) primes $p$. We
*** using "q" as the argument instead; is this correct? ***
@@@@@
*** no brace for \mathfrak , before:
q{{(k/2)$\ is unramified at $p$, $H^0(I_p,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))=A'_{{\mathfrak q}}(k/2)$, which is ${\mathfrak q}$-divisible. Therefore
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)$, which is ${\mathfrak q}$-divisible. Therefore
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-divisible. Therefore
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ (which, remember, is the same as
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$) injects into $H^1(I_p,A'_{{\mathfrak q}}(k/2))$. It
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$. It
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,T'_{{\mathfrak q}}(k/2))$\ that
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ that
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ of the restriction of $c$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))\simeq H^1(I_p,A[{\mathfrak q}](k/2))$\ is zero. Hence
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is zero. Hence
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is also
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(k/2))$\ is equal to (not just contained in)
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is equal to (not just contained in)
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(k/2))$\ to
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, so we have shown that ${\mathrm {res}}_p(\gamma)\in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(k/2))$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$.
*** using "q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is ${\mathfrak q}$-divisible.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-divisible.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))=\dim H^0(I_p,V'_{{\mathfrak q}}(k/2)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is surjective; this may be done as in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A[{\mathfrak q}](k/2))$\ in $H^1(I_p,A[{\mathfrak q}](k/2))$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ in $H^1(I_p,A[{\mathfrak q}](k/2))$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2)))$, by inflation-restriction. The
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A[{\mathfrak q}](k/2))$\ (this is Lemma 1 of \cite{W}), which we
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ (this is Lemma 1 of \cite{W}), which we
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)$, so $p\mid N$\ implies that
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)$\ is ramified at $p$, hence $\dim
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))=0$\ or $1$. As above, we see that $\dim
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))=\dim H^0(I_p,A[{\mathfrak q}](k/2))$, so we need only
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$, so we need only
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)$. It
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ (hence also on $H^0(I_p,A[{\mathfrak q}](k/2))$) as
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$) as
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A[{\mathfrak q}](k/2))$\ is trivial. Hence ${\mathrm {res}}_p(c)=0$\ so
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is trivial. Hence ${\mathrm {res}}_p(c)=0$\ so
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A_{{\mathfrak q}}(k/2))$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$.
*** using "q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathfrak , before:
q{{$\ is a crystalline representation of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q/{\mathbb Q}_q)$, meaning $D_{\cris}(V'_{{\mathfrak q}})$\ and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q)$, meaning $D_{\cris}(V'_{{\mathfrak q}})$\ and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)$\ and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ have the same dimension, where
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{):=H^0({\mathbb Q}_q,V'_{{\mathfrak q}}\otimes_{{\mathbb Q}_q}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V'_{{\mathfrak q}}\otimes_{{\mathbb Q}_q}
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{\otimes_{{\mathbb Q}_q}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q{
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{[\mathrm {Gal}(\overline{\mathbb Q}_q/{\mathbb Q}_q)]$-module associated to the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q/{\mathbb Q}_q)]$-module associated to the
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q)]$-module associated to the
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$. Since also $q>k$, we may
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,A_{{\mathfrak q}}(k/2))$. For the
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$. For the
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Z{_q$. $h^i(D)=0$\ for all $i\geq2$\ and
*** using "Z" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{{$\ and $D'=T'_{\dR}\otimes
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$. By Lemma 4.5 (c) of \cite{BK},
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}}),
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{),
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}})=\ker(H^1({\mathbb Q}_q,T_{{\mathfrak q}})\rightarrow
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)=\ker(H^1({\mathbb Q}_q,T_{{\mathfrak q}})\rightarrow
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}})\rightarrow
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)\rightarrow
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}})/H^1_e({\mathbb Q}_q,V_{{\mathfrak q}}))
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)/H^1_e({\mathbb Q}_q,V_{{\mathfrak q}}))
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}}))
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{))
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}})=\ker(H^1({\mathbb Q}_q,V_{{\mathfrak q}})\rightarrow
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)=\ker(H^1({\mathbb Q}_q,V_{{\mathfrak q}})\rightarrow
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}})\rightarrow
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)\rightarrow
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,B_{\cris}^{f=1}\otimes_{{\mathbb Q}_q} V_{{\mathfrak q}})).
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q{ V_{{\mathfrak q}})).
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{)).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}).$\ When applying results of
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{).$\ When applying results of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ etc. simply as ${\mathbb Z}_q$-modules,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{_q$-modules,
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-structure.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}}(j))
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}}(j))/H^1_e({\mathbb Q}_q,V_{{\mathfrak q}}(j))\simeq
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))/H^1_e({\mathbb Q}_q,V_{{\mathfrak q}}(j))\simeq
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}}(j))\simeq
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))\simeq
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{_q{{\mathbb Q}_q)/(1-f)(D(j)\otimes_{{\mathbb Z}_q}{\mathbb Q}_q),
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q)/(1-f)(D(j)\otimes_{{\mathbb Z}_q}{\mathbb Q}_q),
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{_q{{\mathbb Q}_q),
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q),
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}}(j))=H^1_f({\mathbb Q}_q,V_{{\mathfrak q}}(j))$, since $j\neq
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))=H^1_f({\mathbb Q}_q,V_{{\mathfrak q}}(j))$, since $j\neq
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V_{{\mathfrak q}}(j))$, since $j\neq
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))$, since $j\neq
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V'_{{\mathfrak q}}(j))=H^1_f({\mathbb Q}_q,V'_{{\mathfrak q}}(j))$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))=H^1_f({\mathbb Q}_q,V'_{{\mathfrak q}}(j))$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,V'_{{\mathfrak q}}(j))$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(j))$.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}}(k/2))\quad\text{and}\quad
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))\quad\text{and}\quad
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2)).\end{displaymath}
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2)).\end{displaymath}
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D'(k/2))\\
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2))@>\pi
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))@>\pi
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2))@>>>H^1({\mathbb Q}_q,A'[{\mathfrak q}](k/2)).
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))@>>>H^1({\mathbb Q}_q,A'[{\mathfrak q}](k/2)).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,A'[{\mathfrak q}](k/2)).
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2)).
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2))$\ is exactly
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is exactly
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2))$. The top right horizontal map is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$. The top right horizontal map is
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{_q,A'[{\mathfrak q}](k/2))$\ is in the image
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is in the image
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T'_{{\mathfrak q}}(k/2))$, by construction, and therefore is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, by construction, and therefore is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D'(k/2))$. By the fullness and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D'(k/2)$\ is isomorphic to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D(k/2)$.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{_q,A[{\mathfrak q}](k/2))$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D(k/2))$\ by the vertical map in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{D(k/2))$\ is surjective,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,T_{{\mathfrak q}}(k/2))$. From
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$. From
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_q,A_{{\mathfrak q}}(k/2))$, as desired.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$, as desired.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$\ with cusp forms of lower level is
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$-part of $c_p$\ for $p\mid N$;
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$, we could have made these problems cancel out, as in Lemma
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{]$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-part of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$\ does not depend on the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$.
*** using "q" as the argument instead; is this correct? ***
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
*** no brace for \mathfrak , before:
q{\mid q$\ with $f\equiv g\pmod{{\mathfrak q}}$. In each case,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$. In each case,
*** using "q" as the argument instead; is this correct? ***
@@@
*** no brace for \mathbb , before:
Q{(\ldots, a_n,
*** using "Q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathbb , before:
Q{(\ldots, a_n, \ldots)$. The third and fourth columns
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{\mid q$\ with $f\equiv g\pmod{{\mathfrak q}}$, and such that the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$, and such that the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$.
*** using "q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathbb , before:
Z{)/(W+W^{\perp})$, defined
*** using "Z" as the argument instead; is this correct? ***
@@@
*** no brace for \mathfrak , before:
q{$\ of~$K$\ of residue
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$\ reductions of~$f$\ and~$g$\ are
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{43}[[q]].\end{displaymath}
*** using "F" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{{\approx\rho_{g,{\mathfrak q}}$\ is irreducible. Since $127$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ is irreducible. Since $127$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{))=0$,~$\overline{f}$\ does not arise from a
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V'_{{\mathfrak q}}(k/2))$\ then the ${\mathfrak q}$-torsion subgroup of
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ then the ${\mathfrak q}$-torsion subgroup of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$\ has ${\mathbb F}_{{\mathfrak q}}$-rank at least $r$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ has ${\mathbb F}_{{\mathfrak q}}$-rank at least $r$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{{\mathfrak q}}$-rank at least $r$.
*** using "F" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-rank at least $r$.
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of $H^1_f({\mathbb Q},A_{{\mathfrak q}}(k/2))$\ is equal to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(k/2))$\ is equal to
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$\ is equal to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$. Admitting these assumptions,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$\ predicted by the
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{$.
*** using "Q" as the argument instead; is this correct? ***
@@@
*** no brace for \mathfrak , before:
q{{=\rho_{g,{\mathfrak q}}$\ is irreducible and does
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ is irreducible and does
*** using "q" as the argument instead; is this correct? ***
@@@
*** no brace for \mathfrak , before:
q{{$\ is
*** using "q" as the argument instead; is this correct? ***
@@@@
*** no brace for \mathbb , before:
Q{,s) = \prod_{i=1}^{d}
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{/{\mathbb Q})$-conjugates of~$f$. Let~$T$\ be the complex torus
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{)$-conjugates of~$f$. Let~$T$\ be the complex torus
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
C{^d/(2\pi i)^{k/2}\mathcal{L}$, where the lattice $\mathcal{L}$
*** using "C" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{{$\ denote the volume of the $(-1)^{(k/2)-1}$
*** using "Q" as the argument instead; is this correct? ***
@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
*** no brace for \mathbb , before:
Q{,k/2){{\Omega_{M_f/{\mathbb Q}}}\qquad\text{and}\qquad
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{{{\qquad\text{and}\qquad
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm}\right)
*** using "a" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Z{$-module of all
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{$
*** using "Q" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{{/(2\pi i)^{((k/2)-1)d}$\ is then
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm}$,
*** using "a" as the argument instead; is this correct? ***
@@@
*** no brace for \mathbb , before:
Q{,k/2){{\Omega_{M_f/{\mathbb Q}}}$\ so the lemma
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{{{$\ so the lemma
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
a{^{\pm}\right)$.\end{theorem_type}
*** using "a" as the argument instead; is this correct? ***
@@
*** no brace for \mathbb , before:
Q{,k/2)/\Omega_{M_f/{\mathbb Q}}$.
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{{$.
*** using "Q" as the argument instead; is this correct? ***
@@@@@@@@@
*** no brace for \mathbb , before:
Z{)$\ and
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{)$. The fourth column contains the odd
*** using "Z" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Z{)/(W+W^{\perp}))$, which
*** using "Z" as the argument instead; is this correct? ***
@@@@@
*** no brace for \mathfrak , before:
q{{(c_{3}(2))>0$. In fact this does happen. Because
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$\ (attached to~$g$\ of level $13$) is unramified at $p=3$,
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{])$\ (the same as $H^0(I_p,A'[{\mathfrak q}])$) is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{])$) is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ by the image of $H^0(I_p,V_{{\mathfrak q}}(k/2))$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(k/2))$.
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(c_p(k/2))>0$\ when $w_p=-1$, which is the case in
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{_p,A[{\mathfrak q}](k/2))$\ is
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{](k/2))$\ is
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(L(f,k/2)/\vol_{\infty})\geq 3$, which computation
*** using "q" as the argument instead; is this correct? ***
@@@@
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q{](3))$\ to
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(3))$\ is not necessarily injective, but its
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of $H^1_f({\mathbb Q},A_{{\mathfrak q}}(2))$\ having
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,A_{{\mathfrak q}}(2))$\ having
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(2))$\ having
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{{\mathfrak q}}$-rank at least~$1$\ (assuming $r\geq 2$), and thus get
*** using "F" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-rank at least~$1$\ (assuming $r\geq 2$), and thus get
*** using "q" as the argument instead; is this correct? ***
@@@@@@@@
*** no brace for \mathbb , before:
Q{,V_{{\mathfrak q}}(2)))=1$. If we assume this is so, and
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(2)))=1$. If we assume this is so, and
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$. It only tells us that the
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{$-torsion subgroup of $H^1_f({\mathbb Q},A_{{\mathfrak q}}(2))$\ has
*** using "q" as the argument instead; is this correct? ***
@
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Q{,A_{{\mathfrak q}}(2))$\ has
*** using "Q" as the argument instead; is this correct? ***
@
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q{{(2))$\ has
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
F{_{{\mathfrak q}}$-rank at least $1$. It could all be in the image of
*** using "F" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$-rank at least $1$. It could all be in the image of
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathbb , before:
Q{,V_{{\mathfrak q}}(2))$. ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$\ appears in the conjectural formula
*** using "Q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{(2))$. ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$\ appears in the conjectural formula
*** using "q" as the argument instead; is this correct? ***
@@
*** no brace for \mathfrak , before:
q{\mid L_q'(f,k/2)$, though it is not
*** using "q" as the argument instead; is this correct? ***
@
*** no brace for \mathfrak , before:
q{{$, as a result of the
*** using "q" as the argument instead; is this correct? ***
@
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q{{$
*** using "q" as the argument instead; is this correct? ***
@@@@@@
*** no brace for \mathfrak , before:
q{$-part of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}$
*** using "q" as the argument instead; is this correct? ***
@@@@@@@@@@@@@@@@@@@@@@@@@@@
*** no brace for \mathbb , before:
Q{)$, {\em Israel J. Math. }{\bf 90 }(1995), 1--66.
*** using "Q" as the argument instead; is this correct? ***
@@@@@@@@@@@@@@@@@@@@@@@@@@@
Reading aux file: /home/was/papers/motive_visibility/dsw_10.aux ...
Processing macros ...++............................................................................................,..........,...............,....,.....................................................
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Translating ...
0/13:top of dsw_10: for dsw_10.html
*** translating preamble ***
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1/13:section:.."Introduction" for node1.html
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2/13:section:.."Motives and Galois representations" for node2.html
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;,.,,,,,,,,,.,,,,,,,,,,.,,,,,,,.,,,,....,,,,..,.,,,,,,,,,,,..........,....,,,,...,.,,,,,,,,,......,....,,,,,,,,,,,,,,.....,.,,,,,,,,,,,,,,,,,,,,,,,,...............,....,...,....,,,,..,.,,,,,,..........,....,,,.,..,....,,,,,,,,,,,.........;...............
5/13:section:.."Congruences of special values" for node5.html
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6/13:section:.."Constructing elements of the Shafarevich-Tate group" for node6.html
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7/13:section:..."Examples and Experiments" for node7.html
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8/13:subsection:.,....."Visible Table~1" for node8.html
;,.,,.,,,,,,,,.,.,,,.,..,,,,,,,,,...,,,,..,,,,,,,,,,,,,,,,,,,.,,,,,,,,,,,,,,,,,.,,,,,,,,,.,,.;...........................................
9/13:subsection:...."How the computation was performed" for node9.html
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10/13:subsection:.,..."Conjecturally nontrivial " for node10.html
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Cannot find matching bracket for 1322,.,.,.,.,....,,,,,,......,.,,,,,,,,,,,,.,,,,,,,,,,,,,,,,,,,,,,......................,....,.,,,,.....,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,.,,,,.,,,,,,,,,,,,,..,,,.,,...,,,,,,,,,,,,.;..........................................................................
11/13:subsection:...."Examples in which hypotheses fail" for node11.html
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12/13:bibliography:.."Bibliography" for node12.html
;,,.....,..,...,....,,.,......,....,.,....,..,...,..,..,..,..,,....,,....,.................................................................;
13/13:sectionstar:.."About this document ..." for node13.html
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