Here are some of the stylistic changes I made:121. I'm tempted to change: "It is consistent with the fact \cite{Fl2} that3this order is necessarily a perfect square."45to67"Our data is consistent with the fact \cite{Fl2} that the part of8$\#\Sha$ coprime to the congruence modulus is necessarily a perfect9square (assuming that $\Sha$ is finite)."1011I'm worried that what you wrote is false, since Sha can be nonsquare12when k=2 (I have examples). However, I haven't read Flach's paper,13and I'm not sure if polarizations play a roll when working with14motives instead of abelian varieties.15162. I changed deRham to "de Rham" throughout.1718193. I changed "It is easy to see that we may choose the20$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that21$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates22$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice23has been made."24to25"Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that26$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates27$T_B^{\pm}$ locally at $\qq$."28294. I changed "We shall now make two further assumptions:" to30"Make two further assumptions:".31325. I changed33"\begin{enumerate}34\item $L(f,k/2)\neq 0$;35\item $L(g,k/2)=0$.36\end{enumerate}37"38$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$39406. I changed41"It is worth pointing out that there are no42examples of $g$ of level one, and positive sign in the functional43equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that44all the normalised cuspidal eigenforms of weight~$k$ and level one45are Galois conjugate) is false. See \cite{CF}."46to47"Note that there are no examples of~$g$ of level one with positive48sign in their functional equation such that $L(g,k/2)=0$, unless49Maeda's conjecture, which asserts that all the newforms of weight~$k$50and level one are Galois conjugate, is false (see \cite{CF})."51527. Remark 5.3 is wrong. The remark, as given, is:53"\begin{remar}\label{sign}54The signs in the functional equations of $L(f,s)$ and $L(g,s)$55are equal, since they are determined by the action of the56involution $W_N$ on the common subspace generated by the57reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.58Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the59common eigenvalue of $W_N$ acting on~$f$ and~$g$.60\end{remar}"61The justification is wrong, because if we allow $\q$ to have residue62characteristic dividing the level, then it is easy to construct63counterexamples to what the remark seems to prove. Here's one:64Up to conjugacy, there are two newforms f and g in S_6(Gamma_0(11)).65We have f = g (mod 11), W_11(f) = f and W_11(g) = -g. Here66a_11(f) = -11^2 and a_11(g) = 11^2. The congruence of Fourier coefficients67modulo 11 does *not* imply that the coefficients of 11^2 in the 11-th Fourier68coefficient of each form are the same. But, modulo anything other than 2 or6911, they do. I suggest that we replace the remark by:70"\begin{remar}\label{sign}71The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are72equal. They are determined by the eigenvalue of the involution $W_N$,73which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and74$b_N$ are each $N^{k/2-1}$ times this sign and $\qq$ has residue75characteristic coprime to $2N$. The common sign in the functional76equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of77$W_N$ acting on~$f$ and~$g$.78\end{remar}79"80I also made some changes to the paragraph after the remark.81828384858: I made variations on all of Mark's suggested changes up till8687> 'Section 6, paragraph 2: "According to the Beilinson-Bloch conjecture...88> the rank of the group CH of Q-rational rational equivalence classes of89> null-homologous codimension k/2 algebraic cycles on the motive M_g"90> Is the phrase "Q-rational rational" redundant? '9192Rational isn't redundant. The two rationals have different meanings.93Very vaguely, I think rational equivalence involves deforming one94cycle to another using a rational variety; another option is numerical95equivalence, which involves the intersection pairing.96979:98Mark says:99> Table/Figure 2: Does not 151^2 also divide L(59k10B,5)?100> If this is excluded because 151 is 1 mod k, then why doesn't a similar101> reason exclude 191^2 from 67k10B? I think I am asking about congruences102> to the level 1 Eisenstein series, and when they occur (as we had never103> tested for them in the zeroth redaction).104105There is a congruence of residue characteristic 151 between 59k10B and106an Eisenstein series of level 59, so there is a reducible mod 151107representation attached to 59k10B, and 151 is eliminated in step 5.108Here's how to see the congruence in MAGMA.109110> M := ModularForms(59,10);111> S := CuspidalSubspace(M);112> time N := Newforms(S);113Time: 10.260114> B := N[2][2];115> time G := CongruenceGroupAnemic(Parent(B),ES(M));116> factor(#G);117[ <29, 1>, <41, 1>, <151, 1>, <181, 1> ]118119> I think I am asking about congruences to the level 1 Eisenstein120> series, and when they occur (as we had never tested for them in the121> zeroth redaction).122123Essentially that's what you're asking.124125In Table 1, when we have two forms f and g, then we know exactly which126prime of a given residue characteristic gives rise to the congruence,127so we can check if it is Eisenstein. This is better than just ruling128out certain characteristics. We haven't systematically checked this129for the forms in Table 1 yet. Here's a program that does it, which I'll130run right now:131132procedure Test(form, qs)133print "Testing ",form;134A := Parent(Newforms(form)[1]);135E := EisensteinSubspace(AmbientSpace(Parent(f)));136g := #CongruenceGroupAnemic(A,E);137print "Eisenstein Cong Number = ", g;138print "Bad number = ", GCD(g, qs);139end procedure;140Test("127k4A",43);141Test("159k4B",5*23);142Test("365k4A",29);143Test("369k4B",13);144Test("453k4A",17);145Test("465k4B",11);146Test("477k4B",73);147Test("567k4B",23);148Test("581k4A",19);149Test("657k4A",5);150Test("681k4A",59);151Test("684k4C",7);152Test("95k6A",31*59);153Test("122k6A",73);154Test("260k6A",17);155156157I just ran this and it proved that there are no Eisenstein congruences158in Table 1. Hmmm. When I ran it, MAGMA used ridiculous amounts of159memory in creating some of the bigger modular forms spaces. Upon160further investigation, I found that the sub<> constructor for vector161spaces in MAGMA is now messed up, in that it uses way too much memory162if the input is not in echelon form. I wrote a new version of the163function Quotient() in ModSym/core.m that works around this problem.164Mark, if you're having the same problem get core.m from165meccah:/usr/local/Magma2.9/package/Geomtry/ModSym/.16616716810. I'm pushing the weight 8 examples up to level 149, to include169the impressively large examples that Mark pointed out.17017111. "Incidentally, the question of higher powers dividing B never172appeared in your tables: further data show ..."173174I'm not sure what I want to do about this. Mark, do you want to add a175section discussing this? It doesn't interest me, so I don't feeling176like writing such a section...17717812. Intro, para 7. I commented out the sentence "In proving the local179conditions at primes dividing the level, and also in examining the180local Tamagawa factors at these primes, we make use of a higher weight181level-lowering result due to Jordan and Livn\'e \cite{JL}." since it182seems very out of place in that paragraph.18318413. I modified the wording of the first sentence of (2) of the proof185of Theorem 6.1 so that an equation didn't stick out into the margine,186and so the wording is cleaner.187