Here are some of the stylistic changes I made: 1. I'm tempted to change: "It is consistent with the fact \cite{Fl2} that this order is necessarily a perfect square." to "Our data is consistent with the fact \cite{Fl2} that the part of $\#\Sha$ coprime to the congruence modulus is necessarily a perfect square (assuming that $\Sha$ is finite)." I'm worried that what you wrote is false, since Sha can be nonsquare when k=2 (I have examples). However, I haven't read Flach's paper, and I'm not sure if polarizations play a roll when working with motives instead of abelian varieties. 2. I changed deRham to "de Rham" throughout. 3. I changed "It is easy to see that we may choose the $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice has been made." to "Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that $\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates $T_B^{\pm}$ locally at $\qq$." 4. I changed "We shall now make two further assumptions:" to "Make two further assumptions:". 5. I changed "\begin{enumerate} \item $L(f,k/2)\neq 0$; \item $L(g,k/2)=0$. \end{enumerate} " $$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$ 6. I changed "It is worth pointing out that there are no examples of $g$ of level one, and positive sign in the functional equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that all the normalised cuspidal eigenforms of weight~$k$ and level one are Galois conjugate) is false. See \cite{CF}." to "Note that there are no examples of~$g$ of level one with positive sign in their functional equation such that $L(g,k/2)=0$, unless Maeda's conjecture, which asserts that all the newforms of weight~$k$ and level one are Galois conjugate, is false (see \cite{CF})." 7. Remark 5.3 is wrong. The remark, as given, is: "\begin{remar}\label{sign} The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are equal, since they are determined by the action of the involution $W_N$ on the common subspace generated by the reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$. Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of $W_N$ acting on~$f$ and~$g$. \end{remar}" The justification is wrong, because if we allow $\q$ to have residue characteristic dividing the level, then it is easy to construct counterexamples to what the remark seems to prove. Here's one: Up to conjugacy, there are two newforms f and g in S_6(Gamma_0(11)). We have f = g (mod 11), W_11(f) = f and W_11(g) = -g. Here a_11(f) = -11^2 and a_11(g) = 11^2. The congruence of Fourier coefficients modulo 11 does *not* imply that the coefficients of 11^2 in the 11-th Fourier coefficient of each form are the same. But, modulo anything other than 2 or 11, they do. I suggest that we replace the remark by: "\begin{remar}\label{sign} The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are equal. They are determined by the eigenvalue of the involution $W_N$, which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and $b_N$ are each $N^{k/2-1}$ times this sign and $\qq$ has residue characteristic coprime to $2N$. The common sign in the functional equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of $W_N$ acting on~$f$ and~$g$. \end{remar} " I also made some changes to the paragraph after the remark. 8: I made variations on all of Mark's suggested changes up till > 'Section 6, paragraph 2: "According to the Beilinson-Bloch conjecture... > the rank of the group CH of Q-rational rational equivalence classes of > null-homologous codimension k/2 algebraic cycles on the motive M_g" > Is the phrase "Q-rational rational" redundant? ' Rational isn't redundant. The two rationals have different meanings. Very vaguely, I think rational equivalence involves deforming one cycle to another using a rational variety; another option is numerical equivalence, which involves the intersection pairing. 9: Mark says: > Table/Figure 2: Does not 151^2 also divide L(59k10B,5)? > If this is excluded because 151 is 1 mod k, then why doesn't a similar > reason exclude 191^2 from 67k10B? I think I am asking about congruences > to the level 1 Eisenstein series, and when they occur (as we had never > tested for them in the zeroth redaction). There is a congruence of residue characteristic 151 between 59k10B and an Eisenstein series of level 59, so there is a reducible mod 151 representation attached to 59k10B, and 151 is eliminated in step 5. Here's how to see the congruence in MAGMA. > M := ModularForms(59,10); > S := CuspidalSubspace(M); > time N := Newforms(S); Time: 10.260 > B := N[2][2]; > time G := CongruenceGroupAnemic(Parent(B),ES(M)); > factor(#G); [ <29, 1>, <41, 1>, <151, 1>, <181, 1> ] > I think I am asking about congruences to the level 1 Eisenstein > series, and when they occur (as we had never tested for them in the > zeroth redaction). Essentially that's what you're asking. In Table 1, when we have two forms f and g, then we know exactly which prime of a given residue characteristic gives rise to the congruence, so we can check if it is Eisenstein. This is better than just ruling out certain characteristics. We haven't systematically checked this for the forms in Table 1 yet. Here's a program that does it, which I'll run right now: procedure Test(form, qs) print "Testing ",form; A := Parent(Newforms(form)[1]); E := EisensteinSubspace(AmbientSpace(Parent(f))); g := #CongruenceGroupAnemic(A,E); print "Eisenstein Cong Number = ", g; print "Bad number = ", GCD(g, qs); end procedure; Test("127k4A",43); Test("159k4B",5*23); Test("365k4A",29); Test("369k4B",13); Test("453k4A",17); Test("465k4B",11); Test("477k4B",73); Test("567k4B",23); Test("581k4A",19); Test("657k4A",5); Test("681k4A",59); Test("684k4C",7); Test("95k6A",31*59); Test("122k6A",73); Test("260k6A",17); I just ran this and it proved that there are no Eisenstein congruences in Table 1. Hmmm. When I ran it, MAGMA used ridiculous amounts of memory in creating some of the bigger modular forms spaces. Upon further investigation, I found that the sub<> constructor for vector spaces in MAGMA is now messed up, in that it uses way too much memory if the input is not in echelon form. I wrote a new version of the function Quotient() in ModSym/core.m that works around this problem. Mark, if you're having the same problem get core.m from meccah:/usr/local/Magma2.9/package/Geomtry/ModSym/. 10. I'm pushing the weight 8 examples up to level 149, to include the impressively large examples that Mark pointed out. 11. "Incidentally, the question of higher powers dividing B never appeared in your tables: further data show ..." I'm not sure what I want to do about this. Mark, do you want to add a section discussing this? It doesn't interest me, so I don't feeling like writing such a section... 12. Intro, para 7. I commented out the sentence "In proving the local conditions at primes dividing the level, and also in examining the local Tamagawa factors at these primes, we make use of a higher weight level-lowering result due to Jordan and Livn\'e \cite{JL}." since it seems very out of place in that paragraph. 13. I modified the wording of the first sentence of (2) of the proof of Theorem 6.1 so that an equation didn't stick out into the margine, and so the wording is cleaner.