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