CoCalc Shared Fileswww / papers / motive_visibility / comment_5.txt
Author: William A. Stein
1Here are some of the stylistic changes I made:
2
31. I'm tempted to change: "It is consistent with the fact \cite{Fl2} that
4this order is necessarily a perfect square."
5
6to
7
8"Our data is consistent with the fact \cite{Fl2} that the part of
9$\#\Sha$ coprime to the congruence modulus is necessarily a perfect
10square (assuming that $\Sha$ is finite)."
11
12I'm worried that what you wrote is false, since Sha can be nonsquare
13when k=2 (I have examples).  However, I haven't read Flach's paper,
14and I'm not sure if polarizations play a roll when working with
16
172. I changed deRham to "de Rham" throughout.
18
19
203. I changed "It is easy to see that we may choose the
21$\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
22$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
23$T_B^{\pm}$ locally at $\qq$. Let us suppose that such a choice
25to
26"Choose $\delta_f^{\pm}\in T_B^{\pm}$ in such a way that
27$\ord_{\qq}(\aaa^{\pm})=0$, i.e., $\delta_f^{\pm}$ generates
28$T_B^{\pm}$ locally at $\qq$."
29
304. I changed "We shall now make two further assumptions:" to
31   "Make two further assumptions:".
32
335. I changed
34"\begin{enumerate}
35\item $L(f,k/2)\neq 0$;
36\item $L(g,k/2)=0$.
37\end{enumerate}
38"
39$$L(f,k/2)\neq 0 \qquad\text{and}\qquad L(g,k/2)=0.$$
40
416. I changed
42"It is worth pointing out that there are no
43examples of $g$ of level one, and positive sign in the functional
44equation, such that $L(g,k/2)=0$, unless Maeda's conjecture (that
45all the normalised cuspidal eigenforms of weight~$k$ and level one
46are Galois conjugate) is false. See \cite{CF}."
47to
48"Note that there are no examples of~$g$ of level one with positive
49sign in their functional equation such that $L(g,k/2)=0$, unless
50Maeda's conjecture, which asserts that all the newforms of weight~$k$
51and level one are Galois conjugate, is false (see \cite{CF})."
52
537. Remark 5.3 is wrong.   The remark, as given, is:
54"\begin{remar}\label{sign}
55The signs in the functional equations of $L(f,s)$ and $L(g,s)$
56are equal, since they are determined by the action of the
57involution $W_N$ on the common subspace generated by the
58reductions $\pmod{\qq}$ of $\delta_f^{\pm}$ and $\delta_g^{\pm}$.
59Specifically, the sign is $(-1)^{k/2}w_N$, where $w_N$ is the
60common eigenvalue of $W_N$ acting on~$f$ and~$g$.
61\end{remar}"
62The justification is wrong, because if we allow $\q$ to have residue
63characteristic dividing the level, then it is easy to construct
64counterexamples to what the remark seems to prove.  Here's one:
65Up to conjugacy, there are two newforms f and g in S_6(Gamma_0(11)).
66We have f = g (mod 11), W_11(f) = f and W_11(g) = -g.  Here
67a_11(f) = -11^2 and a_11(g) = 11^2.   The congruence of Fourier coefficients
68modulo 11 does *not* imply that the coefficients of 11^2 in the 11-th Fourier
69coefficient of each form are the same. But, modulo anything other than 2 or
7011, they do.  I suggest that we replace the remark by:
71"\begin{remar}\label{sign}
72The signs in the functional equations of $L(f,s)$ and $L(g,s)$ are
73equal. They are determined by the eigenvalue of the involution $W_N$,
74which is determined by $a_N$ and $b_N$ modulo~$\qq$, because $a_N$ and
75$b_N$ are each $N^{k/2-1}$ times this sign and $\qq$ has residue
76characteristic coprime to $2N$.  The common sign in the functional
77equation is $(-1)^{k/2}w_N$, where $w_N$ is the common eigenvalue of
78$W_N$ acting on~$f$ and~$g$.
79\end{remar}
80"
81I also made some changes to the paragraph after the remark.
82
83
84
85
868: I made variations on all of Mark's suggested changes up till
87
88> 'Section 6, paragraph 2: "According to the Beilinson-Bloch conjecture...
89> the rank of the group CH of Q-rational rational equivalence classes of
90> null-homologous codimension k/2 algebraic cycles on the motive M_g"
91> Is the phrase "Q-rational rational" redundant? '
92
93Rational isn't redundant.  The two rationals have different meanings.
94Very vaguely, I think rational equivalence involves deforming one
95cycle to another using a rational variety; another option is numerical
96equivalence, which involves the intersection pairing.
97
989:
99Mark says:
100> Table/Figure 2: Does not 151^2 also divide L(59k10B,5)?
101> If this is excluded because 151 is 1 mod k, then why doesn't a similar
102> reason exclude 191^2 from 67k10B? I think I am asking about congruences
103> to the level 1 Eisenstein series, and when they occur (as we had never
104> tested for them in the zeroth redaction).
105
106There is a congruence of residue characteristic 151 between 59k10B and
107an Eisenstein series of level 59, so there is a reducible mod 151
108representation attached to 59k10B, and 151 is eliminated in step 5.
109Here's how to see the congruence in MAGMA.
110
111> M := ModularForms(59,10);
112> S := CuspidalSubspace(M);
113> time N := Newforms(S);
114Time: 10.260
115> B := N[2][2];
116> time G := CongruenceGroupAnemic(Parent(B),ES(M));
117> factor(#G);
118[ <29, 1>, <41, 1>, <151, 1>, <181, 1> ]
119
120> I think I am asking about congruences to the level 1 Eisenstein
121> series, and when they occur (as we had never tested for them in the
122> zeroth redaction).
123
125
126In Table 1, when we have two forms f and g, then we know exactly which
127prime of a given residue characteristic gives rise to the congruence,
128so we can check if it is Eisenstein.  This is better than just ruling
129out certain characteristics.  We haven't systematically checked this
130for the forms in Table 1 yet.  Here's a program that does it, which I'll
131run right now:
132
133procedure Test(form, qs)
134   print "Testing ",form;
135   A := Parent(Newforms(form)[1]);
136   E := EisensteinSubspace(AmbientSpace(Parent(f)));
137   g := #CongruenceGroupAnemic(A,E);
138   print "Eisenstein Cong Number = ", g;
139   print "Bad number = ", GCD(g, qs);
140end procedure;
141Test("127k4A",43);
142Test("159k4B",5*23);
143Test("365k4A",29);
144Test("369k4B",13);
145Test("453k4A",17);
146Test("465k4B",11);
147Test("477k4B",73);
148Test("567k4B",23);
149Test("581k4A",19);
150Test("657k4A",5);
151Test("681k4A",59);
152Test("684k4C",7);
153Test("95k6A",31*59);
154Test("122k6A",73);
155Test("260k6A",17);
156
157
158I just ran this and it proved that there are no Eisenstein congruences
159in Table 1.  Hmmm.  When I ran it, MAGMA used ridiculous amounts of
160memory in creating some of the bigger modular forms spaces.  Upon
161further investigation, I found that the sub<> constructor for vector
162spaces in MAGMA is now messed up, in that it uses way too much memory
163if the input is not in echelon form.  I wrote a new version of the
164function Quotient() in ModSym/core.m that works around this problem.
165Mark, if you're having the same problem get core.m from
166meccah:/usr/local/Magma2.9/package/Geomtry/ModSym/.
167
168
16910. I'm pushing the weight 8 examples up to level 149, to include
170the impressively large examples that Mark pointed out.
171
17211. "Incidentally, the question of higher powers dividing B never
173appeared in your tables: further data show ..."
174