CoCalc Public Fileswww / papers / merel2 / changes_v7
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1Dear Loic,
2
3I made a few minor changes to our paper.  If you agree to these,
4and Barry accepts the new version, then we are finally done.
5
6Best,
7William
8
90. I changed
10"images of $\rho$ and $\rho_0$ co\"\i ncide."
11to
12"images of $\rho$ and $\rho_0$ coincide."
13
14(For some reason you wrote "coincide" in a French way.)
15
161. I changed
17
18  "Therefore there exists a field extension $K_2$ of degree
19   $d_2$ dividing $2d_K$ of $\B Q(\sqrt p)$, such that ..."
20
21to
22
23  "Therefore, there exists an extension field $K_2$ of $\B Q(\sqrt p)$,
24   whose degree divides $2d_K$, such that ..."
25
262. I changed
27
28   "Let us define $E$ over $\B Q$ so that $E$ has complex
29    multiplication over $K$."
30
31to
32
33   "Since~$E$ is an elliptic curve over $\bar \B Q$ with complex
34    multiplication by a field of class number one, there is a model
35    for~$E$ that is defined over $\B Q$."
36
373.  I changed
38
39"In this section we explain how we used
40a computer to verify that the hypothesis of Proposition~1 are
41satisfied for $p=11$ and $13 < p < 1000$."
42
43to
44
45"In this section we explain how we used
46a computer to verify that the second hypothesis of Proposition~1 are
47satisfied for $p=11$ and $13 < p < 1000$.  (In the present paper,
48this verification is only required for $p$ that are congruent to~$1$
49modulo~$4$.)"
50
514. I deleted the word "be" from the following sentence:
52
53"Denote by $S_2(\Gamma_0(p);\B Z)$ be the set of modular forms "
54                                 ^^^^
55
565. I deleted "as" in the following:
57
58"if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} = \B C^d$ via"
59
606. I changed
61
62"over a relatively small prime finite field $\B F_\ell$ such
63that~$\ell$ is congruent to~$1$ modulo $p-1$."
64
65to
66
67"over a relatively small finite field $\B F_\ell$ where~$\ell$ is
68congruent to~$1$ modulo $p-1$."
69
707. I deleted the second "For each" in the following:
71
72"For each prime $p<1000$ different than $2,3,5,7, 13$, we
73verified the existence of an ideal that satisfies the three conditions
74given above, as follows.  For each~$p$, we consider each ..."
75
768. I deleted "Finally" from the acknowledgement:
77
78"{\it Acknowledgment} : Finally we would like to thank Barry Mazur for
80
81becomes
82
83"{\it Acknowledgment}: We would like to thank Barry Mazur for