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Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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Dear Loic,
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I made a few minor changes to our paper. If you agree to these,
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and Barry accepts the new version, then we are finally done.
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Best,
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William
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0. I changed
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"images of $\rho$ and $\rho_0$ co\"\i ncide."
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to
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"images of $\rho$ and $\rho_0$ coincide."
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(For some reason you wrote "coincide" in a French way.)
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1. I changed
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"Therefore there exists a field extension $K_2$ of degree
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$d_2$ dividing $2d_K$ of $\B Q(\sqrt p)$, such that ..."
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to
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"Therefore, there exists an extension field $K_2$ of $\B Q(\sqrt p)$,
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whose degree divides $2d_K$, such that ..."
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2. I changed
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"Let us define $E$ over $\B Q$ so that $E$ has complex
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multiplication over $K$."
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to
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"Since~$E$ is an elliptic curve over $\bar \B Q$ with complex
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multiplication by a field of class number one, there is a model
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for~$E$ that is defined over $\B Q$."
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3. I changed
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"In this section we explain how we used
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a computer to verify that the hypothesis of Proposition~1 are
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satisfied for $p=11$ and $13 < p < 1000$."
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to
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"In this section we explain how we used
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a computer to verify that the second hypothesis of Proposition~1 are
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satisfied for $p=11$ and $13 < p < 1000$. (In the present paper,
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this verification is only required for $p$ that are congruent to~$1$
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modulo~$4$.)"
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4. I deleted the word "be" from the following sentence:
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"Denote by $S_2(\Gamma_0(p);\B Z)$ be the set of modular forms "
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^^^^
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5. I deleted "as" in the following:
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"if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} = \B C^d$ via"
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6. I changed
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"over a relatively small prime finite field $\B F_\ell$ such
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that~$\ell$ is congruent to~$1$ modulo $p-1$."
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to
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"over a relatively small finite field $\B F_\ell$ where~$\ell$ is
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congruent to~$1$ modulo $p-1$."
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7. I deleted the second "For each" in the following:
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"For each prime $p<1000$ different than $2,3,5,7, 13$, we
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verified the existence of an ideal that satisfies the three conditions
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given above, as follows. For each~$p$, we consider each ..."
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8. I deleted "Finally" from the acknowledgement:
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"{\it Acknowledgment} : Finally we would like to thank Barry Mazur for
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providing us several useful comments."
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becomes
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"{\it Acknowledgment}: We would like to thank Barry Mazur for
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providing us several useful comments."
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Otherwise, it sounds like we thanked several people, and finally we are
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thanking Barry.
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-- William
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