Edits

[Ashwath Rabindranath]

Hi,

I made the edits I was assigned and sent a merge request to williamstein on git. Hope it works.

Regarding the second referee report, we should start dividing up the second set of comments:
 
Note: We should replace curve everywhere with elliptic curve.

* Page 5, line 2 - "conditions modulo $\mathfrak p$ on" - I assume you

  mean to say "on $E" here. (easy fix)

* Page 5, Prop. 3.2 - "$E$ a curve" -> "$E$ an elliptic curve" (OK, no

  other curves are ever mentioned, but it is still better to be

  precise). (easy fix)

* Page 6, line 3 - "quartic surface" - shouldn't this be a quartic

  _curve_ (a twist of the Klein quartic)? (no idea -- ? )

* Page 6, first line of 3.5 - insert "over" before "$F$". Replace "a

  curve" by "an elliptic curve" (you don't look at homogeneous

  spaces). (easy fix)

* Page 6, Prop. 3.3 - state explicitly that $d$ is assumed to be

  integral and squarefree.  (easy fix)

* Page 6, line 10 from below - in the set of possible $d_1$'s,

  $\sqrt{5}$ and $\varphi \sqrt{5}$ are missing (and indeed, one of

  your twists is by $-\varphi \sqrt{5}$!). (--?)

* Page 6, line 2 from below - this $E$ is different from the $E$'s

  three lines above, so the way this is written is a bit confusing. (--?)

* Page 9, line 6 from below - you did not define $\epsilon_E$ and

  there is no explanation why the condition $\chi(-N) = \epsilon_E$ is

  necessary. Should this be added to Conjecture 3.4? (--?)

* Page 9, line 5 from below - missing braces around $\pm 1$. (easy fix)

* Page 11, first line of Section 4 - I would say "representatives up

  to isomorphism for all...". (easy fix)

* Page 12, Rem. 4.1 - I think you need to add that this holds for

  isogenies to a non-isomorphic curve, since otherwise it would be

  wrong for CM curves with endomorphism algebra F. (yes, this is true)

* Page 12, end of second paragraph - "divide our curves up into

  isogeny classes" doesn't seem to reflect what you are doing, as you

  are really dividing each isogeny class up into isomorphism classes.
(yes, this is true, easy fix)

* Page 12, Section 5 - I think there was a discussion on the sage-nt

  mailing list that involved a mistake in [Cre92] (but I don't

  remember whether it was before or after the submission deadline), so

  please make sure the statement is correct. (any idea?)

* Page 13, second paragraph of Section 7 - #isom should refer to the

  "number of isomorphism classes of curves" and #isog to the "number

  of isogeny classes". (easy fix)

If people are okay with it, I can deal with all the easy fixes in one go.

Ashwath

[R. Andrew Ohana]

On Fri, May 4, 2012 at 2:39 PM, Ashwath Rabindranath
<ashwath.ra...@gmail.com> wrote:
> Hi,
>
> I made the edits I was assigned and sent a merge request to williamstein on
> git. Hope it works.

It did, please see my comments.

Go for it.
>
> Ashwath

I'm almost done with the pdf table (just need to add in the
factorizations of the conductors, and everything should be good), once
that is done, expect to see a pull request from me as well.

--
Andrew

[Ben LeVeque]

I just pushed some edits for the Fisher section and sent a pull request. Let me know how it sounds and I can edit further!

Thanks,

Ben

[Ashwath Rabindranath]

Hi all,

Just made some changes -- note that there were several sloppy uses of "curve" and "classes". I have replaced  them with the more appropriate "elliptic curve" and "isogeny (resp. isomorphism) classes" whenever appropriate. Also made the "easy fixes" pointed out in the previous email based off the second referee report.

I'm not sure if GIT accepted the update requests -- can someone confirm this? I'm still not entirely used to working with GIT :)

Best,
Ashwath

[Ashwath Rabindranath]

I have a saved version of this update, in the event that it failed.

A

[Ashwath Rabindranath]

Since I haven't heard back about the latest edit, I attach a version of the file that was saved three days ago.  We have a week to finish all the remaining edits, so people should remember to make their edits soon. The paper is due Friday, May 18th.

best,
Ashwath

[Ben LeVeque]

Great, thanks Ashwath. Are there edits from the second referee report that haven't been made yet? I'm not sure if we formally divy-ed those up, but I'd be happy to take a look at some of them.

William or Jon, did you happen to read through the Congruence Families section (3.4) I pushed a week or so ago? Are there any changes I should make to that? I want to make sure it's in good shape before I address other points.

Cheers,

Ben

[R. Andrew Ohana]

I made a branch with Ashwath's edits, and made a pull request for William (now I have two, most likely incompatible pull requests waiting on him :-/).

Ashwath, would you mind sending me a copy of referee report from github with an X preceding each fix that you made in your edits. That way I can compile a file full of the changes that have been made, and the ones that still need to be looked at.

--

Andrew

[Jonathan Bober]

I just made some minor edits and marked up the referee report with
notes on all of the changes that have been made. I'll make the rest of
the fixes I need to make later today.

On one comment of the referee's I'm not really sure what to do right
now. The referee says:

"S. 3.7: It would be helpful if you'd define least real/imaginary
periods. Are these just the smallest periods that lie on the
real/imaginary axes?"

To me, "least real period" means "smallest period which is real",
where for a complex number "is real" happens to be synonymous with
"lies on the real axis". So the answer to the referee's question is
"Yes, I was pretty sure that was exactly what I had already said." In
fact, there is some ambiguity there, because it should be _positive_
real/imaginary, but other than that I'm not really sure that what I
wrote is confusing or ambiguous. Is it?

[William Stein]

On Tue, May 15, 2012 at 7:33 PM, Jonathan Bober <jwb...@gmail.com> wrote:
> I just made some minor edits and marked up the referee report with
> notes on all of the changes that have been made. I'll make the rest of
> the fixes I need to make later today.
>
> On one comment of the referee's I'm not really sure what to do right
> now. The referee says:
>
> "S. 3.7: It would be helpful if you'd define least real/imaginary
> periods. Are these just the smallest periods that lie on the
> real/imaginary axes?"
>
> To me, "least real period" means "smallest period which is real",
> where for a complex number "is real" happens to be synonymous with
> "lies on the real axis". So the answer to the referee's question is
> "Yes, I was pretty sure that was exactly what I had already said." In
> fact, there is some ambiguity there, because it should be _positive_
> real/imaginary, but other than that I'm not really sure that what I
> wrote is confusing or ambiguous. Is it?
>

I agree with you. Adding positive would be good. I don't think
changing the paper any further is needed.

William

--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

[Jonathan Bober]

Ok, I've now made the rest of the changes that I was supposed to make.

I think that maybe we are finished now?

[Jonathan Bober]

On Sun, May 13, 2012 at 11:03 AM, Ben LeVeque <ben.l...@gmail.com> wrote:

> Great, thanks Ashwath. Are there edits from the second referee report that
> haven't been made yet? I'm not sure if we formally divy-ed those up, but I'd
> be happy to take a look at some of them.
>
> William or Jon, did you happen to read through the Congruence Families
> section (3.4) I pushed a week or so ago? Are there any changes I should make
> to that? I want to make sure it's in good shape before I address other
> points.

Yes, I did actually look at the section. I think that it looks mostly
ok, and it is acceptable, and it looks like it is now correct, unlike
before.

I'm thinking now, though, that the citation to Fisher is a little
misleading and downplay's Fisher's work. The sentence where we cite
Fisher doesn't state the result of his paper, but what is really the
starting point of his paper. Fisher finds equations for X_E(n) for
some n, which is what allows you to actually search for points, but in
our case it is X_E(7) which is important, and that case was worked out
by Halberstadt and Kraus in 2003 (reference [HK2] in Fisher's paper).

[Noam Elkies]

> "S. 3.7: It would be helpful if you'd define least real/imaginary periods.
> Are these just the smallest periods that lie on the real/imaginary axes?"

Maybe the referee is asking mainly about the possibility that
"the smallest periods that lie on the real/imaginary axes" don't
generate the full period lattice, only a sublattice of index 2
(in which case one might also call the average of the smallest
real and pure-imaginary periods the "imaginary period").

NDE

[Ben LeVeque]

Right, okay. Here's a proposed fix:

Take out citation to Fisher, add a citation to Halberstadt and Kraus below:

"...since it is given by the equation $y^2 + \left(\varphi{} + 1\right)y = x^3 + \left(\varphi{} - 1\right)x^2 + \left(-2 \varphi{}\right)x$. Using work of Halberstadt and Kraus \cite{halberstadt_kraus:X_E7}, Fisher then used a \textsc{Magma} \cite{magma} 

 program..."

If that works, I'll push it.

[Jonathan Bober]

Ah ha! I might expect that to be called the "complex period", I think,
but I don't know the terminology here very well. Regardless, I now
remember being confused on exactly this point, even though I seem to
find it completely clear in retrospect, so I'll try to write something
to clear this up a little bit. Thanks.

[Ben LeVeque]

Alright, I made this change and sent a pull request (and another for the updated biblio.bib file) -- let me know if further edits are in order!

[Jonathan Bober]

I decided to make a few more changes to it, and now I have

"...For any curve $E$, the equation for the twist $X_E(7)$ was found
Halberstadt and Kraus \cite{halb_kraus:XE7}, and also by Fisher
\cite{fisher:families_cong}
through other methods which additionally yield formulas for $X_E(9)$
and $X_E(11)$.

Fisher had already implemented \textsc{Magma} \cite{magma} routines to find
$\ell$-congruent elliptic curves over $\Q$ using these equations and was able
to modify his work for $\Q(\sqrt 5)$. Fortunately, our curve $E$ was then easily
found."

I really don't like that first sentence for some reason, but there it
is. You should pull my latest changes if you want to make any more
changes.

Also, regarding the pull requests: the second pull request you sent
includes the first pull request, because it includes all changes up to
the point that you submit the request. Also, as I just learned, you
can just modify an existing pull request instead of creating a new one
when the first one has not been merged yet.

And as some forewarning: On Friday I am driving to Idaho, so I'll
likely be unable to make any very-last-minute changes. We should send
this in tomorrow, though, I think.

[Ben LeVeque]

Okay, I changed the wording a bit in that sentence:

"...For any elliptic curve, the equation for the correct

twist of $X(7)$ was found both by Halberstadt and Kraus \cite{halb_kraus:XE7} and

by Fisher \cite{fisher:families_cong}, whose methods also yield formulas for the

appropriate twists of $X(9)$ and $X(11)$."

I thought taking out $E$ might clarify it a bit, since we use $E$ and $E'$ above so much. Yeah, it would be great to polish this off today!

[Ashwath Rabindranath]

Could someone send out an email with the list of edits yet to be made? I don't think I have any specific edits that I could have made easily, but I have the time to spend tonight/tomorrow.

A

[Ariah Klages-Mundt]

I also have some time now. As Ashwath mentioned, it'd be helpful to see a list of things that still need to be done.

[Jonathan Bober]

That sounds a lot better.

[Jonathan Bober]

I think that we are basically finished. It looks like the only things
that I did not check off are

Were all the methods you employed necessary to find all elliptic
curves of conductor up to 1831, or would a subset have them have
succeeded (even in hindsight)? If all were really necessary, you
should play this up a bit more in the introduction!

[ ] (ashwath)

* Page 6, line 10 from below - in the set of possible $d_1$'s,
$\sqrt{5}$ and $\varphi \sqrt{5}$ are missing (and indeed, one of
your twists is by $-\varphi \sqrt{5}$!).

[??]

* Page 6, line 2 from below - this $E$ is different from the $E$'s
three lines above, so the way this is written is a bit confusing.

[fixed?]

* Page 12, Section 5 - I think there was a discussion on the sage-nt
mailing list that involved a mistake in [Cre92] (but I don't
remember whether it was before or after the submission deadline), so
please make sure the statement is correct.

I think that these things have probably all been done as well, though.
I don't know why I didn't check off the first one, the next two are in
a paragraph that looks like it has been somewhat rewritten, so I
wasn't really able to tell quickly exactly what they were referring
to, and I hope that someone is certain that the last point is correct.

[Ashwath Rabindranath]

I made the first edit in my first set of changes -- did it fail to appear?

A

[Ariah Klages-Mundt]

* Page 6, line 10 from below - in the set of possible $d_1$'s,
 $\sqrt{5}$ and $\varphi \sqrt{5}$ are missing (and indeed, one of
 your twists is by $-\varphi \sqrt{5}$!).

I'm a little confused about the example in this section right now. According to the previous paragraph, we should have that d_1 is a square-free product of a fixed choice of generators of (2) and the prime above 5 (the primes that fulfill the specific conditions stated in that paragraph). The example states that d_1 is an element of {1, 2, \varphi, 2*\varphi}, but \varphi isn't a generator of the prime above 5. I'm not sure why \varphi is in this set. Also, we are missing a generator for the prime above 5 in this set (and its product with 2) -- I think this is what the referee is talking about, although I'm not exactly sure since he mentioned \sqrt{5}, and I'm not sure why that should be in the set.

Also, I think we need to stipulate that j(E) \neq 0, 1728 before we can say earlier in this section that twisting by d gives an elliptic curve of the form dy^2 = x^3 + ax + b.

[Ben LeVeque]

Alright, great, I think the pull request "Added Halberstadt...." has everything for 3.4 now. I just closed the other request from last night, since I guess it's unnecessary.