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referee_report.txt
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referee_report.txt
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[X] (Andrew will fix) William: The torsion counts seem suspicous
* this was a typo 2->7
[X] (Andrew) double check isogeny degrees
[X] Ari pointed out that the conductor of the Fisher curve (3.1) is wrong.
(Andrew will fix.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
I like this paper. The team behind it uses an impressive array of
computational methods with difficult theoretical underpinnings to
produce some great comprehensive data. Even though many of the
individual techniques they employed had been suggested before, they
have not been developed and deployed systematically with careful
attention to the computational optimization that was clearly necessary
to achieve the goal of this paper.
These results presented are significant and I recommend their
publication. ANTS is a very appropriate place for this work to be
presented. Dembele, whose quaternionic enumeration techniques underlie
much of this paper, has presented aspects of his work at ANTS while
Cremona, whose tables can be viewed as inspiration for the paper under
review, presented a report of his long-term enumeration project over Q
in the plenary session at ANTS VI in Berlin. Similar themes have also
been discussed at ANTS meetings by many other speakers over the years,
and this is a worthy paper to continue this tradition.
Below, please find some assorted corrections/suggestions for the author.
Since Q(sqrt{5}) is a PID, elliptic curves over this field admit
global minimal models. Whenever you mention existence of minimal
discriminants, you are using this existence result. Deserves to be
pointed out.
[X] YEP (ashwath)
page 2, first paragraph: If you order all number fields (not just
totally real ones) by their discriminants, Q(sqrt{-3}) and Q(sqrt{-1})
come before Q(sqrt{5}) . The situation with these fields might be
worth a comment. Cremona and his students has enumerated weight 2 cusp
forms for such fields using modular symbols on the 3-dimensional
hyperbolic upper half-space, but I'm not aware of any systematic
enumeration of elliptic curves. One reason this might be harder is
that these modular forms only have a single period. Thus, your
analytic techniques for determining the period lattices aren't
directly applicable.
[X] for extra page (william)
page 3, top: When you define T, it's not clear that T is to be viewed
as a subalgebra of End V . In fact, V isn't defined at this
point. Maybe move the definition of T after the definition of Tp.
[[I switched the order. Also, I actually
do *not* want to view T as a subalgebra of End(V). I want T to be
an abstract ring that acts on both V and the space of Hilbert
modular forms. ]]
S. 3.4: A twist of the modular curve X(n) parametrizes elliptic curves
with a given mod n representation. Thus, if you've already found your
E0 in your naive search, your search for E would be amount tofinding
rational points on such a curve. I believe Fisher's quartic surface
paramerized pairs of elliptic curves with isomorphic mod n
representations.
[X] (ben l) fix our paper based on fisher; we are wrong.
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?
[X?] (jon will do) OK. [I changed the word 'least' to
'smallest positive'. I'm not sure that this is really any better,
but I think that 'least real period' already means 'smallest period
which is real' (real being a synonym for 'lying on the real axis'.)
I will think about possible writing something different.]
S. 3.7: The relation between the periods of a Hilbert modular forms
and those of the associated elliptic curve is deeper than the
corresponding relation between the periods of elliptic modular forms
and the associated elliptic curves. This is because modular curves
uniformize elliptic curve over Q, but Hilbertmodular surfaces do not
uniformize elliptic curves over real quadratic fields. In fact, the
relations between the periods are not known in the Hilbert modular
case except in the case that the elliptic curve in question is a base
change from one defined over Q. The relevant conjectures here were
formulated by Oda, and you are implicitly using these.
[X] (jon) I suggest just insert it. [I didn't not insert all of this
but instead just added a bit that said something like "Oda conjectured
something and proved something, and stronger conjectures are believed,
for example..."]
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)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The paper reports on computations resulting in a database of
(conjecturally) all elliptic curves over Q(sqrt(5)) with conductor of
norm up to 1831 (which is where the first curve of rank 2
occurs). This involves a lot of hard work, both in developing
algorithms and in implementing them, which the paper can hardly do
justice to within the limit of 15 pages. It is an impressive
achievement.
Here are some minor points that should be fixed.
* Page 5, line 2 - "conditions modulo $\mathfrak p$ on" - I assume you
mean to say "on $E" here.
[fixed]
* Page 5, Prop. 3.2 - "$E$ a curve" -> "$E$ an elliptic curve" (OK, no
other curves are ever mentioned, but it is still better to be
precise).
[fixed]
* Page 6, line 3 - "quartic surface" - shouldn't this be a quartic
_curve_ (a twist of the Klein quartic)?
[paragraph changed, no longer relevant]
* 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).
[fixed]
* Page 6, Prop. 3.3 - state explicitly that $d$ is assumed to be
integral and squarefree.
[fixed]
* 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 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?
[fixed]
* Page 9, line 5 from below - missing braces around $\pm 1$.
[fixed]
* Page 11, first line of Section 4 - I would say "representatives up
to isomorphism for all...".
[fixed]
* 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.
[looks fixed]
* 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.
[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.
* 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".
[looks good]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%