improve introduction

* Dedicate a subsection to our results;
* Account for latest status changes in the N=4 case;
* Wording;
* Weaken a result in the N=4 case a little bit to account for possible
  degenerate cases.
finish the N=5 proof

Still owe some discussions about further efforts (i.e., Chabauty and
Coleman) and empirical results.
remove $$ around single digits

I decided that this is not worthwhile and sometimes even
distracting (upright font among italics in theorem environments).
rewrote the general results subsection (RED ALERT)

@RobinZ16

Almost everything is completely rewritten to ensure styles are
consistent and important steps are not skipped (the usage of each
condition should be stated clearly and unambiguously, eliminating any
handwavy argument --- three trivialities unchecked consists a no-pass).

I stopped checking (and hence rewriting) at the sentence

"If this is true in general, then it follows that all periodic points
for \phi_c (with c rational) in Galois extensions to Q fields with
degree dividing by their period correspond to rational points in C_0(N),
since the first case produces rational trace t = sum_{i=1}^N z_i."

I thought for about five (or ten) minutes without success --- maybe I'm
just too tired after sitting and writing for the past twelve hours, but
I don't want to spend the rest of the night on this, especially because
the computational section is not yet expanded and rewritten.

(In fact, I'm under the impression that (ii) does not hold might
automatically guarantee the trace is rational — this is the case in N =
4, where you don't really need x1 = bar(x3). As long as you have x1 =
bar(something), the trace will be rational. Just a thought.)

I won't be able to proceed without a reply on this matter.
computational section in progress

I'm trimming out some of my earlier conjectures about factorization of
Phi_N(z, c), because once I understand the whole picture and various
details better (thanks to torturing myself by checking every detail),
they look less interesting than before.

I will preserve them in comment form in a separate `trimmed` branch, and
remove them from master.
really trimming out
finish computation section
finish rewriting the general results

* Removed wrong statement (rather, restrict to d = 2);
* Updated introduction to reflect the changes;
* Finish the introduction by refering to the source code repository.
finished a sort of complete draft

Things missing:

* Abstract;
* "Directions from here", the culminating section;
* Proofread.
address issues described in Niccolo's comments

For one-on-one change log (one commit for one comment), see the
one-on-one branch of zmwang/report:
https://github.com/zmwangx/report/tree/one-on-one.

Details
-------

address note 01: why the UBC bound is special
Comment:

    maybe say why it's so special: the constant is supposed to not
    depend on the coefficients of the morphism!

address note 02: exclude the origin in the...

projective space definition.

Comment:

    This definition is wrong, you should throw away the null (n+1)-uple!

address note 03: wording

Comment:

    "over some field K", rather than "in some space"

address note 04: wrong sign in inequality

Comment:

    The sign of the inequality goes the other way!

address note 05: wording

Comment:

    This is a strange choice of words.

address note 06: confusion around the N=4 theorem

Original code:

    Moreover, for each such $c \in \Q$, $\phi_c(z)$ has exactly four
    such periodic points (whereas for any $c \in \Q$, $\phi_c(z)$ has at
    most four periodic points of period 4 defined in some quadratic
    extension to $\Q$).

Comment:

    I don't understand. I thought the c given in the formulas were the
    only c's that give us quadratic 4-periodic points - and they give us
    four of them - while every other c has no quadratic 4-periodic
    points.

Previously, by "period 4" I'm also including exact period 1 and 2, but
obviously this should be confusing to everyone except the original
author (me) at the point of authoring (a week later I might find
it super confusing myself), so I'm trimming this out.

address note 08: wording

Comment:

    Maybe "sketch the notions" is more appropriate, since we're not
    really giving a definition - at least for the normalization.

address note 09: what happens upstairs

Comment:

    This is not really true, after desingularizing many things could
    happen. The point is that the number of rational points on the
    desingularization stays finite and can be easily bounded above and
    below (see
    http://math.stackexchange.com/questions/176844/rational-points-on-singular-curves-and-their-normalization)

I ended up removing the remark, and modified the part that used the
remark. Fortunately, it still works.

address note 10 and 11: references

Comment 10:

    You haven't said what O_X is, you need to give a reference for that.

Comment 11:

    Same here: reference for omega and cohomology group… also I think
    that you should use capital Omega since you mean the sheaf of
    differentials.

address note 14: remove sentence

Comment:

    Let's get rid of this sentence...

address note 15: change the notation for a field...

from K to k.

Comment:

    Formally everything is fine, but you may want to use a different
    letter for the field, since K is usually a number field, while often
    we will use it when K is some transcendent extension.

address note 16: typo

Comment:

    period N, not periodic

refilled the paragraph - nothing changed

address note 17: wording

Comment:

    more like, a refinement appropriately cooked for this specific curve.

address note 18: add reference for BSD

Comment:

    Maybe give a reference for the BSD conjecture?

address note 23: typo

Comment:

    "tried" missing

address note 24: self-depreciating

Comment:

    Don't say that what you're doing is useless! Just say that a very
    small number of rational points would strengthen the conjecture that
    there is no such rational c.

"Useless" remark is removed, superseded by some closing remarks.

Also add todo comments about double checking the known number of
ratinoal points.

address note 25: N=6 corollary

Comment:

    This result by exhaustion is worth it's own title…. maybe call it
    corollary of the conjecture?

Fully expanded.

add reference for the definition of Jacobian

address note 26: add the 6-cycle to computational...

results

Comment:

    you could mention that in fact this gives a quadratic point, as you
    said at the end of section 4.

address notes 12 and 13: geometric genus

Comment 12:

    you may want to explain that computer algebra softwares can compute
    both geometric and arithmetic genus. Also, the affine model needn't
    be smooth.

Comment 13:

    high GEOMETRIC genus. It could happen that a singular affine model
    has high arithmetic genus but geometric genus 1 or 0.

The genus concept is confusing, since in the definition of the
"geometric" genus, the curve is assumed to be smooth (as least that's
what I found in Qing Liu), but in the REAL WORLD, I have the impression
that everyone is using the "geometric" genus to mean the
GEOMETRIC/ARITHMETIC (which are the same) GENUS OF THE NORMALIZATION,
including the software implementations, while the arithmetic genus
retains its original meaning.

I was trying to simplify the situation by saying that we will only talk
about the geometric genus of the normalization from now on, but that
seems to be a failed attempt for people who are already familiar with
the concepts (they will typically ignore the author's idiosyncrasies).

Therefore, in this change set, I tried to make every reference to a
curve (smooth or not?) and a genus (arithmetic or geometric?
normalization?) as explicit as possible.
write direction from here

Haven't included finite field thing yet.
add finite field remarks
improve N/d for d | N to N/(N,d) for any N

And that changes everything!
write abstract

Finally it's about time!
numerous changes, mostly not math

The only substantial math change is regarding the number of rational
points on P(a, b) = 0. I checked the rational points in detail and found
that we know five regular points and three singular points (all of which
are nodes):

Regular: (0, 0), (4, 1/3), (1, 8/3), (6, 9), [0, 1, 0] (infty);
Nodes: (3, 0), [1, 3, 0] (infty), [1, 0, 0] (infty);

So we expect at most 11 rational points on the normalization.
fix another typo: turns out --> turn out
fix footnotes that begin with math

Better begin sentences with English.

Credit to Robin who brought my attention to this.
catch typos with Excalibur

Caught two.