8

Yes, $\omega$ is an invariant differential, also characterized by being nowhere vanishing (including the point at infinity).
At every point, $dx$ and $dy$ together span the cotangent space (which is $1$-dimensional). From the second equation, you see that $dx$ vanishes precisely when $y$ vanishes, and $dy$ vanishes precisely when $P'$ vanishes. This also ...

4

Rational points which are hard to find are those of large height, and in particular large denominator. This method will only find rational points with denominator u when you scale the equations by u, which requires knowing u in advance or looping over all possible u. Try this curve: https://www.lmfdb.org/EllipticCurve/Q/294504803/d/1

4

Your two examples are actually of very different characters. The first has Hodge numbers $h^{2,1} = 3$ and $h^{1,1} = 51$; the second is rigid. This means that in the first case you're looking for a Calabi-Yau threefold with "mirror" Hodge numbers $h^{2,1} = 51$ and $h^{1,1} = 3$, while the mirror of the second may not be a Calabi-Yau threefold at ...

3

For what it's worth, it should not be possible to make $P$ and $nP$ integral on a quasi-minimal equation if $n$ is very large. More precisely, the following theorem holds. It quantifies what Chris Wuthrich said in the comments.
Theorem: If Lang's height conjecture is true (or if the $ABC$-conjecture is true), then there is an absolute constant $C$ such if $E/...

3

You don't say that $c\in\mathbb Z$, but you imply that it is. Since we can absorb cube factors of $c$ into $z$, we may assume that $c$ is cube-free. Then your assertion that $c$ cannot be a power of $2$ is simply the assertion that $c$ cannot be 1, 2, or 4. In general, let $E_c$ be the elliptic curve
$$ E_c : x^3 + y^3 = cz^3. $$
You are asking for a general ...

3

I think the action is $q\star z = -q^{-1} z$, so that you get the "correct" basic theta function. See p. 128 in Fresnel-van der Put or Roquette's book.

2

The paper Isogeny volcanoes by Andrew V. Sutherland contains various improvements for computing isogeny volcanoes. The published version is in ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium, 507–530,
Open Book Ser., 1, Math. Sci. Publ., Berkeley, CA, 2013.

1

This is just a rough draft.
One example which has infinitely many coprime integer soutions using obvious soution.
$$A^3+B^3=cC^3\tag{1}$$
Let $x=A/C$, $y=B/C$ then we get equation $(2).$
$$x^3+y^3 = c\tag{2}$$
Cubic curve $(2)$ can be transformed to ellipric curve $(3).$
$$Y^2 = X^3-432c^2\tag{3}$$
$X = \frac{\large{12c}}{\large{x+y}}$
$Y = \frac{\large{36c(...

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