Elliptic Curve and Differential Form Determine Weierstrass Equation I am reading Fermat's Last Theorem by Diamond, Darmon and Taylor and they state:
"An elliptic curve E over a field F is a proper smooth curve over F of genus
one with a distinguished F-rational point. If $E/F$ is an elliptic curve and if $\omega$ is a non-zero holomorphic differential on E/F then E can be realised in the projective plane by an equation (called a Weierstrass equation) of the form
$$Y^2Z + a_1XYZ + a_3Y Z^2 = X^3 + a_2X^2Z + a_4XZ^2 + a_6Z^3$$
such that the distinguished point is (0 : 1 : 0) (sometimes denoted $\infty$ because
it corresponds to the “point at infinity” in the affine model obtained by setting $Z=1$) and $\omega =\frac{dx}{2y+a_1x+a_3}$."
My question is how does the choice of $\omega$ determine the Weierstrass for $E$? Why state this in terms of differential forms instead of the usual projective embedding?

To ease notation, the Weierstrass equation is generally written using non-homogeneous coordinates $x = X/Z$ and $y = Y/Z$, then
$$E:y^2 + a_1xy+a_3y=x^3+a_2x^2+a_4x+a6$$
If $char(\mathbb{K}) \neq 2$, then we can simplify the equation by completing the square. Thus the substitution
$$y \to\frac{1}{2}(y-a_1x-a_3)$$
gives an equation of the form
$$E:y^2 = 4x^3 + b_2x^2+2b_4 + b6$$
$$b_2 = a_1^2 + 4a_4, \quad\quad b_4 =2a_4+a_1a_3, \quad\quad b_6=a_3^2+4a_6$$
We also define quantities
\begin{align}
b_8 &= a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2\\
c_4 &= b_2^2 - 24b_4\\
c_6 &= -b_2^3 +36b_2b_4-216b_6\\
\Delta &= -b_2^2b_8 - 8b_4^3 - 27b_6^2+9b_2b_4b_6\\
j &=c_4^3/\Delta\\
\omega &=\frac{dx}{2y+a_1x+a_3} = \frac{dy}{3x^2 + 2a_2x+a_4-a_1y}
\end{align}
Where the quantity $\Delta$ is the discriminant of the Weierstrass equation, the quantity $j$ is the $j$-invariant of the elliptic curve, and $\omega$ is the invariant differential associated to the Weierstrass equation.

References:
J. Silverman - The Arithmetic of Elliptic Curves 2-ed. s. $42$ (pdf)
Silverman and Tate's  - Rational Points on Elliptic Curves
Ludwig Bauer - Weierstrass Equations (pdf)

PS: Maybe this is not a clear answer to your questions, but surely the references will be very helpful.
