4 deleted 1 characters in body; added 33 characters in body; deleted 2 characters in body

Here is another example using only algebraic and not transcendental functions. Take three distinct complex numbers $z_1,z_2,z_3$ and let $X$ be the Riemann surface obtained by integrating the form $\sqrt{(z-z_1)(z-z_2)(z-z_3)}dz$. The map $g: X \to\mathbb C$ which sends each germ $\varphi_z\in X$ to $z$ is a covering with infinitely many sheets above the $z$-plane. (The ramification points are exactly above $z_1,z_2,z_3$, and there are infinitely many; each one is of order 2. This is so because near $z_k$ the form is like $(z-z_k)^{1/2}dz$ and therefore the primitive is like $(z-z_k)^{3/2}+cte$.) Now consider the holomorphic map $f: X\to\mathbb C$ which sends each germ $\varphi_z\in X$ to $\varphi_z(z)$. The branch points of $f$ are exactly the same as the branch points of $g$, but this time whilst they are of order 2 for $g$ they are of order 3 . for $f$. (To see why this is so resolve the equation $\zeta=(z-z_k)^{3/2}+cte$ in $z$.) Now comes the important point: For nearly all choices of $z_1,z_2,z_3$ the periods $a_1,a_2,a_3$ of $ydz$ on the elliptic curve $y^2=(z-z_1)(z-z_2)(z-z_3)$ are such that the lattice $\mathbb Q$-linearly independent\mathbb{Z}a_1+\mathbb{Z}a_2+\mathbb{Z}a_3$is dense in the plane. This implies means that the set of the images of the branch points of$f$is dense in the plane. In this way you can construct an example for many other 1-form having periods that are$\mathbb Q$-linearly independent..examples... 3 added 4 characters in body; added 4 characters in body; edited body; added 21 characters in body; deleted 24 characters in body Here is another example using only algebraic and not transcendental functions. Take three distinct complex numbers$z_1,z_2,z_3$and let$X$be the Riemann surface obtained by integrating the form$\sqrt{(z-z_1)(z-z_2)(z-z_3)}dx$. \sqrt{(z-z_1)(z-z_2)(z-z_3)}dz$. The map $g: X \to\mathbb C$ which sends each germ $\varphi_z\in X$ to $z$ is a covering with infinitely many sheets above the $z$-plane. (The ramification points are exactly above $z_1,z_2,z_3$, and there are infinitely many; each one is of order 2. This is so because near $z_k$ the form is like $(z-z_k)^{1/2}dz$ and therefore the primitive is like $(z-z_k)^{3/2}$.) (z-z_k)^{3/2}+cte$.) Now consider the holomorphic map$f: X\to\mathbb C$which sends each germ$\varphi_z\in X$to$\varphi_z(z)$. The branch points of$f$are exactly the same as the branch points of$g$and g$, but this time they are of order 3. (To see that why this is so resolve the equation $\zeta=(z-z_k)^{3/2}+cte$ in $z$.) Now comes the important point: For nearly all choices of $z_1,z_2,z_3$ the periods of $ydz$ on the elliptic curve defined by the equation $y^2=(z-z_1)(z-z_2)(z-z_3)$ are $\mathbb Q$-linearly independent. This implies that the set of the images of the branch points of $f$ is dense in the plane.

In this way you can construct an example for many other 1-form having periods that are $\mathbb Q$-linearly independent...

2 added 53 characters in body

There

Here is another example using only algebraic and not transcendental functions. Take three distinct complex numbers $z_1,z_2,z_3$ and let $X$ be the Riemann surface obtained by integrating the form $\sqrt{(z-z_1)(z-z_2)(z-z_3)}dx$. The map $g: X \to\mathbb C$ which sends each germ $\varphi_z\in X$ to $z$ is a covering with infinitely many sheets above the $z$-plane. (The ramification points are exactly above $z_1,z_2,z_3$, and there infinitely many; each one is of order 2. This is so because near $z_k$ the form is like $(z-z_k)^{1/2}dz$ and therefore the primitive is like $(z-z_k)^{3/2}$.) Now consider the holomorphic map $f: X\to\mathbb C$ which sends each germ $\varphi_z\in X$ to $\varphi_z(z)$. The branch points of $f$ are exactly the same as the branch points of $g$ and they are of order 3. (To see that resolve the equation $\zeta=(z-z_k)^{3/2}+cte$ in $z$.) Now comes the important point: For nearly all choices of $z_1,z_2,z_3$ the periods of $ydz$ on the elliptic curve defined by the equation $y^2=(z-z_1)(z-z_2)(z-z_3)$ are $\mathbb Q$-linearly independent. This implies that the set of the images of the branch points of $f$ is dense in the plane.

In this way you can construct an example for many other 1-form having periods that are $\mathbb Q$-linearly independent...

1