Riemann surfaces

What is the genus of the compact Riemann surface of $w^2 = z^7 - z$?

A compact Riemann surface $X$ of genus $2$ admits a holomorphic map of degree $3$ onto a torus $Y$. What is the total ramification $\sum_{p}(e_p - 1)$?

A compact Riemann surface of genus $4$ is a double cover of the Riemann sphere. How many branch points does the cover have?

Which of these surfaces admit the structure of a Riemann surface? (Choose all that apply.)

Are the tori $\C/(\Z + i\Z)$ and $\C/(\Z + 2i\Z)$ biholomorphic?

On which of these domains is there a continuous function $g$ with $g(z)^2 = z$ for all $z$ in the domain? (Choose all that apply.)

In the proof of the local normal form we write $F(z) = z^k g(z)$ with $g(0) \ne 0$ and need a holomorphic $h$ near $0$ with $h^k = g$. Why does $h$ exist?

Which Lean statement says exactly that there is no continuous square root on $\C \setminus \{0\}$?

Write reduce_tau(tau) that takes a complex number with positive imaginary part and returns the equivalent point of the fundamental domain $\{|\operatorname{Re}\tau| \le \frac12,\ |\tau| \ge 1\}$, by repeating: translate by an integer to make $|\operatorname{Re}\tau| \le \frac12$; if $|\tau| < 1$, replace $\tau$ by $-1/\tau$.