数学代写|MA307 Riemann surface

Let $X=\left{[x: y: z] \in \mathbb{C P}^2 \mid x^3 y+y^3 z+z^3 x=0\right}$ be the smooth projective curve we saw in Homework 1 (it’s called the Klein’s quartic).
(a) Let $\eta=e^{i 2 \pi / 7}$. Show that the map $F: X \rightarrow X$ given by $[x: y$ : $z] \mapsto\left[\eta^2 x: \eta y: \eta^4 z\right]$ is well-defined and a biholomorphism.
(b) Assume that the quotient space $Y=X / \sim$ where $x \sim F(x)$ for all $x \in X$, is a Riemann surface, and the quotient map $\pi: X \rightarrow Y$ is a holomorphic map. Compute the degree of $\pi$, and find its ramification points and their multiplicities.
(c) From the degree-genus formula, it follows that $X$ has genus 3 . Using the Riemann-Hurwitz formula, and your answer in (b), guess what Riemann surface $Y$ is.

Let $\Lambda=\langle z \mapsto z+1, z+i\rangle$ and let $X=\mathbb{C} / \Lambda$. Let $[0] \in X$ be the point corresponding to the $\Lambda$ orbit of $0 \in \mathbb{C}$, and let Aut $_0(X)$ be the group of automorphisms (biholomorphisms) of $X$ that fix $[0]$.
(a) What are the elements of $\operatorname{Aut}_0(X)$ ? (Each such automorphism lifts to an automorphism of $\mathbb{C}$, which is an affine map; determine those affine maps.)
(b) The group $\operatorname{Aut}_0(X)$ acts on $X$ – what is the quotient Riemann surface $X / \operatorname{Aut}_0(X)$ ?

Let $X$ be the hyperelliptic curve defined by the zero set of the (homogenization) of $f(x, y)=y^2-3 x^8-10 x^4-3$, and let $Y$ be the hyperelliptic curve defined by the zero set of the (homogenization) of $g(z, w)=w^2-z^6+1$. Let $X_3$ and $Y_3$ be the corresponding affine curves (i.e. the zero sets of $f$ and $g$ respectively).
(a) Show that the function $f: X_3 \rightarrow Y_3$ defined by
$$
f(x, y)=\left(\left(1+x^2\right) /\left(1-x^2\right), 2 x y /\left(1-x^2\right)^3\right)
$$
extends to a holomorphic map $F: X \rightarrow Y$ of degree 2 which is nowhere ramified.
(b) What is the genus of $X$ and $Y$ ? You need not give a justification for your answer. (Note that the degree-genus formula would not apply since the homogenized polynomials are not nonsingular!)

In the problems below, suppose $X=\mathbb{C} / \Lambda$ where $\Lambda$ is the subgroup of translations corresponding to the lattice generated by $\omega_1, \omega_2$ (that are linearly independent over $\mathbb{R}$ ), exactly as in class. Moreover, let $\mathcal{P}$ be the Weierstrass function on $X$.

Suppose $f$ is an elliptic function on $X$ of degree $d$, and let $A$ be the set of points $c \in \mathbb{C P}^1$ such that $f(z)=c$ has strictly less than $d$ distinct solutions. Show that $A$ is non-empty and finite.

Show that $g(z):=\mathcal{P}^{\prime \prime}(z)-6(\mathcal{P}(z))^2$ is a constant function.

Show that if $f: X \rightarrow \mathbb{C} P^1$ is a holomorphic map (i.e. an elliptic function) of degree two, then
$$
f(z)=\frac{a \mathcal{P}(z-\alpha)+b}{c \mathcal{P}(z-\alpha)+d}
$$
for some $a, b, c, d, \alpha \in \mathbb{C}$ such that $a d-b c=1$.