# Riemann Surfaces

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Paper 1, Section II, F

comment(a) Consider an open $\operatorname{disc} D \subseteq \mathbb{C}$. Prove that a real-valued function $u: D \rightarrow \mathbb{R}$ is harmonic if and only if

$u=\operatorname{Re}(f)$

for some analytic function $f$.

(b) Give an example of a domain $D$ and a harmonic function $u: D \rightarrow \mathbb{R}$ that is not equal to the real part of an analytic function on $D$. Justify your answer carefully.

(c) Let $u$ be a harmonic function on $\mathbb{C}_{*}$ such that $u(2 z)=u(z)$ for every $z \in \mathbb{C}_{*}$. Prove that $u$ is constant, justifying your answer carefully. Exhibit a countable subset $S \subseteq \mathbb{C}_{*}$ and a non-constant harmonic function $u$ on $\mathbb{C}_{*} \backslash S$ such that for all $z \in \mathbb{C}_{*} \backslash S$ we have $2 z \in \mathbb{C}_{*} \backslash S$ and $u(2 z)=u(z)$.

(d) Prove that every non-constant harmonic function $u: \mathbb{C} \rightarrow \mathbb{R}$ is surjective.

Paper 2, Section II, F

commentLet $D \subseteq \mathbb{C}$ be a domain, let $(f, U)$ be a function element in $D$, and let $\alpha:[0,1] \rightarrow D$ be a path with $\alpha(0) \in U$. Define what it means for a function element $(g, V)$ to be an analytic continuation of $(f, U)$ along $\alpha$.

Suppose that $\beta:[0,1] \rightarrow D$ is a path homotopic to $\alpha$ and that $(h, V)$ is an analytic continuation of $(f, U)$ along $\beta$. Suppose, furthermore, that $(f, U)$ can be analytically continued along any path in $D$. Stating carefully any theorems that you use, prove that $g(\alpha(1))=h(\beta(1))$.

Give an example of a function element $(f, U)$ that can be analytically continued to every point of $\mathbb{C}_{*}$ and a pair of homotopic paths $\alpha, \beta$ in $\mathbb{C}_{*}$ starting in $U$ such that the analytic continuations of $(f, U)$ along $\alpha$ and $\beta$ take different values at $\alpha(1)=\beta(1)$.

Paper 3, Section II, F

comment(a) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a polynomial of degree $d>0$, and let $m_{1}, \ldots, m_{k}$ be the multiplicities of the ramification points of $f$. Prove that

$\sum_{i=1}^{k}\left(m_{i}-1\right)=d-1$

Show that, for any list of integers $m_{1}, \ldots, m_{k} \geqslant 2$ satisfying $(*)$, there is a polynomial $f$ of degree $d$ such that the $m_{i}$ are the multiplicities of the ramification points of $f$.

(b) Let $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be an analytic map, and let $B$ be the set of branch points. Prove that the restriction $f: \mathbb{C}_{\infty} \backslash f^{-1}(B) \rightarrow \mathbb{C}_{\infty} \backslash B$ is a regular covering map. Given $z_{0} \notin B$, explain how a closed loop $\gamma$ in $\mathbb{C}_{\infty} \backslash B$ gives rise to a permutation $\sigma_{\gamma}$ of $f^{-1}\left(z_{0}\right)$. Show that the group of all such permutations is transitive, and that the permutation $\sigma_{\gamma}$ only depends on $\gamma$ up to homotopy.

(c) Prove that there is no meromorphic function $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ of degree 4 with branch points $B=\{0,1, \infty\}$ such that every preimage of 0 and 1 has ramification index 2 , while some preimage of $\infty$ has ramification index equal to 3. [Hint: You may use the fact that every non-trivial product of $(2,2)$-cycles in the symmetric group $S_{4}$ is a $(2,2)$-cycle.]

Paper 1, Section II, 24F

commentAssuming any facts about triangulations that you need, prove the Riemann-Hurwitz theorem.

Use the Riemann-Hurwitz theorem to prove that, for any cubic polynomial $f: \mathbb{C} \rightarrow \mathbb{C}$, there are affine transformations $g(z)=a z+b$ and $h(z)=c z+d$ such that $k(z)=g \circ f \circ h(z)$ is of one of the following two forms:

$k(z)=z^{3} \text { or } k(z)=z\left(z^{2} / 3-1\right)$

Paper 2, Section II, 23F

commentLet $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ be a rational function. What does it mean for $p \in \mathbb{C}_{\infty}$ to be a ramification point? What does it mean for $p \in \mathbb{C}_{\infty}$ to be a branch point?

Let $B$ be the set of branch points of $f$, and let $R$ be the set of ramification points. Show that

$f: \mathbb{C}_{\infty} \backslash R \rightarrow \mathbb{C}_{\infty} \backslash B$

is a regular covering map.

State the monodromy theorem. For $w \in \mathbb{C}_{\infty} \backslash B$, explain how a closed curve based at $w$ defines a permutation of $f^{-1}(w)$.

For the rational function

$f(z)=\frac{z(2-z)}{(1-z)^{4}}$

identify the group of all such permutations.

Paper 3, Section II, F

commentLet $\Lambda=\langle\lambda, \mu\rangle \subseteq \mathbb{C}$ be a lattice. Give the definition of the associated Weierstrass $\wp$-function as an infinite sum, and prove that it converges. [You may use without proof the fact that

$\sum_{w \in \Lambda \backslash\{0\}} \frac{1}{|w|^{t}}$

converges if and only if $t>2$.]

Consider the half-lattice points

$z_{1}=\lambda / 2, \quad z_{2}=\mu / 2, \quad z_{3}=(\lambda+\mu) / 2,$

and let $e_{i}=\wp\left(z_{i}\right)$. Using basic properties of $\wp$, explain why the values $e_{1}, e_{2}, e_{3}$ are distinct

Give an example of a lattice $\Lambda$ and a conformal equivalence $\theta: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ such that $\theta$ acts transitively on the images of the half-lattice points $z_{1}, z_{2}, z_{3}$.

Paper 1, Section II, F

commentDefine $X^{\prime}:=\left\{(x, y) \in \mathbb{C}^{2}: x^{3} y+y^{3}+x=0\right\}$.

(a) Prove by defining an atlas that $X^{\prime}$ is a Riemann surface.

(b) Now assume that by adding finitely many points, it is possible to compactify $X^{\prime}$ to a Riemann surface $X$ so that the coordinate projections extend to holomorphic maps $\pi_{x}$ and $\pi_{y}$ from $X$ to $\mathbb{C}_{\infty}$. Compute the genus of $X$.

(c) Assume that any holomorphic automorphism of $X^{\prime}$ extends to a holomorphic automorphism of $X$. Prove that the group Aut $(\mathrm{X})$ of holomorphic automorphisms of $X$ contains an element $\phi$ of order 7 . Prove further that there exists a holomorphic map $\pi: X \rightarrow \mathbb{C}_{\infty}$ which satisfies $\pi \circ \phi=\pi$.

Paper 2, Section II, F

comment(a) Prove that $z \mapsto z^{4}$ as a map from the upper half-plane $\mathbb{H}$ to $\mathbb{C} \backslash\{0\}$ is a covering map which is not regular.

(b) Determine the set of singular points on the unit circle for

$h(z)=\sum_{n=0}^{\infty}(-1)^{n}(2 n+1) z^{n}$

(c) Suppose $f: \Delta \backslash\{0\} \rightarrow \Delta \backslash\{0\}$ is a holomorphic map where $\Delta$ is the unit disk. Prove that $f$ extends to a holomorphic map $\tilde{f}: \Delta \rightarrow \Delta$. If additionally $f$ is biholomorphic, prove that $\tilde{f}(0)=0$.

(d) Suppose that $g: \mathbb{C} \hookrightarrow R$ is a holomorphic injection with $R$ a compact Riemann surface. Prove that $R$ has genus 0 , stating carefully any theorems you use.

Paper 3, Section II, F

commentLet $\Lambda$ be a lattice in $\mathbb{C}$, and $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ a holomorphic map of complex tori. Show that $f$ lifts to a linear map $F: \mathbb{C} \rightarrow \mathbb{C}$.

Give the definition of $\wp(z):=\wp_{\Lambda}(z)$, the Weierstrass $\wp$-function for $\Lambda$. Show that there exist constants $g_{2}, g_{3}$ such that

$\wp^{\prime}(z)^{2}=4 \wp(z)^{3}-g_{2} \wp(z)-g_{3}$

Suppose $f \in \operatorname{Aut}(\mathbb{C} / \Lambda)$, that is, $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ is a biholomorphic group homomorphism. Prove that there exists a lift $F(z)=\zeta z$ of $f$, where $\zeta$ is a root of unity for which there exist $m, n \in \mathbb{Z}$ such that $\zeta^{2}+m \zeta+n=0$.

Paper 1, Section II, F

commentGiven a complete analytic function $\mathcal{F}$ on a domain $G \subset \mathbb{C}$, define the germ of a function element $(f, D)$ of $\mathcal{F}$ at $z \in D$. Let $\mathcal{G}$ be the set of all germs of function elements in $G$. Describe without proofs the topology and complex structure on $\mathcal{G}$ and the natural covering map $\pi: \mathcal{G} \rightarrow G$. Prove that the evaluation map $\mathcal{E}: \mathcal{G} \rightarrow \mathbb{C}$ defined by

$\mathcal{E}\left([f]_{z}\right)=f(z)$

is analytic on each component of $\mathcal{G}$.

Suppose $f: R \rightarrow S$ is an analytic map of compact Riemann surfaces with $B \subset S$ the set of branch points. Show that $f: R \backslash f^{-1}(B) \rightarrow S \backslash B$ is a regular covering map.

Given $P \in S \backslash B$, explain how any closed curve in $S \backslash B$ with initial and final points $P$ yields a permutation of the set $f^{-1}(P)$. Show that the group $H$ obtained from all such closed curves is a transitive subgroup of the group of permutations of $f^{-1}(P)$.

Find the group $H$ for the analytic map $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ where $f(z)=z^{2}+z^{-2}$.

Paper 2, Section II, F

commentState the uniformisation theorem. List without proof the Riemann surfaces which are uniformised by $\mathbb{C}_{\infty}$ and those uniformised by $\mathbb{C}$.

Let $U$ be a domain in $\mathbb{C}$ whose complement consists of more than one point. Deduce that $U$ is uniformised by the open unit disk.

Let $R$ be a compact Riemann surface of genus $g$ and $P_{1}, \ldots, P_{n}$ be distinct points of $R$. Show that $R \backslash\left\{P_{1}, \ldots, P_{n}\right\}$ is uniformised by the open unit disk if and only if $2 g-2+n>0$, and by $\mathbb{C}$ if and only if $2 g-2+n=0$ or $-1$.

Let $\Lambda$ be a lattice and $X=\mathbb{C} / \Lambda$ a complex torus. Show that an analytic map $f: \mathbb{C} \rightarrow X$ is either surjective or constant.

Give with proof an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.

Paper 3, Section II, F

commentDefine the degree of an analytic map of compact Riemann surfaces, and state the Riemann-Hurwitz formula.

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $E=\mathbb{C} / \Lambda$ the associated complex torus. Show that the $\operatorname{map}$

$\psi: z+\Lambda \mapsto-z+\Lambda$

is biholomorphic with four fixed points in $E$.

Let $S=E / \sim$ be the quotient surface (the topological surface obtained by identifying $z+\Lambda$ and $\psi(z+\Lambda)$ ), and let $p: E \rightarrow S$ be the associated projection map. Denote by $E^{\prime}$ the complement of the four fixed points of $\psi$, and let $S^{\prime}=p\left(E^{\prime}\right)$. Describe briefly a family of charts making $S^{\prime}$ into a Riemann surface, so that $p: E^{\prime} \rightarrow S^{\prime}$ is a holomorphic map.

Now assume that, by adding finitely many points, it is possible to compactify $S^{\prime}$ to a Riemann surface $S$ so that $p$ extends to a regular map $E \rightarrow S$. Find the genus of $S$.

Paper 1, Section II, F

commentBy considering the singularity at $\infty$, show that any injective analytic map $f: \mathbb{C} \rightarrow \mathbb{C}$ has the form $f(z)=a z+b$ for some $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$.

State the Riemann-Hurwitz formula for a non-constant analytic map $f: R \rightarrow S$ of compact Riemann surfaces $R$ and $S$, explaining each term that appears.

Suppose $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is analytic of degree 2. Show that there exist Möbius transformations $S$ and $T$ such that

$S f T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$

is the map given by $z \mapsto z^{2}$.

Paper 2, Section II, F

commentLet $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P$ be a fundamental parallelogram whose boundary contains no zeros or poles of $f$. Show that the number of zeros of $f$ in $P$ is the same as the number of poles of $f$ in $P$, both counted with multiplicities.

Suppose additionally that $f$ is even. Show that there exists a rational function $Q(z)$ such that $f=Q(\wp)$, where $\wp$ is the Weierstrass $\wp$-function.

Suppose $f$ is a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$, and $F$ is a meromorphic antiderivative of $f$, so that $F^{\prime}=f$. Is it necessarily true that $F$ is an elliptic function? Justify your answer.

[You may use standard properties of the Weierstrass $\wp$-function throughout.]

Paper 3, Section II, F

commentLet $n \geqslant 2$ be a positive even integer. Consider the subspace $R$ of $\mathbb{C}^{2}$ given by the equation $w^{2}=z^{n}-1$, where $(z, w)$ are coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map to the first factor. Show that $R$ has the structure of a Riemann surface in such a way that $\pi$ becomes an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute the ramification indices at the ramification points.

Assume that, by adding finitely many points, it is possible to compactify $R$ to a Riemann surface $\bar{R}$ such that $\pi$ extends to an analytic map $\bar{\pi}: \bar{R} \rightarrow \mathbb{C}_{\infty}$. Find the genus of $\bar{R}$ (as a function of $n$ ).

Paper 1, Section II, H

comment(a) Let $f: R \rightarrow S$ be a non-constant holomorphic map between Riemann surfaces. Prove that $f$ takes open sets of $R$ to open sets of $S$.

(b) Let $U$ be a simply connected domain strictly contained in $\mathbb{C}$. Is there a conformal equivalence between $U$ and $\mathbb{C}$ ? Justify your answer.

(c) Let $R$ be a compact Riemann surface and $A \subset R$ a discrete subset. Given a non-constant holomorphic function $f: R \backslash A \rightarrow \mathbb{C}$, show that $f(R \backslash A)$ is dense in $\mathbb{C}$.

Paper 2, Section II, H

commentSuppose that $f: \mathbb{C} / \Lambda_{1} \rightarrow \mathbb{C} / \Lambda_{2}$ is a holomorphic map of complex tori, and let $\pi_{j}$ denote the projection map $\mathbb{C} \rightarrow \mathbb{C} / \Lambda_{j}$ for $j=1,2$. Show that there is a holomorphic map $F: \mathbb{C} \rightarrow \mathbb{C}$ such that $\pi_{2} F=f \pi_{1} .$

Prove that $F(z)=\lambda z+\mu$ for some $\lambda, \mu \in \mathbb{C}$. Hence deduce that two complex tori $\mathbb{C} / \Lambda_{1}$ and $\mathbb{C} / \Lambda_{2}$ are conformally equivalent if and only if the lattices are related by $\Lambda_{2}=\lambda \Lambda_{1}$ for some $\lambda \in \mathbb{C}^{*}$.

Paper 3, Section II, H

commentLet $f$ be a non-constant elliptic function with respect to a lattice $\Lambda \subset \mathbb{C}$. Let $P \subset \mathbb{C}$ be a fundamental parallelogram and let the degree of $f$ be $n$. Let $a_{1}, \ldots, a_{n}$ denote the zeros of $f$ in $P$, and let $b_{1}, \ldots, b_{n}$ denote the poles (both with possible repeats). By considering the integral (if required, also slightly perturbing $P$ )

$\frac{1}{2 \pi i} \int_{\partial P} z \frac{f^{\prime}(z)}{f(z)} d z$

show that

$\sum_{j=1}^{n} a_{j}-\sum_{j=1}^{n} b_{j} \in \Lambda$

Let $\wp(z)$ denote the Weierstrass $\wp$-function with respect to $\Lambda$. For $v, w \notin \Lambda$ with $\wp(v) \neq \wp(w)$ we set

$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(v) & \wp(w) \\ \wp^{\prime}(z) & \wp^{\prime}(v) & \wp^{\prime}(w) \end{array}\right)$

an elliptic function with periods $\Lambda$. Suppose $z \notin \Lambda, z-v \notin \Lambda$ and $z-w \notin \Lambda$. Prove that $f(z)=0$ if and only if $z+v+w \in \Lambda$. [You may use standard properties of the Weierstrass $\wp$-function provided they are clearly stated.]

Paper 1, Section II, F

commentLet $f: R \rightarrow S$ be a non-constant holomorphic map between compact connected Riemann surfaces and let $B \subset S$ denote the set of branch points. Show that the map $f: R \backslash f^{-1}(B) \rightarrow S \backslash B$ is a regular covering map.

Given $w \in S \backslash B$ and a closed curve $\gamma$ in $S \backslash B$ with initial and final point $w$, explain how this defines a permutation of the (finite) set $f^{-1}(w)$. Show that the group $H$ obtained from all such closed curves is a transitive subgroup of the full symmetric group of the fibre $f^{-1}(w)$.

Find the group $H$ for $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ where $f(z)=z^{3} /\left(1-z^{2}\right)$.

Paper 2, Section II, F

commentLet $G$ be a domain in $\mathbb{C}$. Define the germ of a function element $(f, D)$ at $z \in D$. Let $\mathcal{G}$ be the set of all germs of function elements in $G$. Define the topology on $\mathcal{G}$. Show it is a topology, and that it is Hausdorff. Define the complex structure on $\mathcal{G}$, and show that there is a natural projection map $\pi: \mathcal{G} \rightarrow G$ which is an analytic covering map on each connected component of $\mathcal{G}$.

Given a complete analytic function $\mathcal{F}$ on $G$, describe how it determines a connected component $\mathcal{G}_{\mathcal{F}}$ of $\mathcal{G}$. [You may assume that a function element $(g, E)$ is an analytic continuation of a function element $(f, D)$ along a path $\gamma:[0,1] \rightarrow G$ if and only if there is a lift of $\gamma$ to $\mathcal{G}$ starting at the germ of $(f, D)$ at $\gamma(0)$ and ending at the germ of $(g, E)$ at $\gamma(1)$.]

In each of the following cases, give an example of a domain $G$ in $\mathbb{C}$ and a complete analytic function $\mathcal{F}$ such that:

(i) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is regular but not bijective;

(ii) $\pi: \mathcal{G}_{\mathcal{F}} \rightarrow G$ is surjective but not regular.

Paper 3, Section II, F

commentLet $\wp(z)$ denote the Weierstrass $\wp$-function with respect to a lattice $\Lambda \subset \mathbb{C}$ and let $f$ be an even elliptic function with periods $\Lambda$. Prove that there exists a rational function $Q$ such that $f(z)=Q(\wp(z))$. If we write $Q(w)=p(w) / q(w)$ where $p$ and $q$ are coprime polynomials, find the degree of $f$ in terms of the degrees of the polynomials $p$ and $q$. Describe all even elliptic functions of degree two. Justify your answers. [You may use standard properties of the Weierstrass $\wp$-function.]

Paper 1, Section II, H

commentIf $X$ is a Riemann surface and $p: Y \rightarrow X$ is a covering map of topological spaces, show that there is a conformal structure on $Y$ such that $p: Y \rightarrow X$ is analytic.

Let $f(z)$ be the complex polynomial $z^{5}-1$. Consider the subspace $R$ of $\mathbb{C}^{2}=\mathbb{C} \times \mathbb{C}$ given by the equation $w^{2}=f(z)$, where $(z, w)$ denotes coordinates in $\mathbb{C}^{2}$, and let $\pi: R \rightarrow \mathbb{C}$ be the restriction of the projection map onto the first factor. Show that $R$ has the structure of a Riemann surface which makes $\pi$ an analytic map. If $\tau$ denotes projection onto the second factor, show that $\tau$ is also analytic. [You may assume that $R$ is connected.]

Find the ramification points and the branch points of both $\pi$ and $\tau$. Compute also the ramification indices at the ramification points.

Assuming that it is possible to add a point $P$ to $R$ so that $X=R \cup\{P\}$ is a compact Riemann surface and $\tau$ extends to a holomorphic map $\tau: X \rightarrow \mathbb{C}_{\infty}$ such that $\tau^{-1}(\infty)=\{P\}$, compute the genus of $X .$

Paper 2, Section II, H

commentState and prove the Valency Theorem and define the degree of a non-constant holomorphic map between compact Riemann surfaces.

Let $X$ be a compact Riemann surface of genus $g$ and $\pi: X \rightarrow \mathbb{C}_{\infty}$ a holomorphic map of degree two. Find the cardinality of the set $R$ of ramification points of $\pi$. Find also the cardinality of the set of branch points of $\pi$. [You may use standard results from lectures provided they are clearly stated.]

Define $\sigma: X \rightarrow X$ as follows: if $p \in R$, then $\sigma(p)=p$; otherwise, $\sigma(p)=q$ where $q$ is the unique point such that $\pi(q)=\pi(p)$ and $p \neq q$. Show that $\sigma$ is a conformal equivalence with $\pi \sigma=\pi$ and $\sigma \sigma=$ id.

Paper 3, Section II, H

commentState the Uniformization Theorem.

Show that any domain of $\mathbb{C}$ whose complement has more than one point is uniformized by the unit disc $\Delta .$ [You may use the fact that for $\mathbb{C}_{\infty}$ the group of automorphisms consists of Möbius transformations, and for $\mathbb{C}$ it consists of maps of the form $z \mapsto a z+b$ with $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$.

Let $X$ be the torus $\mathbb{C} / \Lambda$, where $\Lambda$ is a lattice. Given $p \in X$, show that $X \backslash\{p\}$ is uniformized by the unit $\operatorname{disc} \Delta$.

Is it true that a holomorphic map from $\mathbb{C}$ to a compact Riemann surface of genus two must be constant? Justify your answer.

Paper 1, Section II, I

comment(i) Let $f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with radius of convergence $r$ in $(0, \infty)$. Show that there is at least one point $a$ on the circle $C=\{z \in \mathbb{C}:|z|=r\}$ which is a singular point of $f$, that is, there is no direct analytic continuation of $f$ in any neighbourhood of $a$.

(ii) Let $X$ and $Y$ be connected Riemann surfaces. Define the space $\mathcal{G}$ of germs of function elements of $X$ into $Y$. Define the natural topology on $\mathcal{G}$ and the natural $\operatorname{map} \pi: \mathcal{G} \rightarrow X$. [You may assume without proof that the topology on $\mathcal{G}$ is Hausdorff.] Show that $\pi$ is continuous. Define the natural complex structure on $\mathcal{G}$ which makes it into a Riemann surface. Finally, show that there is a bijection between the connected components of $\mathcal{G}$ and the complete holomorphic functions of $X$ into $Y$.

Paper 2, Section II, I

comment(i) Show that the open unit $\operatorname{disc} D=\{z \in \mathbb{C}:|z|<1\}$ is biholomorphic to the upper half-plane $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$.

(ii) Define the degree of a non-constant holomorphic map between compact connected Riemann surfaces. State the Riemann-Hurwitz formula without proof. Now let $X$ be a complex torus and $f: X \rightarrow Y$ a holomorphic map of degree 2 , where $Y$ is the Riemann sphere. Show that $f$ has exactly four branch points.

(iii) List without proof those Riemann surfaces whose universal cover is the Riemann sphere or $\mathbb{C}$. Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic map such that there are two distinct elements $a, b \in \mathbb{C}$ outside the image of $f$. Assuming the uniformization theorem and the monodromy theorem, show that $f$ is constant.

Paper 3, Section II, I

commentLet $\Lambda=\mathbb{Z}+\mathbb{Z} \lambda$ be a lattice in $\mathbb{C}$ where $\operatorname{Im}(\lambda)>0$, and let $X$ be the complex torus $\mathbb{C} / \Lambda .$

(i) Give the definition of an elliptic function with respect to $\Lambda$. Show that there is a bijection between the set of elliptic functions with respect to $\Lambda$ and the set of holomorphic maps from $X$ to the Riemann sphere. Next, show that if $f$ is an elliptic function with respect to $\Lambda$ and $f^{-1}\{\infty\}=\emptyset$, then $f$ is constant.

(ii) Assume that

$f(z)=\frac{1}{z^{2}}+\sum_{\omega \in \Lambda \backslash\{0\}}\left(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right)$

defines a meromorphic function on $\mathbb{C}$, where the sum converges uniformly on compact subsets of $\mathbb{C} \backslash \Lambda$. Show that $f$ is an elliptic function with respect to $\Lambda$. Calculate the order of $f$.

Let $g$ be an elliptic function with respect to $\Lambda$ on $\mathbb{C}$, which is holomorphic on $\mathbb{C} \backslash \Lambda$ and whose only zeroes in the closed parallelogram with vertices $\{0,1, \lambda, \lambda+1\}$ are simple zeroes at the points $\left\{\frac{1}{2}, \frac{\lambda}{2}, \frac{1}{2}+\frac{\lambda}{2}\right\}$. Show that $g$ is a non-zero constant multiple of $f^{\prime}$.

Paper 1, Section II, I

comment(i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$.

(ii) Let $U=\{z \in \mathbf{C}: \operatorname{Re}(z)>0\} \subset \mathbf{C}$; explain carefully how a holomorphic function $f$ may be defined on $U$ satisfying the equation

$\left(f(z)^{2}-1\right)^{2}=z$

Let $\mathcal{F}$ denote the connected component of the space of germs $\mathcal{G}$ (of holomorphic functions on $\mathbf{C} \backslash\{0\})$ corresponding to the function element $(U, f)$, with associated holomorphic $\operatorname{map} \pi: \mathcal{F} \rightarrow \mathbf{C} \backslash\{0\}$. Determine the number of points of $\mathcal{F}$ in $\pi^{-1}(w)$ when (a) $w=\frac{1}{2}$, and (b) $w=1$.

[You may assume any standard facts about analytic continuations that you may need.]

Paper 2, Section II, I

commentLet $X$ be the algebraic curve in $\mathbb{C}^{2}$ defined by the polynomial $p(z, w)=z^{d}+w^{d}+1$ where $d$ is a natural number. Using the implicit function theorem, or otherwise, show that there is a natural complex structure on $X$. Let $f: X \rightarrow \mathbb{C}$ be the function defined by $f(a, b)=b$. Show that $f$ is holomorphic. Find the ramification points and the corresponding branching orders of $f$.

Assume that $f$ extends to a holomorphic map $g: Y \rightarrow \mathbb{C} \cup\{\infty\}$ from a compact Riemann surface $Y$ to the Riemann sphere so that $g^{-1}(\infty)=Y \backslash X$ and that $g$ has no ramification points in $g^{-1}(\infty)$. State the Riemann-Hurwitz formula and apply it to $g$ to calculate the Euler characteristic and the genus of $Y$.

Paper 3, Section II, I

commentLet $\Lambda$ be the lattice $\mathbb{Z}+\mathbb{Z} i, X$ the torus $\mathbb{C} / \Lambda$, and $\wp$ the Weierstrass elliptic function with respect to $\Lambda$.

(i) Let $x \in X$ be the point given by $0 \in \Lambda$. Determine the group

$G=\{f \in \operatorname{Aut}(X) \mid f(x)=x\}$

(ii) Show that $\wp^{2}$ defines a degree 4 holomorphic map $h: X \rightarrow \mathbb{C} \cup\{\infty\}$, which is invariant under the action of $G$, that is, $h(f(y))=h(y)$ for any $y \in X$ and any $f \in G$. Identify a ramification point of $h$ distinct from $x$ which is fixed by every element of $G$.

[If you use the Monodromy theorem, then you should state it correctly. You may use the fact that $\operatorname{Aut}(\mathbb{C})=\{a z+b \mid a \in \mathbb{C} \backslash\{0\}, b \in \mathbb{C}\}$, and may assume without proof standard facts about $\wp$.]

Paper 1, Section II, 23G

commentSuppose that $R_{1}$ and $R_{2}$ are Riemann surfaces, and $A$ is a discrete subset of $R_{1}$. For any continuous map $\alpha: R_{1} \rightarrow R_{2}$ which restricts to an analytic map of Riemann surfaces $R_{1} \backslash A \rightarrow R_{2}$, show that $\alpha$ is an analytic map.

Suppose that $f$ is a non-constant analytic function on a Riemann surface $R$. Show that there is a discrete subset $A \subset R$ such that, for $P \in R \backslash A, f$ defines a local chart on some neighbourhood of $P$.

Deduce that, if $\alpha: R_{1} \rightarrow R_{2}$ is a homeomorphism of Riemann surfaces and $f$ is a non-constant analytic function on $R_{2}$ for which the composite $f \circ \alpha$ is analytic on $R_{1}$, then $\alpha$ is a conformal equivalence. Give an example of a pair of Riemann surfaces which are homeomorphic but not conformally equivalent.

[You may assume standard results for analytic functions on domains in the complex plane.]

Paper 2, Section II, 23G

commentLet $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\tau$ is a fixed complex number with non-zero imaginary part. Suppose that $f$ is a meromorphic function on $\mathbb{C}$ for which the poles of $f$ are precisely the points in $\Lambda$, and for which $f(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. Assume moreover that $f^{\prime}(z)$ determines a doubly periodic function with respect to $\Lambda$ with $f^{\prime}(-z)=-f^{\prime}(z)$ for all $z \in \mathbb{C} \backslash \Lambda$. Prove that:

(i) $f(-z)=f(z)$ for all $z \in \mathbb{C} \backslash \Lambda$.

(ii) $f$ is doubly periodic with respect to $\Lambda$.

(iii) If it exists, $f$ is uniquely determined by the above properties.

(iv) For some complex number $A, f$ satisfies the differential equation $f^{\prime \prime}(z)=6 f(z)^{2}+A$.

Paper 3, Section II, G

commentState the Classical Monodromy Theorem for analytic continuations in subdomains of the plane.

Let $n, r$ be positive integers with $r>1$ and set $h(z)=z^{n}-1$. By removing $n$ semi-infinite rays from $\mathbb{C}$, find a subdomain $U \subset \mathbb{C}$ on which an analytic function $h^{1 / r}$ may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface $R$ for the complete analytic function $h^{1 / r}$.

Let $Z$ denote the set of $n$th roots of unity and assume that the natural analytic covering map $\pi: R \rightarrow \mathbb{C} \backslash Z$ extends to an analytic map of Riemann surfaces $\tilde{\pi}: \tilde{R} \rightarrow \mathbb{C}_{\infty}$, where $\tilde{R}$ is a compactification of $R$ and $\mathbb{C}_{\infty}$ denotes the extended complex plane. Show that $\tilde{\pi}$ has precisely $n$ branch points if and only if $r$ divides $n$.

Paper 1, Section II, G

commentGiven a lattice $\Lambda \subset \mathbb{C}$, we may define the corresponding Weierstrass $\wp$-function to be the unique even $\Lambda$-periodic elliptic function $\wp$ with poles only on $\Lambda$ and for which $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$. For $w \notin \Lambda$, we set

$f(z)=\operatorname{det}\left(\begin{array}{ccc} 1 & 1 & 1 \\ \wp(z) & \wp(w) & \wp(-z-w) \\ \wp^{\prime}(z) & \wp^{\prime}(w) & \wp^{\prime}(-z-w) \end{array}\right)$

an elliptic function with periods $\Lambda$. By considering the poles of $f$, show that $f$ has valency at most 4 (i.e. is at most 4 to 1 on a period parallelogram).

If $w \notin \frac{1}{3} \Lambda$, show that $f$ has at least six distinct zeros. If $w \in \frac{1}{3} \Lambda$, show that $f$ has at least four distinct zeros, at least one of which is a multiple zero. Deduce that the meromorphic function $f$ is identically zero.

If $z_{1}, z_{2}, z_{3}$ are distinct non-lattice points in a period parallelogram such that $z_{1}+z_{2}+z_{3} \in \Lambda$, what can be said about the points $\left(\wp\left(z_{i}\right), \wp^{\prime}\left(z_{i}\right)\right) \in \mathbb{C}^{2}(i=1,2,3) ?$

Paper 2, Section II, G

commentGiven a complete analytic function $\mathcal{F}$ on a domain $U \subset \mathbb{C}$, describe briefly how the space of germs construction yields a Riemann surface $R$ associated to $\mathcal{F}$ together with a covering map $\pi: R \rightarrow U$ (proofs not required).

In the case when $\pi$ is regular, explain briefly how, given a point $P \in U$, any closed curve in $U$ with initial and final points $P$ yields a permutation of the set $\pi^{-1}(P)$.

Now consider the Riemann surface $R$ associated with the complete analytic function

$\left(z^{2}-1\right)^{1 / 2}+\left(z^{2}-4\right)^{1 / 2}$

on $U=\mathbb{C} \backslash\{\pm 1, \pm 2\}$, with regular covering map $\pi: R \rightarrow U$. Which subgroup of the full symmetric group of $\pi^{-1}(P)$ is obtained in this way from all such closed curves (with initial and final points $P)$ ?

Paper 3, Section II, G

commentShow that the analytic isomorphisms (i.e. conformal equivalences) of the Riemann sphere $\mathbb{C}_{\infty}$ to itself are given by the non-constant Möbius transformations.

State the Riemann-Hurwitz formula for a non-constant analytic map between compact Riemann surfaces, carefully explaining the terms which occur.

Suppose now that $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is an analytic map of degree 2 ; show that there exist Möbius transformations $S$ and $T$ such that

$S f T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$

is the map given by $z \mapsto z^{2}$.

Paper 1, Section II, G

comment(a) Let $X=\mathbb{C} \cup\{\infty\}$ be the Riemann sphere. Define the notion of a rational function $r$ and describe the function $f: X \rightarrow X$ determined by $r$. Assuming that $f$ is holomorphic and non-constant, define the degree of $r$ as a rational function and the degree of $f$ as a holomorphic map, and prove that the two degrees coincide. [You are not required to prove that the degree of $f$ is well-defined.]

Let $A=\left\{a_{1}, a_{2}, a_{3}\right\}$ and $B=\left\{b_{1}, b_{2}, b_{3}\right\}$ be two subsets of $X$ each containing three distinct elements. Prove that $X \backslash A$ is biholomorphic to $X \backslash B$.

(b) Let $Z \subset \mathbb{C}^{2}$ be the algebraic curve defined by the vanishing of the polynomial $p(z, w)=w^{2}-z^{3}+z^{2}+z$. Prove that $Z$ is smooth at every point. State the implicit function theorem and define a complex structure on $Z$, so that the maps $g, h: Z \rightarrow \mathbb{C}$ given by $g(z, w)=w, h(z, w)=z$ are holomorphic.

Define what is meant by a ramification point of a holomorphic map between Riemann surfaces. Give an example of a ramification point of $g$ and calculate the branching order of $g$ at that point.

Paper 2, Section II, G

comment(a) Let $\Lambda=\mathbb{Z}+\mathbb{Z} \tau$ be a lattice in $\mathbb{C}$, where the imaginary part of $\tau$ is positive. Define the terms elliptic function with respect to $\Lambda$ and order of an elliptic function.

Suppose that $f$ is an elliptic function with respect to $\Lambda$ of order $m>0$. Show that the derivative $f^{\prime}$ is also an elliptic function with respect to $\Lambda$ and that its order $n$ satisfies $m+1 \leqslant n \leqslant 2 m$. Give an example of an elliptic function $f$ with $m=5$ and $n=6$, and an example of an elliptic function $f$ with $m=5$ and $n=9$.

[Basic results about holomorphic maps may be used without proof, provided these are accurately stated.]

(b) State the monodromy theorem. Using the monodromy theorem, or otherwise, prove that if two tori $\mathbb{C} / \Lambda_{1}$ and $\mathbb{C} / \Lambda_{2}$ are conformally equivalent then the lattices satisfy $\Lambda_{2}=a \Lambda_{1}$, for some $a \in \mathbb{C} \backslash\{0\}$.

[You may assume that $\mathbb{C}$ is simply connected and every biholomorphic map of $\mathbb{C}$ onto itself is of the form $z \mapsto c z+d$, for some $c, d \in \mathbb{C}, c \neq 0$.]

Paper 3, Section II, G

comment(i) Let $f(z)=\sum_{n=1}^{\infty} z^{2^{n}}$. Show that the unit circle is the natural boundary of the function element $(D(0,1), f)$, where $D(0,1)=\{z \in \mathbb{C}:|z|<1\}$.

(ii) Let $X$ be a connected Riemann surface and $(D, h)$ a function element on $X$ into $\mathbb{C}$. Define a germ of $(D, h)$ at a point $p \in D$. Let $\mathcal{G}$ be the set of all the germs of function elements on $X$ into $\mathbb{C}$. Describe the topology and the complex structure on $\mathcal{G}$, and show that $\mathcal{G}$ is a covering of $X$ (in the sense of complex analysis). Show that there is a oneto-one correspondence between complete holomorphic functions on $X$ into $\mathbb{C}$ and the connected components of $\mathcal{G}$. [You are not required to prove that the topology on $\mathcal{G}$ is secondcountable.]

1.II.23H

commentDefine the terms Riemann surface, holomorphic map between Riemann surfaces and biholomorphic map.

Show, without using the notion of degree, that a non-constant holomorphic map between compact connected Riemann surfaces must be surjective.

Let $\phi$ be a biholomorphic map of the punctured unit disc $\Delta^{*}=\{0<|z|<1\} \subset \mathbb{C}$ onto itself. Show that $\phi$ extends to a biholomorphic map of the open unit disc $\Delta$ to itself such that $\phi(0)=0$.

Suppose that $f: R \rightarrow S$ is a continuous holomorphic map between Riemann surfaces and $f$ is holomorphic on $R \backslash\{p\}$, where $p$ is a point in $R$. Show that $f$ is then holomorphic on all of $R$.

[The Open Mapping Theorem may be used without proof if clearly stated.]

2.II.23H

commentExplain what is meant by a divisor $D$ on a compact connected Riemann surface $S$. Explain briefly what is meant by a canonical divisor. Define the degree of $D$ and the notion of linear equivalence between divisors. If two divisors on $S$ have the same degree must they be linearly equivalent? Give a proof or a counterexample as appropriate, stating accurately any auxiliary results that you require.

Define $\ell(D)$ for a divisor $D$, and state the Riemann-Roch theorem. Deduce that the dimension of the space of holomorphic differentials is determined by the genus $g$ of $S$ and that the same is true for the degree of a canonical divisor. Show further that if $g=2$ then $S$ admits a non-constant meromorphic function with at most two poles (counting with multiplicities).

[General properties of meromorphic functions and meromorphic differentials on $S$ may be used without proof if clearly stated.]

3.II $. 22$

commentDefine the degree of a non-constant holomorphic map between compact connected Riemann surfaces and state the Riemann-Hurwitz formula.

Show that there exists a compact connected Riemann surface of any genus $g \geqslant 0$.

[You may use without proof any foundational results about holomorphic maps and complex algebraic curves from the course, provided that these are accurately stated. You may also assume that if $h(s)$ is a non-constant complex polynomial without repeated roots then the algebraic curve $C=\left\{(s, t) \in \mathbb{C}^{2}: t^{2}-h(s)=0\right\}$ is path connected.]

4.II.23H

commentLet $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\operatorname{Im} \tau>0$. The Weierstrass function $\wp$ is the unique meromorphic $\Lambda$-periodic function on $\mathbb{C}$, such that the only poles of $\wp$ are at points of $\Lambda$ and $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$.

Show that $\wp$ is an even function. Find all the zeroes of $\wp^{\prime}$.

Suppose that $a$ is a complex number such that $2 a \notin \Lambda$. Show that the function

$h(z)=(\wp(z-a)-\wp(z+a))(\wp(z)-\wp(a))^{2}-\wp^{\prime}(z) \wp^{\prime}(a)$

has no poles in $\mathbb{C} \backslash \Lambda$. By considering the Laurent expansion of $h$ at $z=0$, or otherwise, deduce that $h$ is constant.

[General properties of meromorphic doubly-periodic functions may be used without proof if accurately stated.]

1.II.23F

commentDefine a complex structure on the unit sphere $S^{2} \subset \mathbb{R}^{3}$ using stereographic projection charts $\varphi, \psi$. Let $U \subset \mathbb{C}$ be an open set. Show that a continuous non-constant map $F: U \rightarrow S^{2}$ is holomorphic if and only if $\varphi \circ F$ is a meromorphic function. Deduce that a non-constant rational function determines a holomorphic map $S^{2} \rightarrow S^{2}$. Define what is meant by a rational function taking the value $a \in \mathbb{C} \cup\{\infty\}$ with multiplicity $m$ at infinity.

Define the degree of a rational function. Show that any rational function $f$ satisfies $(\operatorname{deg} f)-1 \leqslant \operatorname{deg} f^{\prime} \leqslant 2 \operatorname{deg} f$ and give examples to show that the bounds are attained. Is it true that the product $f . g$ satisfies $\operatorname{deg}(f . g)=\operatorname{deg} f+\operatorname{deg} g$, for any non-constant rational functions $f$ and $g$ ? Justify your answer.

2.II.23F

commentA function $\psi$ is defined for $z \in \mathbb{C}$ by

$\psi(z)=\sum_{n=-\infty}^{\infty} \exp \left(\pi i\left(n+\frac{1}{2}\right)^{2} \tau+2 \pi i\left(n+\frac{1}{2}\right)\left(z+\frac{1}{2}\right)\right)$

where $\tau$ is a complex parameter with $\operatorname{Im}(\tau)>0$. Prove that this series converges uniformly on the subsets $\{|\operatorname{Im}(z)| \leqslant R\}$ for $R>0$ and deduce that $\psi$ is holomorphic on $\mathbb{C}$.

You may assume without proof that

$\psi(z+1)=-\psi(z) \quad \text { and } \quad \psi(z+\tau)=-\exp (-\pi i \tau-2 \pi i z) \psi(z)$

for all $z \in \mathbb{C}$. Let $\ell(z)$ be the logarithmic derivative $\ell(z)=\frac{\psi^{\prime}(z)}{\psi(z)}$. Show that

$\ell(z+1)=\ell(z) \quad \text { and } \quad \ell(z+\tau)=-2 \pi i+\ell(z)$

for all $z \in \mathbb{C}$. Deduce that $\psi$ has only one zero in the parallelogram $P$ with vertices $\frac{1}{2}(\pm 1 \pm \tau)$. Find all of the zeros of $\psi .$

Let $\Lambda$ be the lattice in $\mathbb{C}$ generated by 1 and $\tau$. Show that, for $\lambda_{j}, a_{j} \in \mathbb{C}$ $(j=1, \ldots, n)$, the formula

$f(z)=\lambda_{1} \frac{\psi^{\prime}\left(z-a_{1}\right)}{\psi\left(z-a_{1}\right)}+\ldots+\lambda_{n} \frac{\psi^{\prime}\left(z-a_{n}\right)}{\psi\left(z-a_{n}\right)}$

gives a $\Lambda$-periodic meromorphic function $f$ if and only if $\lambda_{1}+\ldots+\lambda_{n}=0$. Deduce that $\frac{d}{d z}\left(\frac{\psi^{\prime}(z-a)}{\psi(z-a)}\right)$ is $\Lambda$-periodic.

3.II.22F

comment(i) Let $R$ and $S$ be compact connected Riemann surfaces and $f: R \rightarrow S$ a non-constant holomorphic map. Define the branching order $v_{f}(p)$ at $p \in R$ showing that it is well defined. Prove that the set of ramification points $\left\{p \in R: v_{f}(p)>1\right\}$ is finite. State the Riemann-Hurwitz formula.

Now suppose that $R$ and $S$ have the same genus $g$. Prove that, if $g>1$, then $f$ is biholomorphic. In the case when $g=1$, write down an example where $f$ is not biholomorphic.

[The inverse mapping theorem for holomorphic functions on domains in $\mathbb{C}$ may be assumed without proof if accurately stated.]

(ii) Let $Y$ be a non-singular algebraic curve in $\mathbb{C}^{2}$. Describe, without detailed proofs, a family of charts for $Y$, so that the restrictions to $Y$ of the first and second projections $\mathbb{C}^{2} \rightarrow \mathbb{C}$ are holomorphic maps. Show that the algebraic curve

$Y=\left\{(s, t) \in \mathbb{C}^{2}: t^{4}=\left(s^{2}-1\right)(s-4)\right\}$

is non-singular. Find all the ramification points of the $\operatorname{map} f: Y \rightarrow \mathbb{C} ;(s, t) \mapsto s$.

4.II.23F

commentLet $R$ be a Riemann surface, $\widetilde{R}$a topological surface, and $p: \widetilde{R} \rightarrow R$ a continuous map. Suppose that every point $x \in \widetilde{R}$admits a neighbourhood $\widetilde{U}$such that $p$ maps $\widetilde{U}$homeomorphically onto its image. Prove that $\widetilde{R}$has a complex structure such that $p$ is a holomorphic map.

A holomorphic map $\pi: Y \rightarrow X$ between Riemann surfaces is called a covering map if every $x \in X$ has a neighbourhood $V$ with $\pi^{-1}(V)$ a disjoint union of open sets $W_{k}$ in $Y$, so that $\pi: W_{k} \rightarrow V$ is biholomorphic for each $W_{k}$. Suppose that a Riemann surface $Y$ admits a holomorphic covering map from the unit $\operatorname{disc}\{z \in \mathbb{C}:|z|<1\}$. Prove that any holomorphic map $\mathbb{C} \rightarrow Y$ is constant.

[You may assume any form of the monodromy theorem and basic results about the lifts of paths, provided that these are accurately stated.]

$3 . \mathrm{II} . 22 \mathrm{~F} \quad$

commentDefine the branching order $v_{f}(p)$ at a point $p$ and the degree of a non-constant holomorphic map $f$ between compact Riemann surfaces. State the Riemann-Hurwitz formula.

Let $W_{m} \subset \mathbb{C}^{2}$ be an affine curve defined by the equation $s^{m}=t^{m}+1$, where $m \geqslant 2$ is an integer. Show that the projective curve $\bar{W}_{m} \subset \mathbb{P}^{2}$ corresponding to $W_{m}$ is non-singular and identify the points of $\bar{W}_{m} \backslash W_{m}$. Let $F$ be a continuous map from $\bar{W}_{m}$ to the Riemann sphere $S^{2}=\mathbb{C} \cup\{\infty\}$, such that the restriction of $F$ to $W_{m}$ is given by $F(s, t)=s$. Show that $F$ is holomorphic on $\bar{W}_{m}$. Find the degree and the ramification points of $F$ on $\bar{W}_{m}$ and their branching orders. Determine the genus of $\bar{W}_{m}$.

[Basic properties of the complex structure on an algebraic curve may be used without proof if accurately stated.]

1.II.23F

commentLet $\Lambda=\mathbb{Z}+\mathbb{Z} \tau$ be a lattice in $\mathbb{C}$, where $\tau$ is a fixed complex number with positive imaginary part. The Weierstrass $\wp$-function is the unique meromorphic $\Lambda$-periodic function on $\mathbb{C}$ such that $\wp$ is holomorphic on $\mathbb{C} \backslash \Lambda$, and $\wp(z)-1 / z^{2} \rightarrow 0$ as $z \rightarrow 0$.

Show that $\wp(-z)=\wp(z)$ and find all the zeros of $\wp^{\prime}$ in $\mathbb{C}$.

Show that $\wp$ satisfies a differential equation

$\wp^{\prime}(z)^{2}=Q(\wp(z))$

for some cubic polynomial $Q(w)$. Further show that

$Q(w)=4\left(w-\wp\left(\frac{1}{2}\right)\right)\left(w-\wp\left(\frac{1}{2} \tau\right)\right)\left(w-\wp\left(\frac{1}{2}(1+\tau)\right)\right)$

and that the three roots of $Q$ are distinct.

[Standard properties of meromorphic doubly-periodic functions may be used without proof provided these are accurately stated, but any properties of the $\wp$-function that you use must be deduced from first principles.]

2.II.23F

commentDefine the terms Riemann surface, holomorphic map between Riemann surfaces, and biholomorphic map.

(a) Prove that if two holomorphic maps $f, g$ coincide on a non-empty open subset of a connected Riemann surface $R$ then $f=g$ everywhere on $R$.

(b) Prove that if $f: R \rightarrow S$ is a non-constant holomorphic map between Riemann surfaces and $p \in R$ then there is a choice of co-ordinate charts $\phi$ near $p$ and $\psi$ near $f(p)$, such that $\left(\psi \circ f \circ \phi^{-1}\right)(z)=z^{n}$, for some non-negative integer $n$. Deduce that a holomorphic bijective map between Riemann surfaces is biholomorphic.

[The inverse function theorem for holomorphic functions on open domains in $\mathbb{C}$ may be used without proof if accurately stated.]

4.II.23F

commentDefine what is meant by a divisor on a compact Riemann surface, the degree of a divisor, and a linear equivalence between divisors. For a divisor $D$, define $\ell(D)$ and show that if a divisor $D^{\prime}$ is linearly equivalent to $D$ then $\ell(D)=\ell\left(D^{\prime}\right)$. Determine, without using the Riemann-Roch theorem, the value $\ell(P)$ in the case when $P$ is a point on the Riemann sphere $S^{2}$.

[You may use without proof any results about holomorphic maps on $S^{2}$ provided that these are accurately stated.]

State the Riemann-Roch theorem for a compact connected Riemann surface $C$. (You are not required to give a definition of a canonical divisor.) Show, by considering an appropriate divisor, that if $C$ has genus $g$ then $C$ admits a non-constant meromorphic function (that is a holomorphic map $C \rightarrow S^{2}$ ) of degree at most $g+1$.

1.II.23H

commentLet $\Lambda$ be a lattice in $\mathbb{C}$ generated by 1 and $\tau$, where $\tau$ is a fixed complex number with $\operatorname{Im} \tau>0$. The Weierstrass $\wp$-function is defined as a $\Lambda$-periodic meromorphic function such that

(1) the only poles of $\wp$ are at points of $\Lambda$, and

(2) there exist positive constants $\varepsilon$ and $M$ such that for all $|z|<\varepsilon$, we have

$\left|\wp(z)-1 / z^{2}\right|<M|z|$

Show that $\wp$ is uniquely determined by the above properties and that $\wp(-z)=\wp(z)$. By considering the valency of $\wp$ at $z=1 / 2$, show that $\wp^{\prime \prime}(1 / 2) \neq 0$.

Show that $\wp$ satisfies the differential equation

$\wp^{\prime \prime}(z)=6 \wp^{2}(z)+A,$

for some complex constant $A$.

[Standard theorems about doubly-periodic meromorphic functions may be used without proof provided they are accurately stated, but any properties of the $\wp$-function that you use must be deduced from first principles.]

2.II.23H

commentDefine the terms function element and complete analytic function.

Let $(f, D)$ be a function element such that $f(z)^{n}=p(z)$, for some integer $n \geqslant 2$, where $p(z)$ is a complex polynomial with no multiple roots. Let $F$ be the complete analytic function containing $(f, D)$. Show that every function element $(\tilde{f}, \tilde{D})$ in $F$ satisfies $\tilde{f}(z)^{n}=p(z) .$

Describe how the non-singular complex algebraic curve

$C=\left\{(z, w) \in \mathbb{C}^{2} \mid w^{n}-p(z)=0\right\}$

can be made into a Riemann surface such that the first and second projections $\mathbb{C}^{2} \rightarrow \mathbb{C}$ define, by restriction, holomorphic maps $f_{1}, f_{2}: C \rightarrow \mathbb{C}$.

Explain briefly the relation between $C$ and the Riemann surface $S(F)$ for the complete analytic function $F$ given earlier.

[You do not need to prove the Inverse Function Theorem, provided that you state it accurately.]

3.II.22H

commentExplain what is meant by a meromorphic differential on a compact connected Riemann surface $S$. Show that if $f$ is a meromorphic function on $S$ then $d f$ defines a meromorphic differential on $S$. Show also that if $\eta$ and $\omega$ are two meromorphic differentials on $S$ which are not identically zero then $\eta=h \omega$ for some meromorphic function $h$. Show that zeros and poles of a meromorphic differential are well-defined and explain, without proof, how to obtain the genus of $S$ by counting zeros and poles of $\omega$.

Let $V_{0} \subset \mathbb{C}^{2}$ be the affine curve with equation $u^{2}=v^{2}+1$ and let $V \subset \mathbb{P}^{2}$ be the corresponding projective curve. Show that $V$ is non-singular with two points at infinity, and that $d v$ extends to a meromorphic differential on $V$.

[You may assume without proof that that the map

$(u, v)=\left(\frac{t^{2}+1}{t^{2}-1}, \frac{2 t}{t^{2}-1}\right), \quad t \in \mathbb{C} \backslash\{-1,1\},$

is onto $V_{0} \backslash\{(1,0)\}$ and extends to a biholomorphic map from $\mathbb{P}^{1}$ onto $V$.]

4.II.23H

Define what is meant by the degree of a non-constant holomorphic map between compact connected Riemann surfaces, and state the Riemann-Hurwitz formula.

Let $E_{\Lambda}=\mathbb{C} / \Lambda$ be an elliptic curve defined by some lattice $\Lambda$. Show that the map

$\psi: z+\Lambda \in E_{\Lambda} \rightarrow-z+\Lambda \in E_{\Lambda}$

is biholomorphic, and that there are four points in $E_{\Lambda}$ fixed by $\psi$.

Let $S=E_{\Lambda} / \sim$ be the quotient surface (the topological surface obtained by identifying $z+\Lambda$ and $\psi(z+\Lambda)$, for each $z)$ and let $\pi: E_{\Lambda} \rightarrow S$ be the corresponding projection map. Denote by $E_{\Lambda}^{0} \subset E_{\Lambda}$