The Lumenary · First-year graduate

Riemann surfaces

Name: ______________________

Date: ______________________

You will learn to

Define Riemann surfaces by holomorphic atlases and construct them from charts, from algebraic equations such as $w^2 = z^3 - z$ , and as quotients $\C/\Lambda$
Prove the local normal form of holomorphic maps and that a non-constant holomorphic map between compact Riemann surfaces is a branched covering of a well-defined degree
Prove the Riemann–Hurwitz formula and use it to compute the genus of hyperelliptic and Fermat curves
Classify complex tori up to biholomorphism and explain why they form a one-complex-parameter family
State the uniformisation theorem and the Riemann–Roch theorem and use them in examples

1.
What is the genus of the compact Riemann surface of $w^2 = z^7 - z$ ?
2.
A compact Riemann surface $X$ of genus $2$ admits a holomorphic map of degree $3$ onto a torus $Y$ . What is the total ramification $\sum_{p}(e_p - 1)$ ?
3.
A compact Riemann surface of genus $4$ is a double cover of the Riemann sphere. How many branch points does the cover have?
4.
Which of these surfaces admit the structure of a Riemann surface? (Choose all that apply.)
- A.The torus
- B.The sphere
- C.The closed orientable surface of genus $2$
- D.The Klein bottle
- E.The real projective plane
- F.The open disc
5.
Are the tori $\C/(\Z + i\Z)$ and $\C/(\Z + 2i\Z)$ biholomorphic?
- A.No: $\tau = i$ and $\tau = 2i$ both lie in the standard fundamental domain and are distinct, so they are not $SL(2, \Z)$ -equivalent
- B.Yes: all tori are homeomorphic, hence biholomorphic
- C.Yes: $z \mapsto 2z$ maps one lattice onto the other
- D.No: their fundamental parallelograms have different areas
6.
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.)
- A. $\C \setminus (-\infty, 0]$
- B. $\C \setminus \{0\}$
- C.The upper half-plane
- D.The annulus $1 < |z| < 2$
- E.The Riemann surface $\{(z, w) : w^2 = z, z \ne 0\}$ , with $g(z, w) = w$
7.
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?
- A.Near $0$ , $g$ takes values in a small disc around $g(0) \ne 0$ avoiding $0$ , where a holomorphic logarithm exists; take $h = \exp(\frac1k \log g)$
- B.Every holomorphic function has a holomorphic $k$ -th root
- C.By the fundamental theorem of algebra
- D.Because $X$ is compact
8.
Which Lean statement says exactly that there is no continuous square root on $\C \setminus \{0\}$ ?
- A.¬ ∃ g : ℂ → ℂ, ContinuousOn g {z | z ≠ 0} ∧ ∀ z, z ≠ 0 → g z ^ 2 = z
- B.¬ ∃ g : ℂ → ℂ, ∀ z, z ≠ 0 → g z ^ 2 = z
- C.¬ ∃ g : ℂ → ℂ, Continuous g ∧ ∀ z, g z ^ 2 = z
- D.∃ g : ℂ → ℂ, ContinuousOn g {z | z ≠ 0} ∧ ¬ ∀ z, z ≠ 0 → g z ^ 2 = z
9.
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 $-1/\tau$ .
Computational — Practice

Answer key — Riemann surfaces

1.
$z^7 - z = z(z^6 - 1)$ has $7$ distinct roots, and since $7$ is odd, $\infty$ is also a branch point: $r = 8$ . Riemann–Hurwitz for the double cover gives $2g - 2 = -4 + 8$ , so $g = 3$ . (In general $w^2 = f(z)$ with $f$ squarefree of degree $d$ has genus $\lfloor (d-1)/2 \rfloor$ .)
2.
$2 \cdot 2 - 2 = 3(2 \cdot 1 - 2) + R$ gives $2 = 0 + R$ , so $R = 2$ : for instance two simple ramification points, or one point with $e = 3$ .
3.
$2 \cdot 4 - 2 = -4 + r$ gives $r = 10 = 2g + 2$ . Such a surface is $w^2 = f(z)$ with $f$ of degree $9$ or $10$ .
4.
A surface admits a complex structure if and only if it is orientable: holomorphic maps preserve orientation, and conversely every orientable surface has a conformal structure. So the torus, the sphere, the genus- $2$ surface and the open disc do (the disc in two inequivalent ways, $\mathbb D$ and $\C$ ), while the Klein bottle and $\R P^2$ do not.
5.
Both $\tau = i$ and $\tau = 2i$ lie in the fundamental domain, and $2i$ lies in its interior. A point of the interior is $SL(2, \Z)$ -equivalent to no other point of the domain, so $2i$ is not equivalent to $i$ . So the lattices are not homothetic and the tori are not biholomorphic; numerically, $j(i) = 1728 \ne j(2i) = 66^3$ .
6.
A continuous square root exists on $\C \setminus (-\infty, 0]$ and on the upper half-plane (simply connected, avoiding $0$ ), and tautologically on the Riemann surface $w^2 = z$ . It does not exist on $\C \setminus \{0\}$ or on the annulus, because each contains a loop winding once round $0$ , along which a continuous square root would change sign.
7.
Because $g$ is continuous with $g(0) \ne 0$ , on a small disc $|z| < r$ its values lie in a disc $D$ not containing $0$ . On $D$ there is a holomorphic branch of $\log$ , and $h = \exp(\frac1k\log g)$ is holomorphic with $h^k = g$ and $h(0) \ne 0$ . Then $\zeta = z h(z)$ is a local coordinate in which $F = \zeta^k$ .
8.
The claim is: no $g$ is simultaneously continuous on $\C \setminus \{0\}$ and a square root there — statement (a). Dropping continuity (b) makes it false; requiring continuity everywhere (c) gives a weaker true statement; (d) is a trivial existential.
9.
```
def reduce_tau(tau):
    while True:
        tau -= round(tau.real)
        if abs(tau) < 1 - 1e-15:
            tau = -1 / tau
        else:
            return tau
```
Why it terminates: $\operatorname{Im}(-1/\tau) = \operatorname{Im}\tau/|\tau|^2 > \operatorname{Im}\tau$ when $|\tau| < 1$ , and the imaginary parts of the points of an $SL(2, \Z)$ -orbit are bounded above with only finitely many values above any given level. The output determines the torus $\C/(\Z + \Z\tau)$ up to biholomorphism — up to the identifications on the boundary of the domain.