Some Remarks on Plane Curves - Elliptic Curves

Definition 2.1 (Rational plane affine curve). A plane affine curve $C = {f (x, y) = 0} \subset 𝔸^{2}$ is rational if it has a rational parametrisation, i.e. $\exists ϕ (t), ψ (t) \in K (t)$ such that:

(i) $𝔸^{1} \to 𝔸^{2}$ , $t \mapsto (ϕ (t), ψ (t))$ is injective on $𝔸^{1} ∖ {finite set}$ .
(ii) $f (ϕ (t), ψ (t)) = 0$ .

Example 2.2.

(a) Any (non-singular) plane conic is rational.

Substitute $y = t (x + 1)$ . We get $x^{2} + t^{2} {(x + 1)}^{2} = 1$ , hence $(x + 1) ((x - 1) + t^{2} (x + 1)) = 0$ , hence $x = - 1$ or $x = \frac{1 - t^{2}}{1 + t^{2}}$ .

Therefore this has a rational parametrisation
$(x, y) = (\frac{1 - t^{2}}{1 + t^{2}}, \frac{2 t}{1 + t^{2}}) .$
(b) Any singular plane cubic is rational

rational parametrisation $(x, y) = (t^{2}, t^{3})$ .

rational parametrisation: $(x, y) = (e x e r c i s e, e x e r c i s e)$ .
(c) Corollary 1.6 shows that elliptic curves are not rational.

Remark 2.3. The genus $g (C) \in ℤ_{\geq 0}$ is an invariant of a smooth projective curve $C$ .

If $K = ℂ$ then $g (C) = genus of Riemann surface$ .
A smooth plane curve $C \subset ℙ^{2}$ of degree $d$ has genus $g (C) = \frac{(d - 1) (d - 2)}{2}$ .

Proposition 2.4. Assuming that:

$K = \bar{K}$
$C$ a smooth projective curve

Then

(i) $C$ is rational (see Definition 2.1) if and only if $g (C) = 0$ .
(ii) $C$ is an elliptic curve (see Definition 1.5) if and only if $g (C) = 1$ .

Proof.

(i) Omitted.
(ii) $\Rightarrow$ : Check $C$ a smooth plane curve (exercise). Then use Remark 2.3.
$\Leftarrow$ : See later.

□

Order of vanishing

We write

{ord}_{p} (f)

for the order of vanishing of

f \in K {(C)}^{\times}

P

(negative of

f

has a pole).

Fact:

{ord}_{p} : K {(C)}^{\times} \to ℤ

is a discrete valuation, i.e.

{ord}_{p} (f_{1} f_{2}) = {ord}_{p} (f_{1}) + {ord}_{p} (f_{2})

and

{ord}_{p} (f_{1} + f_{2}) \geq \min ({ord}_{p} (f_{1}), {ord}_{p} (f_{2}))

Example 2.5. $C = {g = 0} \subset 𝔸^{2}$ , $g \in K [x, y]$ irreducible.

$K (C) = Frac \frac{K [x, y]}{(g)}$ .

g = g_{0} + g_{1} (x, y) + g_{2} (x, y) + \dots

where $g_{i}$ are homogeneous of degree $i$ .

Suppose $P = (0, 0) \in C$ is a smooth point, i.e. $g_{0} = 0$ , $g_{1} (x, y) = α x + β y$ with $α, β$ not both zero.

Fact: $γ x + δ y \in K (C)$ is a uniformiser at $P$ if and only if $α δ - β γ \neq 0$ .

Example 2.6.

{y^{2} = x (x - 1) (x - λ)} \subset 𝔸^{2}

where $λ \neq 0, 1$ . Projective closure ( $x = \frac{X}{Z}$ , $y = \frac{Y}{Z}$ ):

{Y^{2} Z = X (X - Z) (X - λ Z)} \subset ℙ^{2} .

Let $P = (0 : 1 : 0)$ .

Aim: Compute ${ord}_{p} (x)$ and ${ord}_{p} (y)$ .

Put $w = \frac{Z}{Y}$ , $t = \frac{X}{Y}$ . So

w = t (t - w) (t - λ w) (∗)

Now $P$ is the point $(t, w) = (0, 0)$ . This is a smooth point with

{ord}_{p} (t) = {ord}_{p} (t - w) = {ord}_{p} (t - λ w) = 1 .

( $*$ ) implies ${ord}_{p} (w) = 3$ . Therefore ${ord}_{p} (x) = {ord}_{p} (t ∕ w) = 1 - 3 = - 2$ and ${ord}_{p} (y) = {ord}_{p} (1 ∕ w) = - 3$ .

Riemann Roch Spaces

Definition (Divisor). A divisor is a formal sum of points on $C$ , say

D = \sum_{P \in C} n_{p} P

where $n_{p} \in ℤ$ and $n_{p} = 0$ for all but finitely many $P \in C$ .

We write $\deg D = \sum n_{P}$ .

We say $D$ is effective (written $D \geq 0$ ) if $n_{P} \geq 0$ for all $P$ .

i.e. the

K

-vector space of rational functions on

C

with “poles no worse than specified by

D

”.

Proposition 2.7. Assuming that:

$K = \bar{K}$ , $char K \neq 2$
$C \subset ℙ^{2}$ a smooth plane cubic
$P \in C$ a point of inflection

Then we may change coordinates such that

C : Y^{2} Z = X (X - Z) (X - λ Z)

for some $λ \neq 0, 1$ and $P = (0 : 1 : 0)$ .

Proof. We change coordinates such that $P = (0 : 1 : 0)$ , $T_{p} (C) = {Z = 0}$ , and $C : {F (X, Y, Z) = 0} \subset ℙ^{2}$ .

$P \in C$ part of inflection implies $F (t, 1, 0) = const t^{3}$ , i.e. $F$ has no terms $X^{2} Y$ , $X Y^{2}$ or $Y^{3}$ .

Therefore

F \in ⟨ Y^{2} Z, X Y Z, Y Z^{2}, X^{3}, X^{2} Z, X Z^{2}, Z^{3} ⟩ .

The $Y^{2} Z$ coefficient must be $\neq 0$ otherwise $P \in C$ is singular, and the coefficient of $X^{3}$ is $\neq 0$ otherwize $Z | F$ .

We are free to rescale $X$ , $Y$ , $Z$ and $F$ . Then without loss of generality $C$ is defined by

Y^{2} Z + a_{1} X Y Z + a_{3} Y Z^{2} = X^{3} + a_{2} X^{z} Z + a_{4} X Z^{2} + a_{6} Z^{3} .

Weierstrass form.

Substituting $Y \leftarrow Y - \frac{1}{2} a_{1} X - \frac{1}{2} a_{3} Z$ , we may suppose $a_{1} = a_{3} = 0$ . Now

Y^{2} Z = Z^{3} f (\frac{X}{Z})

for some monic cubic polynomial $f$ .

$C$ is smooth, so $f$ has distinct roots. Without loss of generality say $0, 1, λ$ . Then $C$ is given by

Y^{2} Z = X (X - Z) (X - λ Z) . □

Remark. It may be shown that the points of inflection on a smooth plane curve

C = {F (X_{1}, X_{2}, X_{3}) = 0} \subset ℙ^{2}

are given by

F = \underset{Hessian}{\underset{⏟}{\det (\frac{\partial^{2} F}{\partial X_{i} \partial X_{j}})}} = 0,

2.1 The degree of a morphism

Let

ϕ : C_{1} \to C_{2}

be a non-constant morphism of smooth projective curves.

Definition (Ramification index). Suppose $P \in C_{1}$ , $Q \in C_{2}$ , $ϕ : P \mapsto Q$ . Let $t \in K (C_{2})$ be a uniformiser at $Q$ .

The ramification index of $ϕ$ at $P$ is

e_{ϕ} (P) = {ord}_{P} (ϕ^{*} t)

(always $\geq 1$ , independent of choice of $t$ ).

Theorem 2.8. Assuming that:

$ϕ : C_{1} \to C_{2}$ a non-constant morphism of smooth projective curves

Then

\sum_{P \in ϕ^{- 1} Q} e_{ϕ} (P) = \deg ϕ \forall Q \in C_{2} .

Moreover, if $ϕ$ is separable then $e_{ϕ} (P) = 1$ for all but finitely many $P \in C_{1}$ . In particular:

(i) $ϕ$ is surjective (on $\bar{K}$ -points)
(ii) $# ϕ^{- 1} (Q) \leq \deg ϕ$
(iii) If $ϕ$ is separable then equality holds in (ii) for all but finitely many $Q \in C_{2}$ .

Remark 2.9. Let $C$ be an algebraic curve. A rational map is given $C --\to ℙ^{n}$ , $P \mapsto (f_{0} (P) : f_{1} (P) : \dots : f_{n} (P))$ where $f_{0}, f_{1}, \dots, f_{n} \in K (C)$ are not all zero.

2 Some Remarks on Plane Curves

Order of vanishing

Riemann Roch Spaces

2.1 The degree of a morphism