Heights - Elliptic Curves - Cambridge III notes

For simplicity, take

K = ℚ

. Write

P \in ℙ^{n} (Q)

P = (a_{0} : a_{1} : \dots : a_{n})

where

a_{0}, \dots, a_{n} \in ℤ

and

\gcd (a_{0}, a_{n}) = 1

Lemma 13.1. Assuming that:

$f_{1}, f_{2} \in ℚ [X_{1}, X_{2}]$ coprime homogeneous polynomials of degree $d$
let
$\begin{array}{l} F : ℙ^{1} & \to ℙ^{1} \\ (x_{1} : x_{2}) & \mapsto (f_{1} (x_{1}, x_{2}) : f_{2} (x_{1}, x_{2})) \end{array}$

Then there exist

c_{1}, c_{2} > 0

such that for all

P \in ℙ^{1} (ℚ)

c_{1} H {(P)}^{d} \leq H (F (P)) \leq c_{2} H {(P)}^{d} .

Proof. Without loss of generality $f_{1}, f_{2} \in ℤ [X_{1}, X_{2}]$ .

Upper bound: Write $P = (a_{1} : a_{2})$ , $a_{1}, a_{2} \in ℤ$ coprime.

\begin{array}{l} H (F (P)) & \leq \max (| f_{1} (a_{1}, a_{2}) |, | f_{2} (a_{1}, a_{2}) |) \\ \leq c_{2} \max (| a_{1} |^{d}, | a_{2} |^{d}) \end{array}

where

c_{2} = \max_{i = 1, 2} (sum of absolute values of coefficients of f_{i}) .

Therefore $H (F (P)) \leq c_{2} H {(P)}^{d}$ .

Lower bound: We claim that there exists $g_{i j} \in ℤ [X_{1}, X_{2}]$ homogeneous of degree $d - 1$ and $κ \in ℤ_{> 0}$ such that

\sum_{j = 1}^{2} g_{i j} f_{j} = κ X_{i}^{2 d - 1}, i = 1, 2 . (†)

Indeed, running Euclid’s algorithm on $f_{1} (X, 1)$ and $f_{2} (X, 1)$ gives $r, s \in ℚ [X]$ of degree $< d$ such that

r (X) f_{1} (X, 1) + s (X) f_{2} (X, 1) = 1 .

Homogenising and clearing denominators gives ( $†$ ) for $i = 2$ . Likewise for $i = 1$ .

Write $P = (a_{1} : a_{2})$ with $a_{1}, a_{2} \in ℤ$ coprime. ( $†$ ) gives

\sum_{j = 1}^{2} g_{i j} (a_{1}, a_{2}) f_{j} (a_{1}, a_{2}) = κ a_{i}^{2 d - 1}, i = 1, 2 .

Therefore $\gcd (f_{1} (a_{1}, a_{2}), f_{2} (a_{1}, a_{2}))$ divides $\gcd (κ a_{1}^{2 d - 1}, κ a_{2}^{2 d - 1}) = κ$ .

| κ a_{i}^{2 d - 1} | \leq \max_{j = 1, 2} \underset{\leq κ H (F (P))}{\underset{⏟}{| f_{j} (a_{1}, a_{2}) |}} \underset{\leq γ_{i} H {(D)}^{d - 1}}{\underset{⏟}{\sum_{j = 1}^{2} | g_{i j} (a_{1}, a_{2}) |}}

where

γ_{i} = \sum_{j = 1}^{2} (sum of absolute values of coefficients of g_{i j}) .

Therefore

κ | a_{i} |^{2 d - 1} \leq κ H (F (P)) γ_{i} H {(P)}^{d - 1}

H {(P)}^{2 d - 1} \leq \max (γ_{1}, γ_{2}) H (F (P)) H {(P)}^{d - 1}

\underset{= c_{1}}{\underset{⏟}{\frac{1}{\max (γ_{1}, γ_{2})}}} H {(P)}^{d} \leq H (F (P)) . □

Definition (Height of a point). Let $E ∕ ℚ$ be an elliptic curve, $y^{2} = x^{3} + a x + b$ .

Define the height

\begin{array}{l} H : E (ℚ) & \to ℝ_{\geq 1} \\ P & \mapsto {\begin{matrix} H (x) & if P = (x, y) \\ 1 & if P = 0_{E} \end{matrix} \end{array}

Alsdefine logarithmic height

\begin{array}{l} h : E (ℚ) & \to ℝ_{\geq 0} \\ P & \mapsto \log H (P) \end{array}

Lemma 13.2. Assuming that:

$E$ , $E^{'}$ are elliptic curves over $ℚ$
$ϕ : E \to E^{'}$ an isogeny defined over $ℚ$

Then there exists

c > 0

such that

| h (ϕ (P)) - (\deg ϕ) h (P) | \leq c \forall P \in E (ℚ) .

Proof. Recall (Lemma 5.4)

\deg ϕ = \deg ξ

(

= d

say). Lemma 13.1 tells us that there exists

c_{1}, c_{2} > 0

such that

c_{1} H {(P)}^{d} \leq H (ϕ (P)) \leq c_{2} H {(P)}^{d} \forall P \in E (ℚ) .

Taking logs gives

| h (ϕ (P)) - d h (P) | \leq \underset{= c}{\underset{⏟}{\max (\log c_{2} - \log c_{1})}} □

Proof. Put $n = 0$ in above calculation to get

| \frac{1}{4^{m}} h (2^{m} P) - h (P) | < \frac{c}{3} .

Take limit $m \to \infty$ . □

Proof. $ĥ (P)$ bounded means we have a bound on $h (P)$ (by Lemma 13.3). So only finitely many possibilities for $x$ . Each $x$ gives $\leq 2$ choices for $y$ . □

Proof. By ?? 66, there exists $c > 0$ such that

| h (ϕ P) - (\deg ϕ) h (P) | < c \forall P \in E (ℚ) .

Replace $P$ by $2^{n} P$ , divide by $4^{n}$ and take limit $n \to \infty$ . □

Proof. Let $E$ have Weierstrass equation $y^{2} = x^{3} + a x + b$ , $a, b \in ℤ$ . Let $P, Q, P + Q, P - Q$ have $x$ coordinates $x_{1}, \dots, x_{4}$ . By Lemma 5.8, there exists $W_{0}, W_{1}, W_{2} \in ℤ [x_{1}, x_{2}]$ of degree $\leq 2$ in $x_{2}$ such that

(1 : x_{3} + x_{4} : x_{3} x_{4}) = (\underset{= {(x_{1} - x_{2})}^{2}}{\underset{⏟}{W_{0}}} : W_{1} : W_{2}) .

Write $x_{i} = \frac{r_{i}}{s_{i}}$ with $r_{i}, s_{i} \in ℤ$ coprime.

\begin{array}{l} \underset{\gcd = 1}{\underset{⏟}{(s_{3} s_{4} : r_{3} s_{4} + r_{4} s_{3} : r_{3} r_{4})}} & = ({(r_{1} s_{2} - r_{2} s_{1})}^{2} : \dots) \\ H (P + Q) H (P - Q) & = \max (| r_{3} |, | s_{3} |) \max (| r_{4} |, | s_{4} |) \\ \leq 2 \max (| s_{3} s_{4} |, | r_{3} s_{4} + r_{4} s_{3} |, | r_{3} r_{4} |) \\ \leq 2 \max (| r_{1} s_{2} - r_{2} s_{1} |^{2}, \dots) \\ \leq c \max {(| r_{1} |^{2}, | s_{1} |)}^{2} \max {(| r_{2} |, | s_{2} |)}^{2} \\ = c H {(P)}^{2} H {(Q)}^{2} \end{array}

where $c$ depends on $E$ , but not on $P$ and $Q$ . □

Proof. Lemma 13.6 and $| h (2 P) - 4 h (P) |$ bounded gives that there exists $c \in ℝ$ such that

h (P + Q) + h (P - Q) \leq 2 h (P) + 2 h (Q) + c \forall P, Q \in E (ℚ) .

Replacing $P$ , $Q$ by $2^{n} P$ , $2^{n} Q$ , dividing by $4^{n}$ and taking the limit $n \to \infty$ gives

ĥ (P + Q) + ĥ (P - Q) \leq 2 ĥ (P) + 2 ĥ (Q) .

Replacing $P, Q$ by $P + Q$ , $P - Q$ and $ĥ (2 P) = 4 ĥ (P)$ gives the reverse inequality. Therefore $ĥ$ satisfies the parallelogram law, and hence $ĥ$ is a quadratic form. □

Remark. For $K$ a number field and $P = (a_{0} : a_{1} : \dots : a_{n}) \in ℙ^{n} (K)$ , define

H (P) = \prod_{v} \max_{0 \leq i \leq n} | a_{i} |_{v}

where the product is over all places $v$ , and the absolute values are normalised such that

\prod_{v} | λ |_{v} = 1 \forall λ \in K^{*} .

Using this definition, all results in this section generalise when $ℚ$ is replaced by a number field $K$ .

13 Heights