Exercise 111.65.4 (Heights). Let $K$ be a field. Let $h_ n : \mathbf{P}^ n(K) \to \mathbf{R}$, $n \geq 0$ be a collection of functions satisfying the $2$ axioms we discussed in the lectures. Let $X$ be a projective variety over $K$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module and recall that we have constructed in the lectures an associated height function $h_\mathcal {L} : X(K) \to \mathbf{R}$. Let $\alpha : X \to X$ be an automorphism of $X$ over $K$.

1. Prove that $P \mapsto h_\mathcal {L}(\alpha (P))$ differs from the function $h_{\alpha ^*\mathcal{L}}$ by a bounded amount. (Hint: recall that if there is a morphism $\varphi : X \to \mathbf{P}^ n$ with $\mathcal{L} = \varphi ^*\mathcal{O}_{\mathbf{P}^ n}(1)$, then by construction $h_\mathcal {L}(P) = h_ n(\varphi (P))$ and play around with that. In general write $\mathcal{L}$ as a difference of two of these.)

2. Assume that $h_\mathcal {L}(P) - h_\mathcal {L}(\alpha (P))$ is unbounded on $X(K)$. Show that $h_\mathcal {N}$ with $\mathcal{N} = \mathcal{L} \otimes \alpha ^*\mathcal{L}^{\otimes -1}$ is unbounded on $X(K)$.

3. Assume $X$ is an elliptic curve and that $\mathcal{L}$ is a symmetric ample invertible module on $X$ such that $h_\mathcal {L}$ is unbounded on $X(K)$. Show that there exists an invertible module $\mathcal{N}$ of degree $0$ such that $h_\mathcal {N}$ is unbounded. (Hints: Recall that $X$ is an abelian variety of dimension $1$. Thus $h_\mathcal {L}$ is quadratic up to a constant by results in the lectures. Choose a suitable point $P_0 \in X(K)$. Let $\alpha : X \to X$ be translation by $P_0$. Consider $P \mapsto h_\mathcal {L}(P) - h_\mathcal {L}(P + P_0)$. Apply the results you proved above.)

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