I am trying to understand a special case of Theorem 11.1 here.
Given a projective variety $V$ over $\mathbb Q$ and a divisor $D$ on $V$, one can associate a class of functions $\phi_{V, D} : V(\overline{\mathbb{Q}}) \to \mathbb R$ modulo the bounded functions. Essentially (when $D$ is "base point free"), one sets $\phi_{V, D} = h \circ \phi_D$ where $\phi_D : V \to \mathbb P^n, x \mapsto [f_0(x) : ... f_n(x)]$ is the morphism associated to a basis $f_0, ..., f_n$ of the Riemann-Roch space $L(D)$, and $h$ is the "height" function on $\mathbb P^n$.
Now, let $V = E$ be an elliptic curve, and $D = P_0$ be a single point. By Riemann--Roch, $L(P_0)$ has dimension 1, so $\phi_D$ is a constant morphism, and so is $h_{E, P_0}$.
Now the additivity property (see theorem 11.1 above) should give that actually $h_{E, D}$ is bounded for every $D$ ! What am I missing here?