In Lawrence and Venkatesh's paper on the Mordell conjecture, they prove that there are finitely many $K$-rational points on a hyperbolic curve $X$, where $K$ is a number field, by showing that there exists an analytic function on the $p$-adic curve $X(K_v)$, where $v$ is some arbitrary place above $p$, a rational prime of good reduction for the curve $X$.

LV prove the existence of such a $p$-adic analytic function by showing that there is some analytic map $\Phi : X(K_v)\longrightarrow \mathcal{F}$, called the $p$-adic period map, which has a certain flag variety $\mathcal{F}$ as its target, with the property that:

- The image of $\Phi$ in $\mathcal{F}$ has Zariski closure of dimension greater than that of the Zariski closure of the image of $\Phi|_{X(K)}$.

They deduce that there has to exist a non-zero $p$-adic analytic function on $X(K_v)$ that vanishes on $X(K)$. My question is: why is this a consequence of the previous statement, i.e. why does the inequality of dimensions imply the existence of such a function?

If this is somehow helpful, this is explicitly stated in Lemma 3.3 of their paper.