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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 $X(K_v)$ vanishing on $X(K)$, where $v$ is some arbitrary place above $p$, a 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.

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    $\begingroup$ Dear @kindasorta, Please note that the $p$-adic period map is defined on a residue disk, but in general not on the whole $X(K_v)$. $\endgroup$
    – Doug Liu
    Commented Sep 21, 2023 at 14:15

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Let $V_K,V_{K_v}$ be the Zariski closures of images of $\Phi$ and $\Phi|_{X(K)}$, respectively. Since $V_{K_v}$ has higher dimension than $V_K$, it is a proper subvariety, which means there must be some nonzero regular function $f$ on $V_{K_v}$ which vanishes on $V_K$. Since $X(K_v)$ is Zariski dense in $V_{K_v}$ (by definition), $f$ must be nonzero on some element of $\Phi(X(K_v))$. That is to say, $f\circ\Phi$ is nonzero on $X(K_v)$, but it will vanish on $X(K)$. Hence $f\circ\Phi$ is the desired function.

Edit: the above doesn't quite work, since $V_{X(K)}$ need not be affine. But also the statement as written in the post is different from the one cited in Lemma 3.3 (indeed the statement in the OP is wrong - as $X(K_v)$ is compact, it admits no nonconstant analytic functions).

The statement in the lemma instead amounts to a local version of this statement: let $\Omega$ be the $v$-adic unit disk in $X(K_v)$ and let $S\subseteq\Omega$ be a subset such that $\Phi(S)$ is contained in a proper subvariety of the Zariski closure of $\Phi(X(K_v))$ (for instance, in the setting of the question, $S=\Omega\cap X(K)$). Then there is a nonzero analytic function on $\Omega$ which vanishes on $S$.

Using the argument I gave above, we can prove this locally: by taking an affine subset $U$ of $V_{K_v}$, we can find a regular function which vanishes on its intersection with $\Phi(S)$. It implies that $\Phi^{-1}(U)\cap S$ is contained in a vanishing locus of an analytic function on $\Phi^{-1}(U)$. This should be sufficient for finiteness as $S$ is covered by finitely many such subsets.

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  • $\begingroup$ $\mathbb{P}^1$ is a proper subvariety of $\mathbb{P}^n$, but there is no non-zero regular algebraic or analytic function $f$ on $\mathbb{P}^n$ that vanishes on your favorite embedding of $\mathbb{P}^1$. Could you elaborate on why such $f$ exists? In which category does it live? $\endgroup$
    – kindasorta
    Commented Feb 12, 2023 at 19:15
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    $\begingroup$ @kindasorta Oops, you are right, my argument would have worked if we were working with affine varieties, but that is not the case here. By working locally on $V$'s I believe we can deduce that $X(K)$ is locally contained in a vanishing locus of such an analytic function. I have looked at the paper and it seems that Lemma 3.3 isn't claiming anything more - they only make a claim about subsets of the disk $\Omega_v$, not the entire projective variety $\endgroup$
    – Wojowu
    Commented Feb 12, 2023 at 19:29
  • $\begingroup$ The disk $\Omega$ is a residue disk on the $p$-adic curve $X(K_v)$. My worry is that $V_K$ and $V_{K_v}$ could be projective, in which case it is unclear to me how one comes up with such a function on $V_{K_v}$. If such a function exists then it is clear that it may be pulled back to a nonzero function on $X(K_v)$. $\endgroup$
    – kindasorta
    Commented Feb 12, 2023 at 19:48
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    $\begingroup$ @kindasorta In this case, unless $X(K_v)$ is empty, then properness should be equivalent to compactness. And while you are right that general hyperbolic curves are not always proper, it ought to be an assumption in your problem - non-proper hyperbolic curves can have infinitely many rational points over $K$, it is integral points over $O_K$ which they have only finitely many of. And anyway, considering $X$ proper is enough to see the statement is not true in general. $\endgroup$
    – Wojowu
    Commented Feb 13, 2023 at 15:58
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    $\begingroup$ @kindasorta I am not! I wanted to add some remarks about that in the answer, but I was not confident in their validity and they didn't seem necessary. My idea was that perhaps after showing that this set is locally contained in a vanishing locus, perhaps one can argue cohomologically (using essentially resolutions of Cousin's problems over $K_v$) that one can glue the functions defining respective loci into a single one. However it is not at all obvious that the functions I produce can be chosen "compatibly" (it is true a posteriori by finiteness, but that doesn't seem clear from the argument) $\endgroup$
    – Wojowu
    Commented Feb 13, 2023 at 16:07

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