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Let $\mathbb{F}$ be a finite field and let $q$ be its number of elements. Let $(C,\mathcal{O}_C)$ be a geometrically irreducible smooth projective curve over $\mathbb{F}$ and let $K=\mathbb{F}(C)$ be its function field. Let $\operatorname{Frob_q}$ be the $q$-Frobenius on $K$. For a closed point $P$ on $C$, let $\mathcal{O}_P$ denotes the discrete valuation subring of $K$ of elements that are regular at $P$.

My question is the following: if $S$ is a set of closed point of $C$ and $x\in \mathcal{O}_P+(\operatorname{id}-\operatorname{Frob}_q)(K)$ for all $P\in S$, do we have $x\in \left(\bigcap_{P\in S}\mathcal{O}_P\right)+(\operatorname{id}-\operatorname{Frob}_q)(K)$?

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    $\begingroup$ If $S$ is not the whole of $C$ then yes, $x\in O_P+y_P-y_P^q$, construct some $f$ whose principal part at each $P\in S$ is the same as $y_P$. $\endgroup$
    – reuns
    Commented Nov 7, 2020 at 15:56
  • $\begingroup$ @reuns Thank you for your comment. Is there a result that guarentee me the existence of $f$? For instance, when $S$ is $C\setminus \{Q\}$ for a closed point $Q$? $\endgroup$
    – Stabilo
    Commented Nov 7, 2020 at 16:11
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    $\begingroup$ Riemann-Roch ${}{}$ $\endgroup$
    – reuns
    Commented Nov 7, 2020 at 16:54
  • $\begingroup$ Many thanks ! (the terminology I was looking for is the Strong Approximation Theorem) $\endgroup$
    – Stabilo
    Commented Nov 7, 2020 at 17:23

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