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Hello,

my question probably isn't too hard, but I can't find the answer.

Let $K$ be a number field, $\mathcal{X} /\mathcal{O}_K$ be an arithmetic surface, which is a regular model for a projective, smooth curve $X/K$ and let $(\mathcal{L}, ||\cdot ||)$ be a hermitian line bundle on $X$. Then every point $P\in X(K)$ admits a homomorphism $f_P : Spec \mathcal{O}_K \to \mathcal{X}$.

This is what is said and I don't understand why $f_P$ must exist and what it does.

Thanks in advance for answers!

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  • $\begingroup$ This is nothing specifically to do with Arakelov theory, and you want $\mathcal{X}$ to be proper over $\mathcal{O}_K$. That said, which part do you not understand? Are you happy that a $K$-point of $X$ is the same as a morphism $\textbf{Spec }K \to X$, and an $\mathcal{O}_K$-point of $\mathcal{X}$ is the same as a morphism $\textbf{Spec }\mathcal{O}_K \to \mathcal{X}$? If so, you just need to think about why, on a projective variety, having a $K$-point is the same as having an $\mathcal{O}_K$ point. Then look up the valuative criterion of properness. $\endgroup$
    – Martin Bright
    Commented Dec 19, 2012 at 8:51
  • $\begingroup$ Wow, thanks. This is probably everything I needed. Though I can't find where in the lecture notes that I am reading it is required that $\mathcal{X}$ should be proper over $\mathcal{O}_K$, but maybe I just missed that. $\endgroup$
    – Linda
    Commented Dec 19, 2012 at 9:37
  • $\begingroup$ I deleted the tag "arakelov-theory". $\endgroup$
    – Linda
    Commented Dec 19, 2012 at 9:38
  • $\begingroup$ As Martin said, this requires $\mathcal X$ to be projective. Anywayt, the adjective projective is probably implicit in the use of the word model, and hopefully included in its definition. $\endgroup$
    – ACL
    Commented Dec 19, 2012 at 11:27

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