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This question is about the analogy between number fields and function fields. It's a soft question and the title misrepresents the question.

Consider the projective line over a field. This has many endomorphisms of degree $>1$, e.g., $x\mapsto x^n$.

Now, a well-known analogy states that $\mathbf{P}^1_{\mathbf{F}_p}$ is "similar" to Spec $\mathbf{Z}$.

But $\mathbf{Z}$ has no endomorphisms. I don't know really how to phrase this question, but here the analogy seems to break down and I was just wondering why.

If I'm not mistaken, for any number ring $O_K$, every endomorphism is an automorphism.

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    Aut of A^1 is not k... :-)2012-11-17

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Geometrically, every point of $\text{Spec } \mathbb{Z}$ has a different residue field, so there's no reason to expect endomorphisms that take any point to a different point. On the other hand, every point of $\mathbb{A}^1(k)$ (more closely analogous than the projective line) has the same residue field if $k$ is algebraically closed.

The reason this analogy isn't called a theorem is that it breaks down sometimes. Don't worry too much about it. Just be grateful for the parts that don't break down (e.g. the analogy between branched covers and extensions of number fields).

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    Good analogies aren't $p$erfect matches: otherwise one could prove a theorem stating and equivalence or isomorphism of some kind and the analogy would lose most of its interest!2012-11-17