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The analogy between $O_K$ ($K$ a number field) and affine curves over a field has been very fruitful. It also knows many variations: the field over which the curve is defined may have positive or zero characterstic; it may be algebraically closed or not; it may be viewed locally (by various notions of "locally"); it may be viewed through the "field with one element" (if I understand that program) and so forth.

Often when I've dealt with this analogy the case is that the geometric analog of a question is easier to deal with than the arithmetic one, and is strongly suggestive for the veracity of the arithmetic statement.

My question is: what are some examples of where this analogy fails? For example, when something holds in the geometric case, and it is tempting to conjecture it's true in the arithmetic case, but it turns out to be false. If you can attach an opinion for as to why the analogy doesn't go through in your example that would be extra nice, but not necessary.

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The abc-conjecture without the epsilon in it? – KConrad Oct 28 '10 at 8:46
Oh, for a reference, in Lang's Algebra he gives examples showing the epsilon is needed for the abc-conjecture over Z. – KConrad Oct 28 '10 at 8:48
Periodicity of zeta functions (may be cheating). – S. Carnahan Oct 28 '10 at 8:53
Scott, what do you have in mind: that it has periodic in s? That wouldn't seem tempting to conjecture for the arithmetic case. – KConrad Oct 28 '10 at 12:49
Example 1: vanishing of ${\rm{H}}^1(k,G)$ for global function fields $k$ and connected semisimple $k$-groups $G$ that are simply connected. (Over number fields one has to account for effect of real places.) Example 2: if a nonzero element of a global function field is an $n$th power in all completions then it is globally an $n$th power, but this is false for certain number fields and certain $n$. (Over number fields one has to account for the effect of 2-adic places.) In both cases, the phenomena underlying failure over number fields were known before the proofs were found over fn fields. – BCnrd Oct 28 '10 at 20:27

Of the commenters on this question, two are authors (with Harald Helfgott) of the very nice paper "Root numbers and ranks in positive characteristic", which gives an example (under parity conjecture) of a non-isotrivial 1-parameter family of elliptic curves over a global function field $K = \mathbf{F}_q(u)$ (any odd $q$) such that each fiber $E_t$ for $t \in K$ has rank strictly greater than that of the generic fiber. This is conjectured to be impossible in the number field case.

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If $A/K$ is an abelian variety, $v$ is a place of $K$, $h$ is the global height function and $\lambda_v$ is the local height function at $v$, then comparing $h(P)$ and $\lambda_v(P)$ for $P \in A(K)$ varies a lot depending on the situation. $\lambda_v(P) = O(1)$ if $K$ is a function field of characteristic zero, $\lambda_v(P) = O(h(P)^{1/2})$ (usually) in positive characteristic and this cannot be improved, and $\lambda_v(P) = O(\log(h(P)))$ conjecturally for number fields (and is definitely not $O(1)$).

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