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Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"

There seems to be a general philosophy that problems over function fields are easier to deal with than those over number fields. Can someone actually elaborate on this analogy between number fields and function fields? I'm not sure where I can find information about this. Ring of integers being Dedekind domains, finite residue field, RH over function fields easier to deal with, anything else? Being quite ignorant about this analogy, I am actually not even convinced that why working over function fields "should" give insights about questions about number fields.

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    $\begingroup$ A more specific version of this question I have is "what is the relationship between ramification in one setting and ramification in the other?" $\endgroup$
    – Qiaochu Yuan
    Commented Oct 20, 2009 at 4:30
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    $\begingroup$ There is now some material collected on the nLab, here: ncatlab.org/nlab/show/function+field+analogy $\endgroup$
    – Urs Schreiber
    Commented Jul 20, 2014 at 20:31

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There's a really nice table in section 2.6 of these notes from a seminar that Bjorn Poonen ran at Berkeley a few years ago.

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  • $\begingroup$ Really nice indeed! $\endgroup$
    – Andrew Critch
    Commented Oct 20, 2009 at 6:19
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    $\begingroup$ I like this table so much that I'm commenting in addition to upvoting. $\endgroup$
    – JSE
    Commented Oct 23, 2009 at 15:52
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I am actually not even convinced that why working over function fields "should" give insights about questions about number fields.

It's perfectly respectable not to be convinced of this; the analogy has a long history of "working," but it's quite rare to be able to prove anything about number fields by proving the analogous function field statement. More common is that you prove something over function fields and the proof gives you ideas for an analogous proof over number fields; this is especially likely when the main players in your function field proof are things appearing in the right column of Poonen's table (referenced in David Brown's answer.)

And of course one must keep in mind that the analogy isn't perfect: This paper of Conrad, Conrad, and Helfgott is a nice example of a function-field phenomenon that should not occur over a number field.

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    $\begingroup$ Perhaps it's appropriate to point out that, sometimes, the function field theorems are enough for the number field case as well! The example in my mind is Ngo's proof of the fundamental lemma for the function field case, and Waldspurger's theorem that this implies number field case (though you could argue this is all local, I suppose). $\endgroup$
    – Bhargav
    Commented Oct 23, 2009 at 17:47
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Sure, here's a overview.

Suppose you have a ring R over a field k, then, by the magic of algebraic geometry, you can think about it in a geometric way. You do this by defining points as epimorphisms R \to k and finding out that a lot of geometric intuition plays out nicely.

Now if you start with a field, the above procedure gives you just a single point, so it's more interesting to find ring inside it — people usually take the ring of integers inside the field, which is uniquely defined.

Now the amazing thing is that you can perform this exact procedure either on number fields like Q or on function fields like F_p(t) and it gives you a very similar geometric structure.

For example, you can talk about completion of your ring by some maximal ideal and this corresponds to considering infinitesimal geometry around a single point. For number fields that would be something like Q_p while for function fields that would be F_p[[t]]. Not if you think how Q_p is basically F_p formally extended by p you notice the techniques wors the same in both cases.

E.g. the theory of ramification is basically the theory of extending either F_p[[t]] or Q_p. (There are important differences though — F_p[[t]] can be extended with F_{p^2}[[t]])

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From the valuation theoretic point of view both types of fields (nearly) look the same. If we compare Q with Fp[X], then the set of all (real-valued) absolute values on these fields is made up by the p-adic valuations (coming from prime ideals) plus one additional valuation, the "infinite prime ideal" which in the case of Q is the standard absolute value and in the case of Fp[X] is the (negative of) the degree valuation. But here is one significant difference: While the infinite prime in Fp[X] is also a discrete valuation, the infinite prime in Q is not discrete. Very important in this context is the product formula which says that for a fixed non-zero element a of a global field the product of the absolute value of a over all (finite AND infinite) places is equal to 1.

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    $\begingroup$ Poonen's table is awesome. $\endgroup$
    – user717
    Commented Oct 20, 2009 at 7:02
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Here is an interesting survey by Buium on the surprising analogies conc. differential equations, here a very interesting article by Esnault on char 0/char p analogies and how to make use of them.

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