Global fields consist of finite extensions of $\mathbb{Q}$ (algebraic number fields) and finite extensions of $\mathbb{F}_q(x)$ (function fields in 1 variable over a finite field). The latter are isomorphic to the category of curves over $\mathbb{F}_q(x)$, and they can be generalized to function fields in $n$ variables over $\mathbb{F}_q(x)$, which are isomorphic to the category of varieties over $\mathbb{F}_q(x)$. Is there an analogous generalization of algebraic number fields?
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$\begingroup$ There is an axiomatic characterization of global fields by Artin and Whaples. Classfield theory as done by Chevalley can be carried out provided these axioms are true. Therefore number fields and function fields in one variable are special. I do not think classfield theory can be carried out for function fields in more than one variable, for instance. $\endgroup$– RegenbogenCommented Mar 21, 2010 at 12:58
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1$\begingroup$ Whilst the standard global fields are special in many ways, there is actually a well-developed class field theory for these higher-dimensional fields, with both idelic and ideal-theoretic descriptions due to Parshin, Kato, Saito in the 80s and many others since. $\endgroup$– dkeCommented Mar 21, 2010 at 14:33
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1$\begingroup$ In fact, this question put me in mind of Kato's survey "Generalized class field theory" from the 1990 ICM, since in the introduction, Kato in his own inimitable way explains how human beings need to start doing arithmetic in fields which are finitely generated over their prime subfields rather than merely in the one-dimensional classical global fields. $\endgroup$– dkeCommented Mar 21, 2010 at 14:36
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$\begingroup$ A little comment on your question: Studying varieties of dimension greater than 1 is not the same thing as studying their function fields. $\endgroup$– Joel DodgeCommented Mar 22, 2010 at 1:55
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The function fields (in one or more variables) over $\mathbb{F}_q$ are precisely the infinite, finitely generated fields of characteristic $p$. Thus an at least reasonable characteristic $0$ analogue is given by the (necessarily infinite!) finitely generated fields of characteristic $0$. In other words, function fields in finitely many (possibly zero) variables over a number field $K$.
Indeed there has been much work on generalizing arithmetic geometric statements over global fields to arithmetic geometric statements over arbitrary infinite, finitely generated fields. The one which springs most readily to my mind is the following generalization of the Mordell-Weil theorem due to Lang and Neron: the group of rational points on any abelian variety over any finitely generated field is a finitely generated abelian group.
answered Mar 21, 2010 at 9:30
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$\begingroup$ Wait, Prof. Clark, maybe I'm not reading what you wrote correctly, but it's a theorem of Zariski that given a field k, any finitely generated k-algebra that is also a field is necessarily algebraic, but aren't function fields always transcendental? $\endgroup$ Commented Mar 21, 2010 at 10:04
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$\begingroup$ Either that or I'm wrong, rather! $\endgroup$ Commented Mar 21, 2010 at 10:06
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3$\begingroup$ Finitely generated as a field extension is not the same as finitely generated as a $k$-algebra. $\endgroup$ Commented Mar 21, 2010 at 10:38