Global field

In mathematics, the term global field refers to either of the following:

*a number field, i.e., a finite extension of Q or
*the function field of an algebraic curve over a finite field, i.e., a finitely generated field of characteristic "p">0 of transcendence degree 1.

There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the "product formula" for non-zero elements "x"::$prod\_v\; |x|\_v\; =\; 1.$

The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for local zeta-functions settled by André Weil in 1940. The terminology may be due to Weil, who wrote his "Basic Number Theory" (1967) in part to work out the parallelism.

It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.

References

* J.W.S. Cassels, "Global fields", in J.W.S. Cassels and A. Frohlich (edd), "Algebraic number theory", Academic Press, 1973. Chap.II, pp.45-84.
* J.W.S. Cassels, "Local fields", Cambridge University Press, 1986, ISBN 0-521-31525-5. P.56.