Published online by Cambridge University Press: 26 February 2010
Let be a smooth projective, geometrically irreducible curve over a finite field
. We fix a rational point ∞on
, and consider the ring A of functions on
regular away from ∞. We set k to be the function field of
and k∞ its completion at ∞. After taking algebraic closure we obtain the field
whose elements will be called “numbers”.