Valuations on arithmetic surfaces

Abstract

In this paper, we give the definition of the height of a valuation and the definition of the big field Cp,G, where p is a prime and G ⊂ R is an additive subgroup containing 1. We conclude that Cp,G is a field and Cp,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m n ∈ Z, let Vm,n be an R-vector space of dimension n − m + 1, whose coordinates are indexed from m to n. We generalize the definition of Cp,G, where p is a prime and G ⊂ Vm,n is an additive subgroup containing 1. We also conclude that Cp,G is a field if m 0 n.