2016/07/02 by Kosters, Michiel, Wan, Daqing
#11G20 #11R37 (primary) #12F05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1607.00523
This paper contains three parts. In the first part, we give a thorough overview of the theory of Artin-Schreier-Witt extensions: this theory allows one to understand the Z/pnZ-extensions of any field K of characteristic p via p-typical Witt vectors. Let Wn(K) be the ring of p-typical Witt vectors of K of length n and let \wp = F-id: Wn(K)\longrightarrow Wn(K), where F is the Frobenius map and id is the identity map. Artin-Schreier-Witt theory tells us that the abelian group Wn(K)/\wp Wn(K) represents the set of Z/pnZ-extensions of K. Since this theory is hard to find in literature, we have included a complete treatment in the paper. In the second part of the paper, we study Zp-extensions of a local field K=k((T)) of characteristic p>0 where k is a finite field. Local class field theory and Artin-Schreier-Witt theory give us the Schmid-Witt symbol [ , ): W(K)/\wp W(K) × \widehatK^* → W(Fp)=Zp, which contains the ramification information of Zp-extensions of K. We present a new simplified formula for [ , ). This formula allows one to compute ramification groups, conductors and discriminants in an easy way. In the third part, we study Zp-extensions of global function fields over a finite field. First, we give a formula for computing the genus in such a tower. We show that a previously obtained lower bound for the genus growth in a Zp-extension is incorrect and we give a sharp lower bound. We also study when the genus behaves in a `stable' way. Finally, we find unique representatives of Zp-extensions of the rational function field k(X), and compute the genus in such a tower.