2009/01/06 by Jérémy Le Borgne, Borgne, Jérémy Le · 1 citation
Mathematics · #11S15 #11S25 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:11S15 #msc:11S25
paper · pdf · doi:10.48550/arxiv.0901.0682
12 pages
arxiv created 2009/11/10 · arxiv updated 2009/12/01
Let p a prime number, Qp the field of p-adic numbers, K a finite extension of Qp, K an algebraic closure, and Cp the completion of Qp, on which the valuation on Qp extends. In his proof of the Ax-Sen-Tate theorem, Ax shows that if x in Cp satisfies v(sx - x) > A for all s in the absolute Galois group of K G, then there is a y in K such that v(x-y) >= A - C, with the constant C = p/(p-1)2. Ax questions the optimality of this constant, which we study here. Introducing the extension of K by pn-th roots of the uniformizer and relying on Tate's and Colmez's works, we find the optimal constant and some more information about elements in Cp satisfying v(sx - x) >= A for all s in G, we compute the first cohomology group of G with coefficients in the ring of integers of K.