Sen's Filtration Theorem And Its Applications

The equivalent relation between ramification filtration and Lie filtration is the con-tent of Sen's filtration theorem.In this article,we give the proofofthe filtration theorem and deduce some interesting conclusions from it.Then we construct some examples to verify these conclusions.In the first chapter,we give a brief history of ramification filtration and Lie filtra-tion.After that,we give our innovations.In the second chapter,firstly,we recall ramification theory in local fields:In view of groups,we recall the structure of ramification groups and their correspondence with higher unit groups;In view of fields,we introduce Lubin-Tate module to charaterize maximal totally ramified abelian extension.Next,we recall some definitions and results related to Lie filtration:In view of groups,we introduce some analytic pro-p-groups with special properties;In view of manifolds,we introduce the correspondence between Lie groups and Lie algebras.In the third chapter,firstly,we report Sen's proof of the filtration theorem in two parts.From the proof,we find that the information contained in ramification groups but not in the groups of Lie filtration can determine the information of the Galois groups.Also,we find a redundant assumption in the original proof.Then,by use of the filtration theorem,we obtain the structure of ramification groups in totally ramified p-adic Lie extensions and properties of ramification jumps.Finally,we give how the ramification filtration and Lie filtration could reflet the ramification properties of fields.In the fourth chapter,we construct three examples of abelian extensions and com-pute their ramification filtrations and Lie filtrations.The first one is a cyclotomic ex-tension.We take the Lubin-Tate extension as our second example.In the last example,we consider general Zp-extensions,and give a simpler proof of the main theorem in this case.In the end,we extend our problems and give some future outlooks.