By Kiran S. Kedlaya
Over the past 50 years the speculation of p-adic differential equations has grown into an energetic zone of study in its personal correct, and has very important functions to quantity thought and to computing device technological know-how. This publication, the 1st complete and unified creation to the topic, improves and simplifies current effects in addition to together with unique fabric. in accordance with a path given by way of the writer at MIT, this contemporary remedy is offered to graduate scholars and researchers. workouts are integrated on the finish of every bankruptcy to aid the reader evaluation the fabric, and the writer additionally offers exact references to the literature to assist additional examine.
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Additional resources for p-adic Differential Equations
Example text
For any field F, there is a trivial norm of F defined by | f |triv = 1 f = 0, 0 f = 0. This norm is nonarchimedean, and F is complete under it. The trivial case will always be allowed unless explicitly excluded; it is often a useful input into a highly nontrivial construction, as in the next few examples. 2. Let F be any field, and let F((t)) denote the field of formal Laurent series. The t-adic valuation vt on F is defined as follows: for f = i i ci t ∈ F((t)), vt ( f ) is the least i for which ci = 0.
The additive value group of F is the set of negative logarithms of the multiplicative value group. , isomorphic to Z), we say F is discretely valued. 3 Norms on modules 17 o F = { f ∈ F : v( f ) ≥ 0}, m F = { f ∈ F : v( f ) > 0}, κ F = o F /m F . Note that o F is a local ring (the valuation ring of F), m F is the maximal ideal of o F , and κ F is a field (the residue field of F). , not nonarchimedean) absolute values on fields. 4 (Ostrowski). Let F be a field equipped with a norm | · |. Then | · | fails to be nonarchimedean if and only if the sequence |1|, |2|, |3|, .
In the process, we introduce a general framework which will allow us to consider Newton polygons in a wider range of circumstances; it is based on a version of Hensel’s lemma that applies in not necessarily commutative rings. As a first application, we fill in a few missing proofs from Chapter 1. 1 Introduction to Newton polygons We start with the possibly familiar notion of the Newton polygon associated with a polynomial over a nonarchimedean ring. 1. Let R be a ring equipped with a nonarchimedean submuli tiplicative (semi)norm | · |.