Author: Eiko

Time: 2025-02-01 23:03:11 - 2025-02-01 23:03:11 (UTC)

References: p-adic Differential Equations by Kiran S. Kedlaya

Rings of functions on discs and annuli

Let $K$ be a complete field of characteristic $0$ for a norm $|\cdot |$ with residue field $\kappa$ of characteristic $p$ (which is allowed to be zero as well o.o). The norm is normalized $|p|=p^{-1}$ (if $p\ne 0$).

Various Power Series Rings

Recall in rigid analytic geometry of ${\mathbb{P}}^1$, the most important examples of affinoid subset $F\subset {\mathbb{P}}^1$ are the ring domain $\{|z|=1\}$ and the thick ring domain $\{r_1\le |z|\le r_2\}$. The ring of functions on these domains are given by power series rings with convergence conditions, similar to a Tate algebra.

• The classical Tate algebra is the ring of power series $\sum _{n\ge 0}{a}_n{t}^n$ with $|a_n|\stackrel{n\to \mathrm{\infty }}{\to }0$, corresponding to $\mathcal{O}(\{|z|\le 1\})$.

• For $F=\{|z|\le |\beta |\}$, this is just a slight generalization of the Tate algebra, where you can also conveniently think of the Tate algebra of the variable $z/\beta$ instead of $z$.

  $$\begin{array}{rl}\mathcal{O}(F)& =K⟨\frac{z}{\beta }⟩\\ & =\{\sum _{n\ge 0}{a}_n{\beta }^{-n}{t}^n:|a_n|=o(1)\}\\ & =\{\sum _{n\ge 0}{c}_n{t}^n:|c_n{\beta }^n|=o(1)\}\\ & =\{\sum _{n\ge 0}{c}_n{t}^n:v(c_n)-n\cdot v(1/\beta )\stackrel{n\to \mathrm{\infty }}{\to }\mathrm{\infty }\}\end{array}$$

• $F=\{|\alpha |\le |z|\}$. If we do a variable substitution and consider power series ring based at infinity, i.e. with variable $\frac{\alpha }{z}$, the Tate algebra of variable $\frac{\alpha }{z}$ give the ring $\mathcal{O}(F)$.

  $$\begin{array}{rl}\mathcal{O}(F)& =K⟨\frac{\alpha }{z}⟩\\ & =\{\sum _{n\ge 0}{a}_n{\alpha }^n{z}^{-n}:|a_n|=o(1)\}\\ & =\{\sum _{n\ge 0}{c}_n{z}^{-n}:|c_n{\alpha }^{-n}|=o(1)\}\\ & =\{\sum _{n\ge 0}{c}_n{z}^{-n}:v(c_n)-(-n)\cdot v(1/\alpha )\stackrel{n\to \mathrm{\infty }}{\to }\mathrm{\infty }\}\end{array}$$

• For $F=\{|\alpha |\le |z|\le |\beta |\}$, the thick ring domain. This $\mathcal{O}(F)$ is in fact the intersection of $K⟨\frac{z}{\beta }⟩$ and $K⟨\frac{\alpha }{z}⟩$.

  $$\mathcal{O}(F)=K⟨\frac{\alpha }{z},\frac{z}{\beta }⟩=\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:\begin{array}{rl}& v(a_k)-kv(1/\beta )\stackrel{k\to \mathrm{\infty }}{\to }\mathrm{\infty }\\ & v(a_k)-kv(1/\alpha )\stackrel{k\to -\mathrm{\infty }}{\to }\mathrm{\infty }\end{array}\}$$

  or equivalently as

  $$K⟨\frac{\alpha }{z},\frac{z}{\beta }⟩=\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:\begin{array}{rl}& |a_k{\alpha }^k|=o(1)\\ & |a_k{\beta }^k|=o(1)\end{array}\}.$$

  Which pictorially is a V-shaped region, your function need to eventually go above the V-shape and the height to the V-shape needs to go to infinity on both sides. We also conveniently denote this ring as $K⟨\frac{\alpha }{z},\frac{z}{\beta }⟩$.

• For $F=\{|z|=1\}$, the ring domain, we have

  $$\begin{array}{rl}\mathcal{O}(F)& =K⟨z,z^{-1}⟩\\ & =\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:|a_n|\stackrel{|n|\to \mathrm{\infty }}{\to }0\}\\ & =\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:v(a_n)-n\cdot v(1/1)\stackrel{|n|\to \mathrm{\infty }}{\to }\mathrm{\infty }\}\end{array}$$

Some Other Intermediate Rings

• We define the ring with finite Gauss norm as

  $$K[[t/\beta ]{]}_0:=\{\sum _{i=0}^{\mathrm{\infty }}{a}_i{t}^i:sup|a_i|{\beta }^i<\mathrm{\infty }\}.$$

  Clearly we have for $\delta <\beta$ that

  $$K⟨t/\beta ⟩\subset K[[t/\beta ]{]}_0\subset K⟨t/\delta ⟩$$

  Note that we can also write it as

  $$K[[t]{]}_0={\mathfrak{o}}_K[[t]]{\otimes }_{{\mathfrak{o}}_K}K$$

  where ${\mathfrak{o}}_K$ is the ring of integers of $K$.

• It is also possible to mix the two type of boundaries, producing rings like

  $$K⟨\frac{\alpha }{z},\frac{z}{\beta }]{]}_0=\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:\begin{array}{rl}& sup|a_k|{\beta }^k<\mathrm{\infty }\\ & |a_k|{\alpha }^k=o(1)\end{array}\}$$

  $$K[[\frac{\alpha }{z},\frac{z}{\beta }]{]}_0=\{\sum _{n\in \mathbb{Z}}{a}_n{t}^n:\begin{array}{rl}& sup|a_k|{\beta }^k<\mathrm{\infty }\\ & sup|a_k|{\alpha }^k<\mathrm{\infty }\end{array}\}$$

  For which we have the simple property, for $\delta \in [\alpha ,\beta )$

  $$K⟨\alpha /t,t/\beta ⟩\subset K⟨\alpha /t,t/\beta ]{]}_0\subset K⟨\alpha /t,t/\delta ⟩$$

Newton Polygon

Given an element $x$ of $K⟨\alpha /t,t/\beta ⟩$, we can associate a Newton polygon to it.

1. The full Newton Polygon $\mathrm{NP}(x)$ is the lower convex hull of the set of points $(i,v(a_i))$.

2. The essential Newton Polygon $\mathrm{ENP}(x)={\mathrm{NP}}_{[\log |\alpha |,\log |\beta |]}(x)$ is generated by the subset of slopes within $[\log |\alpha |,\log |\beta |]$.

3. Denote by $d_{\lambda }(f)=inf(v(a_i)-\lambda i)$ the distance of the Newton polygon to the line $y=\lambda x$.

Properties

Note: all the $\log$ are respect to the base of the valuation. $\log |\alpha |$ can also be more easily understood as $v(1/\alpha )$.

• $\mathrm{ENP}(x)$ has finite length.

• $d_{(\cdot )}(x)$ is continuous and piecewise linear on $[\log |\alpha |,\log |\beta |]$.

• $|x{|}_{\lambda }$ is continuous on $[\log |\alpha |,\log |\beta |]$ and log-convex, for $t\in [0,1]$ we have

  $$|x{|}_{t\lambda +(1-t)\mu }\le |x{|}_{\lambda }^t|x{|}_{\mu }^{1-t}.$$

  Or equivalently, for $x\in K⟨\alpha /t,t/\beta ⟩$ we have that $d_{(\cdot )}(x)$ is concave, i.e.

  $$d_{t\lambda +(1-t)\mu }(x)\ge t{d}_{\lambda }(x)+(1-t)d_{\mu }(x).$$

  Proof:

  $$\begin{array}{rl}inf[v(a_i)-(t\lambda +(1-t)\mu )i]& =inf[t(v(a_i)-\lambda i)+(1-t)(v(a_i)-\mu i)]\\ & \ge tinf(v(a_i)-\lambda i)+(1-t)inf(v(a_i)-\mu i)\end{array}$$

  Taking $|x{|}_{\lambda }=p^{-d_{\lambda }(x)}$ we obtain the log-convexity.

• Since $|x{|}_{\lambda }$ is log-convex, there exists a point ${\lambda }_0$ (if we allow $\pm \mathrm{\infty }$) such that $|x{|}_{{\lambda }_0}$ is minimal, $|x{|}_{\lambda }$ is decreasing for $\lambda <{\lambda }_0$ and increasing for $\lambda >{\lambda }_0$.

  If $\alpha =0$, this ${\lambda }_0=-\mathrm{\infty }$ and we have $|x{|}_{\lambda }$ is increasing on $(-\mathrm{\infty },\log |\beta |]$.

Frechet Completeness

The rings $K⟨\alpha /t,t/\beta ⟩$ and $K⟨\alpha /t,t/\beta ]{]}_0$ are Frechet complete with respect to any Gauss norm $|\cdot {|}_{\lambda }$ with slope $\lambda \in [\log |\alpha |,\log |\beta |]$.

Unit Structure

• A non-zero element $f\in K⟨t/\beta ⟩$ is a unit iff $|f-c{|}_{\log |\beta |}<|f{|}_{\log |\beta |}$ for some unit $c\in K^{\times }$.

• A nonzero element $f\in K⟨\alpha /t,t/\beta ⟩$ is a unit iff there exists a term $c{t}^i$ such that $|f-c{t}^i{|}_{\lambda }<|f{|}_{\lambda }$ for $\lambda \in \{\log |\alpha |,\log |\beta |\}$.

• A nonzero element $f\in K⟨\alpha /t,t/\beta ]{]}_0$ is a unit iff there exists a term $c{t}^i$ such that $|f-c{t}^i{|}_{\lambda }<|f{|}_{\lambda }$ for $\lambda \in [\log |\alpha |,\log |\beta |)$.

Factorization

In $K⟨\alpha /t,t/\beta ⟩$ or $K⟨\alpha /t,t/\beta ]{]}_0$, if the series $x$ has a unique term maximizing $|x_m|{\rho }^m$ for some $\rho \in [\alpha ,\beta ]$ or the supremum of $|x_i|{\rho }^i$ is achievable with least achieving index $m$, then by moving the term $x_m{t}^m$ to center we can obtain the following factorization:

$$x=x_m{t}^m\cdot g(t)\cdot h(t^{-1})$$

with $h(t^{-1})\in K⟨\alpha /t⟩\iff h\in K⟨t/{\alpha }^{-1}⟩$, $|h(t^{-1})-1|<1$ and $g(t)\in K⟨t/\beta ⟩$ or $K⟨t/\beta ]{]}_0$ with $|g{|}_{\log |\beta |}=1$ (If you are getting unique index $m$, $|g{|}_{\log |\beta |}<1$).

Analytic Elements

Sitting between $K⟨t/\beta ⟩\subset K[[t/\beta ]{]}_0$, the set of elements in $K[[t/\beta ]{]}_0$ that comes from limit of rational functions $K(t)$ is called the set of analytic elements, denoted $K[[t/\beta ]{]}_{\mathrm{an}}$.