Last modified date: <%+ tp.file.last_modified_date() %>
- Tags: - #todo/untagged - Refs: - Some useful p-adic formulas - Witt vector - Links: - #todo/create-links
p-adic
See disambiguating completion and localization
p-adic integers
-
Idea: invert all primes except \(p\), so allow \(a/b\) where \(p\) does not divide \(b\), i.e. \(v_p(a/b) \geq 0\).
-
Start with \({\mathbf{Q}}\) and take the Cauchy completion with respect to the p-adic absolute value to obtain \({ {\mathbf{Q}}_p }\).
- Yields alternative metric space completions of \({\mathbf{Q}}\) that are not \({\mathbf{R}}\).
-
Take the ring of integers to define \({ {\mathbf{Z}}_{\widehat{p}} }= {\mathcal{O}}_{{ {\mathbf{Q}}_p }}\).
- Note that \({ {\mathbf{Z}}_{\widehat{p}} }\hookrightarrow{ {\mathbf{Q}}_p }\), \({\mathbf{Z}}\hookrightarrow{ {\mathbf{Z}}_{\widehat{p}} }\), \({\mathbf{Q}}\hookrightarrow{ {\mathbf{Q}}_p }\) are all dense embeddings.
-
Can be realized as an inverse imit exhibiting it as an adic completion \begin{align*} {\mathbf{Z}}_p := \varprojlim_{n} \,{\mathbf{Z}}/\left\langle{p^n}\right\rangle \subseteq \prod_n {\mathbf{Z}}/p^n{\mathbf{Z}} \end{align*}
-
One can take the algebraic closure to obtain \(\operatorname{cl}^{\mathrm{alg}} { {\mathbf{Q}}_p }\) and extend \(v_p\). The resulting space is not Cauchy-complete, so complete to obtain \({ {\mathbf{C}}_p }\). The result is algebraically closed by not locally compact.
-
Interesting properties of any ultrametric space \((X, d)\):
- All triangles are isosceles: if \(d(x,y) \neq d(y,z)\) then \(d(x, z) = \max\left\{{d(x,y), d(y,z)}\right\}\), so two sides must have the same length.
- Each point is the center of the disc it’s in: if \(x\in {\mathbb{D}}_r(p)\) then \({\mathbb{D}}_r(x) = {\mathbb{D}}_r(p)\).
- Every disc is clopen.
- Two discs are either equal or disjoint, and if disjoint then \(d({\mathbb{D}}_r(x), {\mathbb{D}}_s(y)) = d(x,y)\), i.e. the infimum is obtained by the distance between any two points in either disc.
- The induced topology is zero-dimensional, i.e. there is a basis of clopen sets.
- \(X\) is totally disconnected.
- \(\left\{{x_k}\right\}\) is Cauchy iff consecutive differences vanish, i.e. \(d(x_k, x_{k+1}) \to 0\).
- If \(X\) is Cauchy-complete, \(\sum a_k\) converges in \(X\) iff \(a_k\to 0\).
-
See spherically complete. a stronger notion than Cauchy completeness, needed for \(p{\hbox{-}}\)adic functional analysis.
In terms of geometry and valuations
See formal disk
-
Can be realized in terms of valuations; the p-adic integers form a ball: \begin{align*} { {\mathbf{Z}}_{\widehat{p}} }= \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}v_p(x) \geq 0}\right\} = \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert}_p \leq 1}\right\} = {\mathbb{B}}_p \end{align*}
- One can recover \({\mathbf{Z}}= \bigcap_{p\in \operatorname{Spec}{\mathbf{Z}}} {\mathbb{B}}_p = \bigcap_{p\in \operatorname{Spec}{\mathbf{Z}}} { {\mathbf{Z}}_{\widehat{p}} }\) as the intersection of all balls.
-
\({ {\mathbf{Z}}_{\widehat{p}} }\) is a local ring; the maximal ideal is an open interior of a ball: \begin{align*}{\mathfrak{m}}_{{ {\mathbf{Z}}_{\widehat{p}} }} = p{ {\mathbf{Z}}_{\widehat{p}} }= \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}v_p(x) > 0}\right\} = \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert}_p < 1}\right\} = {\mathbb{B}}_p^\circ= \left\{{{a\over b}\in {\mathbf{Q}}{~\mathrel{\Big\vert}~}a\in p{\mathbf{Z}}}\right\} \end{align*}
-
The units form a sphere: \begin{align*} { {\mathbf{Z}}_{\widehat{p}} }^{\times}= {\mathbb{S}}^1_p = \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}v_p(x) = 0}\right\} = \left\{{x\in { {\mathbf{Q}}_p }{~\mathrel{\Big\vert}~}{\left\lvert {x} \right\rvert}_p = 1}\right\} \end{align*}
Facts
-
There is a residue field \({ {\mathbf{Z}}_{\widehat{p}} }/p^n{ {\mathbf{Z}}_{\widehat{p}} }\cong {\mathbf{Z}}/p{\mathbf{Z}}\cong { \mathbf{F} }_p\) canonically.
-
\({ {\mathbf{Z}}_{\widehat{p}} }\) is a local but not complete ring.
-
\({\mathbf{Q}}\cap{ {\mathbf{Z}}_{\widehat{p}} }= { L_p {\mathbf{Z}}}\subseteq { {\mathbf{Q}}_p }\) is the localization at p.
-
\({ {\mathbf{Q}}_p }\cong { {\mathbf{Z}}_{\widehat{p}} }{ \left[ { \scriptstyle \frac{1}{p} } \right] }\) as algebras.
-
Topological properties of \({ {\mathbf{Q}}_p }\):
- Every two elements can be separated by ball of non-integer distance, and the norm only takes on integer values. Thus all balls are clopen and \({ {\mathbf{Q}}_p }\) is totally disconnected.
- \({ {\mathbf{Z}}_{\widehat{p}} }\) is compact and \({ {\mathbf{Q}}_p }\) is locally compact; \({ {\mathbf{Z}}_{\widehat{p}} }\) is totally bounded.
- \({ {\mathbf{Z}}_{\widehat{p}} }\) is a profinite group.
-
Open question: can characterize \({ {\mathbf{Z}}_{\widehat{p}} }= { L_p {\mathbf{Z}}}{ {}_{ \widehat{\left\langle{p}\right\rangle} } }\), i.e. first localize at the prime ideal \(p\) and then complete at the maximal ideal \(p\).
Expansions and series representations
- Series representations: \(a\in { {\mathbf{Z}}_{\widehat{p}} }\implies a = (a_n) = f_a(p) \coloneqq\sum_{i\geq 0} a_i x^i \mathrel{\Big|}_{x=p}\) where \(0\leq a_i \leq p^{i-1}\) and \(a_i \cong a_{i+1}\operatorname{mod}p^n\) are successive lifts of \(a_0\) modulo higher powers of \(p\).
- \(p{\hbox{-}}\)adic expansions: the series representation is redundant since \(a_i\) constrains \(a_{i+1}\) to only the \(p\) possible values in \({\mathbf{Z}}/p^{i+1}\) that are congruent to \(a_i \operatorname{mod}p^i\). So one can always write \(a_{i+1} = a_i + p^i b_i\) for \(b_i = {a_{i+1} - a_i \over p^n}\). The sequence \((b_i)\) with \(b_0 = a_1\) and \(b_i = {a_{i+1} - a_i \over p^n}\) is the p-adic expansion of \(p\).
Examples
-
Redundant representations in \({ {\mathbf{Z}}_{\widehat{p}} }\) for \(p=7\):
-
The more canonical \(p{\hbox{-}}\)adic expansions:
Lifts
It almost never happens that a variety in characteristic p can be lifted to characteristic zero together with its Frobenius endomorphism.
