Witt vector

We derive the ring of ${\displaystyle p}$-adic integers ${\displaystyle \mathbb {Z} _{p}}$ from the finite field ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ using a construction which naturally generalizes to the Witt vector construction.

The ring ${\displaystyle \mathbb {Z} _{p}}$ of ${\displaystyle p}$-adic integers can be understood as the projective limit of ${\displaystyle \mathbb {Z} /p^{i}\mathbb {Z} .}$ Specifically, it consists of the sequences ${\displaystyle (n_{0},n_{1},\ldots )}$ with ${\displaystyle n_{i}\in \mathbb {Z} /p^{i+1}\mathbb {Z} ,}$ such that ${\displaystyle n_{j}\equiv n_{i}{\bmod {p}}^{i+1}}$ for ${\displaystyle j\geq i.}$ That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections ${\displaystyle \mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} .}$

The elements of ${\displaystyle \mathbb {Z} _{p}}$ can be expanded as (formal) power series in ${\displaystyle p}$

${\displaystyle a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots ,}$

where ${\displaystyle a_{i}}$ are usually taken from the integer interval ${\displaystyle [0,p-1]=\{0,1,\ldots ,p-1\}.}$ Of course, this power series usually will not converge in ${\displaystyle \mathbb {R} }$ using the standard metric on the reals, but it will converge in ${\displaystyle \mathbb {Z} _{p},}$ with the ${\displaystyle p}$-adic metric. We will sketch a method of defining ring operations for such power series.

Letting ${\displaystyle a+b}$ be denoted by ${\displaystyle c}$, one might consider the following definition for addition:

${\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}+c_{1}p&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p&&{\bmod {p}}^{2}\\c_{0}+c_{1}p+c_{2}p^{2}&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p+(a_{2}+b_{2})p^{2}&&{\bmod {p}}^{3}\end{aligned}}}$

and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set ${\displaystyle [0,p-1].}$

There is a better coefficient subset of ${\displaystyle \mathbb {Z} _{p}}$which does yield closed formulas, the Teichmuller representatives: zero together with the ${\displaystyle (p-1)^{\text{th}}}$roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives ${\displaystyle [0,p-1]}$) as roots of ${\displaystyle x^{p-1}-1=0}$ through Hensel lifting, the ${\displaystyle p}$-adic version of Newton's method. For example, in ${\displaystyle \mathbb {Z} _{5},}$ to calculate the representative of ${\displaystyle 2,}$ one starts by finding the unique solution of ${\displaystyle x^{4}-1=0}$ in ${\displaystyle \mathbb {Z} /25\mathbb {Z} }$ with ${\displaystyle x\equiv 2{\bmod {5}}}$; one gets ${\displaystyle 7.}$ Repeat this in ${\displaystyle \mathbb {Z} /125\mathbb {Z} ,}$ with the conditions ${\displaystyle x^{4}-1=0}$ and ${\displaystyle x\equiv 7{\bmod {2}}5}$ gives ${\displaystyle 57,}$ and so on; the resulting Teichmüller representative is the sequence ${\displaystyle (2,7,57,\ldots ).}$ The existence of a lift in each step is guaranteed by the greatest common divisor ${\displaystyle (x^{p-1}-1,(p-1)x^{p-2})=1}$ in every ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} .}$

This algorithm shows that for every ${\displaystyle j\in [0,p-1]}$, there is exactly one Teichmuller representative with ${\displaystyle a_{0}=j}$, which we denote ${\displaystyle \omega (j).}$ Indeed, this defines the Teichmüller character ${\displaystyle \omega$ :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}} satisfying ${\displaystyle m\circ \omega =\mathrm {id} _{\mathbb {F} _{p}}}$ if we denote ${\displaystyle m:\mathbb {Z} _{p}\to \mathbb {Z} _{p}/p\mathbb {Z} _{p}\cong \mathbb {F} _{p}.}$ Note that ${\displaystyle \omega }$ is not additive, as the sum need not be a representative. Despite this, if ${\displaystyle \omega (k)=\omega (i)+\omega (j){\bmod {p}}}$ in ${\displaystyle \mathbb {Z} _{p},}$ then ${\displaystyle i+j=k}$ in ${\displaystyle \mathbb {F} _{p}.}$

Because of this one-to-one correspondence given by ${\displaystyle \omega }$, one can expand every ${\displaystyle p}$-adic integer as a power series in ${\displaystyle p}$ with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as ${\displaystyle \omega (t_{0})=t_{0}+t_{1}p^{1}+t_{2}p^{2}+\cdots .}$ Then, if one has some arbitrary ${\displaystyle p}$-adic integer of the form ${\displaystyle x=x_{0}+x_{1}p^{1}+x_{2}p^{2}+\cdots ,}$ one takes the difference ${\displaystyle x-\omega (x_{0})=x'_{1}p^{1}+x'_{2}p^{2}+\cdots ,}$ leaving a value divisible by ${\displaystyle p}$. Hence, ${\displaystyle x-\omega (x_{0})=0{\bmod {p}}}$. The process is then repeated, subtracting ${\displaystyle \omega (x'_{1})p}$ and proceed likewise. This yields a sequence of congruences

${\displaystyle {\begin{aligned}x&\equiv \omega (x_{0})&&{\bmod {p}}\\x&\equiv \omega (x_{0})+\omega (x'_{1})p&&{\bmod {p}}^{2}\\&\cdots \end{aligned}}}$

So that

${\displaystyle x\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}$

and ${\displaystyle i'>i}$ implies:

${\displaystyle \sum _{j=0}^{i'}\omega ({\bar {x}}_{j})p^{j}\equiv \sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}{\bmod {p}}^{i+1}}$

for

${\displaystyle {\bar {x}}_{i}:=m\left({\frac {x-\sum _{j=0}^{i-1}\omega ({\bar {x}}_{j})p^{j}}{p^{i}}}\right).}$

Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than ${\displaystyle \{0,\ldots ,p-1\}}$. It is clear that

${\displaystyle \sum _{j=0}^{\infty }\omega ({\bar {x}}_{j})p^{j}=x,}$

since

${\displaystyle p^{i+1}\mid x-\sum _{j=0}^{i}\omega ({\bar {x}}_{j})p^{j}}$

for all ${\displaystyle i}$ as ${\displaystyle i\to \infty ,}$ so the difference tends to 0 with respect to the ${\displaystyle p}$-adic metric. The resulting coefficients will typically differ from the ${\displaystyle a_{i}}$ modulo ${\displaystyle p^{i}}$ except the first one.

The Teichmuller coefficients have the key additional property that ${\displaystyle \omega ({\bar {x}}_{i})^{p}=\omega ({\bar {x}}_{i}),}$ which is missing for the numbers in ${\displaystyle [0,p-1]}$. This can be used to describe addition, as follows. Since the Teichmüller character is not additive, ${\displaystyle c_{0}=a_{0}+b_{0}}$ is not true in ${\displaystyle \mathbb {Z} _{p}}$. But it holds in ${\displaystyle \mathbb {F} _{p},}$ as the first congruence implies. In particular,

${\displaystyle c_{0}^{p}\equiv (a_{0}+b_{0})^{p}{\bmod {p}}^{2},}$

and thus

${\displaystyle c_{0}-a_{0}-b_{0}\equiv (a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv {\binom {p}{1}}a_{0}^{p-1}b_{0}+\cdots +{\binom {p}{p-1}}a_{0}b_{0}^{p-1}{\bmod {p}}^{2}.}$

Since the binomial coefficient ${\displaystyle {\binom {p}{i}}}$ is divisible by ${\displaystyle p}$, this gives

${\displaystyle c_{1}\equiv a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}{\bmod {p}}.}$

This completely determines ${\displaystyle c_{1}}$ by the lift. Moreover, the congruence modulo ${\displaystyle p}$ indicates that the calculation can actually be done in ${\displaystyle \mathbb {F} _{p},}$ satisfying the basic aim of defining a simple additive structure.

For ${\displaystyle c_{2}}$ this step is already very cumbersome. Write

${\displaystyle c_{1}=c_{1}^{p}\equiv \left(a_{1}+b_{1}-a_{0}^{p-1}b_{0}-{\frac {p-1}{2}}a_{0}^{p-2}b_{0}^{2}-\cdots -a_{0}b_{0}^{p-1}\right)^{p}{\bmod {p}}.}$

Just as for ${\displaystyle c_{0},}$ a single ${\displaystyle p}$th power is not enough: one must take

${\displaystyle c_{0}=c_{0}^{p^{2}}\equiv (a_{0}+b_{0})^{p^{2}}.}$

However, ${\displaystyle {\binom {p^{2}}{i}}}$ is not in general divisible by ${\displaystyle p^{2},}$ but it is divisible when ${\displaystyle i=pd,}$ in which case ${\displaystyle a^{i}b^{p^{2}-i}=a^{d}b^{p-d}}$ combined with similar monomials in ${\displaystyle c_{1}^{p}}$ will make a multiple of ${\displaystyle p^{2}}$.

At this step, it becomes clear that one is actually working with addition of the form

${\displaystyle {\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}^{p}+c_{1}p&\equiv a_{0}^{p}+a_{1}p+b_{0}^{p}+b_{1}p&&{\bmod {p}}^{2}\\c_{0}^{p^{2}}+c_{1}^{p}p+c_{2}p^{2}&\equiv a_{0}^{p^{2}}+a_{1}^{p}p+a_{2}p^{2}+b_{0}^{p^{2}}+b_{1}^{p}p+b_{2}p^{2}&&{\bmod {p}}^{3}\end{aligned}}}$

This motivates the definition of Witt vectors.

Let ${\displaystyle X}$ be a Witt vector and define

${\displaystyle f_{X}(t)=\prod _{n\geq 1}(1-X_{n}t^{n})=\sum _{n\geq 0}A_{n}t^{n}}$

For ${\displaystyle n\geq 1}$ let ${\displaystyle {\mathcal {I}}_{n}}$ denote the collection of subsets of ${\displaystyle \{1,2,\ldots ,n\}}$ whose elements add up to ${\displaystyle n}$. Then

${\displaystyle A_{n}=\sum _{I\in {\mathcal {I}}_{n}}(-1)^{|I|}\prod _{i\in I}{X_{i}}.}$

We can get the ghost components by taking the logarithmic derivative:

${\displaystyle {\begin{aligned}-t{\frac {d}{dt}}\log f_{X}(t)&=-t{\frac {d}{dt}}\sum _{n\geq 1}\log(1-X_{n}t^{n})\\&=t{\frac {d}{dt}}\sum _{n\geq 1}\sum _{d\geq 1}{\frac {X_{n}^{d}t^{nd}}{d}}\\&=\sum _{n\geq 1}\sum _{d\geq 1}nX_{n}^{d}t^{nd}\\&=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}t^{m}\\&=\sum _{m\geq 1}X^{(m)}t^{m}\end{aligned}}}$

Now we can see ${\displaystyle f_{Z}(t)=f_{X}(t)f_{Y}(t)}$ if ${\displaystyle Z=X+Y}$. So that

${\displaystyle C_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i},}$

if ${\displaystyle A_{n},B_{n},C_{n}}$ are the respective coefficients in the power series ${\displaystyle f_{X}(t),f_{Y}(t),f_{Z}(t)}$. Then

${\displaystyle Z_{n}=\sum _{0\leq i\leq n}A_{n}B_{n-i}-\sum _{I\in {\mathcal {I}}_{n},I\neq \{n\}}(-1)^{|I|}\prod _{i\in I}{Z_{i}}.}$

Since ${\displaystyle A_{n}}$ is a polynomial in ${\displaystyle X_{1},\ldots ,X_{n}}$ and likewise for ${\displaystyle B_{n}}$, we can show by induction that ${\displaystyle Z_{n}}$ is a polynomial in ${\displaystyle X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.}$

If we set ${\displaystyle W=XY}$ then

${\displaystyle -t{\frac {d}{dt}}\log f_{W}(t)=-\sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}.}$

But

${\displaystyle \sum _{m\geq 1}X^{(m)}Y^{(m)}t^{m}=\sum _{m\geq 1}\sum _{d|m}dX_{d}^{m/d}\sum _{e|m}eY_{e}^{m/e}t^{m}}$.

Now 3-tuples ${\displaystyle {m,d,e}}$ with ${\displaystyle m\in \mathbb {Z} ^{+},d|m,e|m}$ are in bijection with 3-tuples ${\displaystyle {d,e,n}}$ with ${\displaystyle d,e,n\in \mathbb {Z} ^{+}}$, via ${\displaystyle n=m/[d,e]}$ (${\displaystyle [d,e]}$ is the least common multiple), our series becomes

${\displaystyle \sum _{d,e\geq 1}de\sum _{n\geq 1}\left(X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{n}=-t{\frac {d}{dt}}\log \prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}}$

So that

${\displaystyle f_{W}(t)=\prod _{d,e\geq 1}\left(1-X_{d}^{\frac {[d,e]}{d}}Y_{e}^{\frac {[d,e]}{e}}t^{[d,e]}\right)^{\frac {de}{[d,e]}}=\sum _{n\geq 0}D_{n}t^{n},}$

where ${\displaystyle D_{n}}$ are polynomials of ${\displaystyle X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.}$ So by similar induction, suppose

${\displaystyle f_{W}(t)=\prod _{n\geq 1}(1-W_{n}t^{n}),}$

then ${\displaystyle W_{n}}$ can be solved as polynomials of ${\displaystyle X_{1},\ldots ,X_{n},Y_{1},\ldots ,Y_{n}.}$