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Demazure, lectures on p-divisible groups, III.1 the Artin-Hasse exponential series

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

Let $p$ be a fixed prime number.

Definition

Let $k$ be a ring. Then the assignation

$${\Lambda }_k:\{\begin{array}{lll}{M}_k& \to & R[[t]]\\ R& \to & 1+\mathrm{tR}[[t]]\end{array}$$\Lambda_k:\begin{cases} M_k&\to&R[[t]] \\ R&\to&1+tR[[t]] \end{cases}

sending a k-ring? to the multiplicative group of formal power series with coefficients in $R$ and constant term $1$ is an affine k-group. As a k-functor? ${\Lambda }_k$ is equivalent to $O_k^{\mathbb{N}}$.

Remark

1. There is an exact sequence $0\to {\Lambda }_k^{(n+1)}\to {\Lambda }_k^{(n)}\to {\alpha }_k\to 0$

2. We have ${\Lambda }_k={\lim }_n{\Lambda }_k/{\Lambda }_k^{(n+1)}$.

3. Each ${\Lambda }_k/{\Lambda }_k^{(n+1)}$ is is an $n$-fold extension of the additive group ${\alpha }_k$.

4. If $k$ is a field then ${\Lambda }_k$ is a unipotent group.

Remark

For $F=(1-t+\cdots )\in \Lambda (k)$ there is an isomorphism of $k$-schemes

$${\varphi }_F:\{\begin{array}{l}{O}_k^{{\mathbb{N}}_{>0}}\to {\Lambda }_k\\ (a_n)\mapsto {\Pi }_{n}F(a_n{t}^n)\end{array}$$\phi_F:\begin{cases} O_k^{\mathbb{N}_{\gt 0}}\to \Lambda_k \\ (a_n)\mapsto \Pi_n F(a_n t^n) \end{cases}

If $k=\mathbb{Q}$ and $F(t)=\exp (-t)$ we have $F(\mathrm{at})F(\mathrm{bt})=F(A(a+b)t)$ and ${\varphi }_F$ is an isomorphism ${\varphi }_F:{\alpha }_k^{{\mathbb{N}}_{>0}}\to {\Lambda }_k$.

If $k$ is a field with characteristic $p$ it is not possible an $F$ such that $F(\mathrm{at})F(\mathrm{bt})=F(\mathrm{ct})$. However there is always a formula $F(\mathrm{at})F(\mathrm{bt})={\Pi }_{i>0}F({\lambda }_i(a,b)t^i$ where ${\lambda }_i(X,Y)\in k[X,Y]$. This is verified by Möbius inversion.

Remark

(Möbius inversion) Let $\mu$ be the Möbius function. Then Möbius inversion gives

$$\exp (-t)={\Pi }_n(1-t^n{)}^{\mu (n)/n}$$exp(-t)=\Pi_n (1-t^n)^{\mu(n)/n}

Definition and Remark

The Artin-Hasse exponential is defined by the morphism

$$E:\{\begin{array}{l}{O}_{{\mathbb{Z}}_{(p)}}^{\mathbb{N}}\to {\Lambda }_{{\mathbb{Z}}_{(p)}}\\ ((a_n),t)\mapsto {\Pi }_{n\ge 0}F(a_n{t}^{p^n})\end{array}$$E:\begin{cases} O^\mathbb{N}_{\mathbb{Z}_{(p)}}\to \Lambda_{\mathbb{Z}_{(p)}} \\ ((a_n),t)\mapsto \Pi_{n\ge 0}F(a_n t^{p^n}) \end{cases}

We have $E((a_i),t)E((b_i),t)=E((S_i(a_0,\dots ,a_i,b_0,\dots ,b_i),t)$ where $s_i\in {\mathbb{Z}}_{(p)}[X_0,\dots ,X_i,Y_0,\dots ,Y_i]$

$P\in \Lambda (R)$, $R\in M_{{\mathbb{Z}}_{(p)}}$ can be uniquely written as

$$P(t)={\Pi }_{(n,p)=1}E((a_n),t^n)$$P(t)=\Pi_{(n,p)=1} E((a_n),t^n)

where $(a_n)\in R^{\mathbb{N}}$

Proposition

The ${\mathbb{Z}}_{(p)}$-group ${\Lambda }_{{\mathbb{Z}}_{(p)}}$ is isomorphic to the n/(n,p)=1-power of the subgroup image of $E$.

By base-change a similar statement applies to ${\Lambda }_{{𝔽}_p}$. This shows that the Artin-Hasse exponential plays over ${𝔽}_p$ a similar role as the usual exponential over $\mathbb{Q}$.