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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

Let pp be a fixed prime number.

Definition

Let kk be a ring. Then the assignation

Λ k:{M k R[[t]] R 1+tR[[t]]\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 RR and constant term 11 is an affine k-group. As a k-functor? Λ k\Lambda_k is equivalent to O k O_k^\mathbb{N}.

Remark
  1. There is an exact sequence 0Λ k (n+1)Λ k (n)α k00\to \Lambda_k^{(n+1)}\to \Lambda_k^{(n)}\to \alpha_k\to 0

  2. We have Λ k=lim nΛ k/Λ k (n+1)\Lambda_k=lim_n \Lambda_k/\Lambda_k^{(n+1)}.

  3. Each Λ k/Λ k (n+1)\Lambda_k /\Lambda_k^{(n+1)} is is an nn-fold extension of the additive group α k\alpha_k.

  4. If kk is a field then Λ k\Lambda_k is a unipotent group.

Remark

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

ϕ F:{O k >0Λ k (a n)Π nF(a nt n)\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=k=\mathbb{Q} and F(t)=exp(t)F(t)=exp(-t) we have F(at)F(bt)=F(A(a+b)t)F(at)F(bt)=F(A(a+b)t) and ϕ F\phi_F is an isomorphism ϕ F:α k >0Λ k\phi_F:\alpha_k^{\mathbb{N}_{\gt 0}}\to \Lambda_k.

If kk is a field with characteristic pp it is not possible an FF such that F(at)F(bt)=F(ct)F(at)F(bt)=F(ct). However there is always a formula F(at)F(bt)=Π i>0F(λ i(a,b)t iF(at)F(bt)=\Pi_{i\gt 0} F(\lambda_i(a,b)t^i where λ i(X,Y)k[X,Y]\lambda_i(X,Y)\in k[X, Y]. This is verified by Möbius inversion.

Remark

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

exp(t)=Π n(1t n) μ(n)/nexp(-t)=\Pi_n (1-t^n)^{\mu(n)/n}
Definition and Remark

The Artin-Hasse exponential is defined by the morphism

E:{O (p) Λ (p) ((a n),t)Π n0F(a nt p n)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,,a i,b 0,,b i),t)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 (p)[X 0,,X i,Y 0,,Y i]s_i\in \mathbb{Z}_{(p)}[X_0,\dots, X_i,Y_0,\dots, Y_i]

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

P(t)=Π (n,p)=1E((a n),t n)P(t)=\Pi_{(n,p)=1} E((a_n),t^n)

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

Proposition

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

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

Revised on May 27, 2012 13:39:07 by Stephan Alexander Spahn (79.227.168.80)