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Section 4.1 \(p\)-divisible groups (Sachi)

Why study \(p\)-divisible groups (Jacob Stix).

  1. Analyse local \(p\)-adic galois action on \(p\)-torsion of elliptic curves, Serre's open image theorem.
    \begin{equation*} \phi_l \colon G_K \to \Aut \lb l \rb \end{equation*}
    Surjective for almost all \(l\text{.}\)
  2. Tool for representing \(p\)-adic cohomology, e.g \(p\)-adic hodge theory.
  3. Describe local properties of moduli spaces of abelian varieties which map to moduli spaces of \(p\)-divisible groups which can be described by semilinear algebra (Serre-Tate).
  4. Explicit local CFT via Lubin-Tate formal groups describing wildly ramified abelian extensions.
  5. The true fundamental group in characteristic \(p\) must include infinitesimal group schemes, \(p\)-divisible groups enter through their tate modules.
Detour, schemes.

There is an (anti)-equivalence of categories

\begin{equation*} \{\text{ring}\} \leftrightarrow\{\text{affine schemes}\}\text{.} \end{equation*}

Moral whatever a scheme is the data of a ring is enough to specify it + homs

\begin{equation*} \Hom_{Ring} (B,A) \leftrightarrow \Hom_{Aff}(\Spec A, \Spec B) \end{equation*}

to specify a base field or base ring play a similar game with \(R\)-algebras and \(R\)-schemes.

Yoneda, schemes are functors: Let \(R\lb T_1,\ldots,T_n\rb \) be a polynomial ring over \(R\text{,}\) we want solutions to

\begin{equation*} f_1= f_2 = \cdots = f_m = 0 \end{equation*}

with coefficients in \(A\) this is asking for a map

\begin{equation*} R\lb T_1, \ldots ,T_n] /(f_i) \to A \end{equation*}

same as

\begin{equation*} \Hom_{R-alg} (R[T_1,\ldots, T_n] /(f_i), A) \end{equation*}

functor \(A\) to this is a functor from \(R\)-algs to sets.

Definition 4.1.1.

For any affine scheme \(A = \Spec B\) we attach a functor \(h_X\) from \(\mathrm{Sch}^\op\) to sets, sending \(\Spec S \mapsto \Hom_{\mathrm{Sch}} (\Spec S, X) = \Hom_{Ring} (B,S) = h_X(\Spec S)\text{.}\) spec S points of X

Example 4.1.2.
\begin{equation*} \aff^n = \Spec \ZZ \lb T_1,\ldots, T_n\rb \end{equation*}
\begin{equation*} \aff^n(T) = \Hom_{\mathrm{Sch}} (T, \aff^n) = \Hom_{Ring}(\ZZ[T_1,\ldots, T_n], S)\cong S^n \end{equation*}
Example 4.1.3.
\begin{equation*} E \colon \Spec k[x,y] / (y^2 - (x^3 + ax +b)),\,k = \QQ \end{equation*}

\(E(\QQ(i)) = \QQ(i)\) points, choosing \(x,y\) satisfying weierstrass equation.

Suppose \(h_X \colon \mathrm{Sch}^\op \to R\) factors through \(\mathrm{Grp} \to \mathrm{Set}\) then this is a group scheme.

Example 4.1.4.
\begin{equation*} \mathbf G_a = \Spec k[t] \end{equation*}
\begin{equation*} S\mapsto \Hom(k[t], S) \cong (S,+) \end{equation*}
Example 4.1.5.
\begin{equation*} \mathbf G_m = \Spec k[t,t\inv] \end{equation*}
\begin{equation*} S\mapsto \Hom(k[t,t\inv], S) \cong (S^\times,\cdot) \end{equation*}
Example 4.1.6.
\begin{equation*} \mu_n = \Spec k[t]/(t^n -1) \end{equation*}
Example 4.1.7.
\begin{equation*} \alpha_{p^n} = \Spec k[t]/(t^{p^n}) \end{equation*}

char \(k = p\)

Cartier Duality \(G\) is a finite group scheme \(/R\) there is a dual

\begin{equation*} G^* (T) = \Hom (G_T, \mathbf G_m) \end{equation*}

\(R\)-scheme \(T\)

\begin{equation*} G \cong (G^*)^* \end{equation*}
Example 4.1.8.
\begin{equation*} \mu_{p^n} \leftrightarrow \ZZ/p^n \end{equation*}
Definition 4.1.9.

Let \(p\) be a prime and \(h\) a non-negative integer. A \(p\)-divisible group of height \(h\) is an inductive system

\begin{equation*} (G_v, i_v) \end{equation*}

where each \(G_v\) is a group scheme \(/R\) of size \(p^{vh}\)

\begin{equation*} i_v \colon G_v \to G_{v+1} \end{equation*}

identifies \(G_v\) with kernel of multiplication by \(p^v\text{.}\)

\begin{equation*} 0 \to G_v \xrightarrow{i_v} G_{v+1} \xrightarrow{[p^v]} G_{v+1} \end{equation*}
Remark 4.1.10.

We can show that \(G_\mu,G_v\) are two levels then

\begin{equation*} 0 \to G_\mu \xrightarrow{i_{\mu, v}} G_{\mu +v } \xrightarrow{[p^\mu]} G_{\mu,v} \end{equation*}

so

\begin{equation*} 0 \to G_\mu \to G_{\mu + v} \to G_v \to 0\text{.} \end{equation*}

The connected etale sequence

A finite flat group scheme \(G\) over a henselian local ring \(R\) admitsa (functorial) decomposition

\begin{equation*} 0 \to G^\circ \to G \to G^\et \to 0 \end{equation*}

connected and etale

There is an equivalence of categories between finite etale gp scheme \(/R\) and its continuous \(\absgal k\) modules when \(R = k\) is a field.

Definition 4.1.11.

An \(n\)-dimensional formal lie group \(/R\) is the formal power series ring

\begin{equation*} A = R\lb \lb x_1, \ldots, x_n \rb\rb \end{equation*}

with a suitable co-multiplication structure.

\begin{equation*} m^* \colon A \to A\widehat \otimes A \end{equation*}
\begin{equation*} m^*(X_i) = (f_i(Y,Z)) \end{equation*}

require

  1. \begin{equation*} F(X,0) = F(0,X) = X \end{equation*}
  2. \begin{equation*} F(X,F(Y,Z)) = F(F(Y,Z),X) = X \end{equation*}
  3. \begin{equation*} F(Y,Z) = F(Z,Y) \end{equation*}

Let \(\psi\) denote multiplication by \(p\) in \(A\) then \(A\) is divisible if \(\psi\) is an isogeny (surj. with finite kernel). Alternatively \(A\) is a finite free \(\psi (A)\)-module.

Example 4.1.13.
\begin{equation*} \mathbf G_m(p) , \, F(X)= Y+Z+YZ \end{equation*}
Example 4.1.14.

\(E\) ordinary elliptic curve \(/\overline{\FF}_p\)

\begin{equation*} E[p] (\overline \FF_p) \end{equation*}

is non-empty

\begin{equation*} E[p] = E[p]^\circ \times E[p]^\et\text{.} \end{equation*}

etale group schemes over alg. closed fields are constant

\begin{equation*} E = E[p]^\circ \times A \end{equation*}

It can't be entirely etale \(\lb p \rb\) would be etale but this induces the 0 map on tangent space so \(E\lb p \rb^\circ \ne 0\text{.}\)

\begin{equation*} |E\lb p \rb | = p^2 \end{equation*}

so each order \(p\text{.}\)

\begin{equation*} A = \ZZ/p \end{equation*}

\(E\) is cartier self dual

\begin{equation*} A^* = \mu_p = E[p]^\circ \end{equation*}

Induct for \(E\lb p^n\rb \text{.}\)