Author: Eiko

Tags: coleman integration, isocrystals, rigid geometry

Time: 2024-09-12 22:12:56 - 2025-01-15 09:49:54 (UTC)

For simplicity here only the affine (affinoid) case is written here but the framework work with general case.

Unipotent Isocrystals

• A unipotent isocrystal on $\bar{A}$ is an $A_K^{†}$-module $M$ and an integrable connection

  $$\mathrm{\nabla }:M\to M{\otimes }_{A_K^{†}}({\mathrm{\Omega }}_{A^{†}}^1\otimes K)$$

  obtained by iterated extensions of trivial connections $(A_K^{†},d)$. In this case the extension is actually a free module.

• A morphism of unipotent isocrystals is a morphism of $A^{†}$-modules $f:M\to N$ such that ${\mathrm{\nabla }}_N\circ f=(f\otimes {\mathrm{\Omega }}_K^1)\circ {\mathrm{\nabla }}_M$. Here ${\mathrm{\Omega }}_K^1={\mathrm{\Omega }}_{A^{†}}^1\otimes K$.

• The category of unipotent isocrystals on $\bar{A}$ is denoted by $\mathcal{U}(\bar{A})$, which as a category only depends on $\bar{A}$ independent of the choice of lift.

Example of Notations

• ${\mathcal{T}}_n$ is the Tate algebra

  $${\mathcal{T}}_n:=\{\sum a_I{t}^I:a_I\in R,|a_I|\to 0\}$$

• $A^{†}$ is the weakly complete finitely generated (wcfg) algebra, a surjective image of Tate algebra

  $$A^{†}:={\mathcal{T}}_n/(f_1,\dots ,f_m)$$

• ${\mathcal{T}}_n^{†}/\pi =\kappa [t_1,\dots ,t_m]$ is the polynomial ring obtained under reduction, so is $\bar{A}=A^{†}/\pi =\kappa [t_1,\dots ,t_m]/(\overline{f_1},\dots ,\overline{f_m})$ a finitely generated $\kappa$-algebra since it is a quotient of Tate algebra.

• The overconvergent differential module ${\mathrm{\Omega }}_{A^{†}}^1$ is not the algebraic module of differential, rather it is the overconvergent module of differentials, which is a $A^{†}$-module given by

  $${\mathrm{\Omega }}_{A^{†}}^1:=\underset{i=1}{\overset{n}{⨁}}{A}^{†}\mathrm{d}{t}_i/(d{f}_1,\dots ,d{f}_m).$$

  The point is that we need to take limit, and in algebraic differential, $\mathrm{d}$ does not cooperate with limit.

Example

Any rank two unipotent isocrystal $M$ is an extension in ${\mathrm{Ext}}_{\mathcal{U}}^1(1,1)$. This splits due to freeness (non-canonical), so it is isomorphic to some $({\mathcal{O}}^2,\mathrm{\nabla })$. Let’s see what happens inside, writing out

If we denote the injection ${\mathcal{O}}_X\to M$ giving out the first frame section $e_1$, and choose a non-canonical section that inverts the surjection $M\to {\mathcal{O}}_X$ which maps $e_2$, then the vertical arrows says

• $\mathrm{\nabla }(e_1)=0$ because the injection commutative diagram

• $\mathrm{\nabla }(e_2)=0$ when you mod out $e_1$, so $\mathrm{\nabla }(e_2)=\omega e_1$ for some $\omega$.

• The requirement that $\mathrm{\nabla }$ is a flat connection gives $\mathrm{\nabla }(\omega e_1)=d\omega \wedge e_1=0$, so $\omega$ is a closed form.

We have found out that $\mathrm{\nabla }=d\cdot I_2+\mathrm{\Lambda }$ where

$$\mathrm{\Lambda }=\left(\begin{array}{cc}0& \omega \\ 0& 0\end{array}\right)$$

In fact this correspondence gives a bijection

$${\mathrm{Ext}}_{\mathcal{U}(\bar{A})}^1(1,1)\cong H_{MW}^1(\bar{A}/K),[\mathrm{\Lambda }]\mapsto [\omega ].$$

In Detail

Let’s consider why the above morphism $\phi$ is an isomorphism in detail. This means

• It is well-defined, i.e. isomorphic unipotent isocrystals give the same cohomology class.

• It is surjective, luckily this is obvious because $\omega$ can be any closed form.

• It is injective, i.e. if $\omega =\mathrm{d}f$ is exact, then the corresponding $\mathrm{\Lambda }$ gives a connection that splits, it is isomorphic to a connection whose matrix is semi-simple.

  When split happens, we have a map $\mathcal{O}\to M:e_2\mapsto a{e}_1+e_2$, where

  $$\begin{array}{rl}a{e}_1+e_2& \leftarrow e_2\\ (\mathrm{d}a+\omega )e_1& \leftarrow 0\end{array}$$

  i.e. we will have $\omega =-\mathrm{d}a$. Conversely if $\omega$ is exact we can construct the split. Thus $\omega$ is exact iff the extension splits.

Tensor Structure

The category $\mathcal{U}(\bar{A})$ is a rigid abelian tensor category, and it can be made into a Tannakian category with the fibre functors. Let $x\in X_{\kappa }(\kappa )$ be a rational point in reduction, the fibre functor at $x$ is a functor from $\mathcal{U}(\bar{A})$ into $K$-vector spaces of flat sections over a local residue disk $U_x$ of $x$,

$${\omega }_x:\mathcal{U}(\bar{A})\to {\mathrm{Vec}}_K,{\omega }_x[(M,\mathrm{\nabla })]:=\{v\in M(U_x),\mathrm{\nabla }v=0\}.$$

• Overconvergence is required to make $\dim M(U_x)=\mathrm{rank}M$. Finding horizontal sections for a unipotent connection is the same as iterated integration, which is feasible because of overconvergence.

• ${\omega }_x$ can also be viewed as a pullback $x^{\ast }$ to an isocrystal on $\mathrm{Spec}(\kappa )$.

• The category $\mathcal{U}(\bar{A})$ together with fibre functor ${\omega }_x$ determines an affine pro-algebraic fundamental group

  $$G=G_x={\pi }_1(\mathcal{U}(\bar{A}),{\omega }_x)$$

  whose functor of points over any $K$-algebra $F$, $G(F)$ is given by

  $$G(F)={\mathrm{Aut}}_{\otimes }({\omega }_x\otimes F),$$

  here ${\omega }_x\otimes F$ is the functor $(\otimes F)\circ {\omega }_x$, The group ${\mathrm{Aut}}_{\otimes }({\omega }_x\otimes F)$ is the group of natural automorphisms for the functor $\alpha :{\omega }_x\otimes F\to {\omega }_x\otimes F$ that commutes with the tensor structure, i.e. ${\alpha }_{M\otimes N}={\alpha }_M\otimes {\alpha }_N$ and ${\alpha }_{\mathcal{O}}=\mathrm{id}$.

• Written explicitly,

  $$\begin{array}{rl}{G}_x(F)& ={\mathrm{Aut}}_{\otimes }({\omega }_x\otimes F)\\ & =\{\mathrm{\forall }M\in \mathcal{U}(\bar{A})\\ & ,{\alpha }_M:{\omega }_x(M)\otimes F\simeq {\omega }_x(M)\otimes F\in \mathrm{NatIsom}({}_F\mathbf{Mod})\\ & ,{\alpha }_{M\otimes N}={\alpha }_M\otimes {\alpha }_N\\ & ,{\alpha }_1=\mathrm{id}\\ & \}\end{array}$$

• There is an equivalence of categories (similar to Riemann-Hilbert correspondence) of finite dimensional $K$-representations of $G$ and unipotent isocrystals on $\bar{A}$

  $${\mathrm{Rep}}_K({\pi }_1(\mathcal{U}(\bar{A}),{\omega }_x))\simeq \mathcal{U}(\bar{A})$$

Lie Algebra Of The Fundamental Group

To extract the Lie algebra of $G(\cdot )$, which is the tangent plane $T_{e}G$ of $G$ at identity, we can use the fact that

$$T_{x}X={\mathrm{Hom}}_{\mathrm{Spec}(K[\epsilon ]/\epsilon )\mapsto x}(\mathrm{Spec}(K[\epsilon ]/{\epsilon }^2),X).$$

Let subscript $\otimes$ denote all the tensor laws ${\alpha }_{M\otimes N}={\alpha }_M\otimes {\alpha }_N$ and ${\alpha }_1=\mathrm{id}$, we have

$$\begin{array}{rl}\mathfrak{g}& =T_{e}G\\ & ={\mathrm{Hom}}_{\mathrm{Spec}(K[\epsilon ]/\epsilon )\mapsto e}(\mathrm{Spec}(K[\epsilon ]/{\epsilon }^2),G)\\ & ={[G(K[\epsilon ]/{\epsilon }^2)\stackrel{{\mathrm{ev}}_{\epsilon =0}}{\to }G(K)]}^{-1}(e)\\ & =\ker [G(K[\epsilon ]/{\epsilon }^2)\stackrel{{\mathrm{ev}}_{\epsilon =0}}{\to }G(K)]\\ & =\ker [\{{\alpha }_M:{\omega }_x(M)\otimes K[\epsilon ]\to {\omega }_x(M)\otimes K[\epsilon ]{\}}_{\otimes }\stackrel{{\mathrm{ev}}_{\epsilon =0}}{\to }G(K)]\\ & ={\{{\alpha }_M=\left(\begin{array}{cc}{1}_{{\omega }_x(M)}& 0\\ \epsilon {\beta }_M& \epsilon 1_{{\omega }_x(M)}\end{array}\right)\}}_{\otimes }\\ & ={\{{\alpha }_M=1_{{\omega }_x(M)\otimes K[\epsilon ]}+\epsilon {\beta }_M:{\beta }_M\in \mathrm{End}({\omega }_x(M))\}}_{\otimes }\\ & ={\{{\beta }_M\in \mathrm{End}({\omega }_x(M))\}}_{\otimes }\end{array}$$

• By standard arguments similar to differentiation, we see that tensor laws ${\alpha }_{M\otimes N}={\alpha }_M\otimes {\alpha }_N$ and ${\alpha }_1=\mathrm{id}$ are equivalent to

  • ${\beta }_{M\otimes N}={\beta }_M\otimes 1_{{\omega }_x(N)}+1_{{\omega }_x(M)}\otimes {\beta }_N$ and

  • ${\beta }_1=0$.

• Since $G$ is unipotent, the Lie algebra $\mathfrak{g}$ is nilpotent.

  The exponential map $\mathfrak{g}\to G$ is therefore algebraically defined.

  This gives us an equivalence of categories of algebraic representations of $G$ and continuous representations of $\mathfrak{g}$.

  $$\mathrm{Rep}(G)\simeq \mathrm{Rep}(\mathfrak{g})$$

The Frobenius Invariant Path

Given two $\kappa$-rational points $x,z\in X_{\kappa }$, the space of functor isomorphisms is what we abstractly call a path space

$$P_{x,z}:={\mathrm{Isom}}_{\otimes }({\omega }_x,{\omega }_z)$$

which is a homogeneous space for the fundamental group $G$, has a natural action of $G_x^{op}\times G_z$, or left action by $G_z$ and right action by $G_x$. These ’path’s gives analytic continuations identifying horizontal sections in different local residue disks $M(U_x{)}^{\mathrm{\nabla }}$ and $M(U_z{)}^{\mathrm{\nabla }}$.

The Frobenius invariant path space is the subspace of $P_{x,z}$ that is fixed by the action of the Frobenius automorphism $\phi$ of $\bar{A}$.

Reference

Besser, Heidelberg Lectures on Coleman Integration