Author: Eiko

Time: 2024-09-13 14:27:14 - 2024-09-19 14:57:37 (UTC)

This is a note for helping me to understand the picture.

Sections To Connections

Start with $n$ sections $s_1,\dots ,s_n$ on a family of abelian varieties $\pi :A\to S$. Assume that they are linearly independent over $\mathbb{Z}$. We want to know on what points, or how many points can they be degenerate.

Integration map

Let $X$ be a curve over ${\mathbb{Q}}_p$, we can form the integration map

$$X({\mathbb{Q}}_p)\times X({\mathbb{Q}}_p)\to H^0(X,{\mathrm{\Omega }}_X^1{)}^{\ast },(x,y)\mapsto {\int }_x^y$$

Family of Curves

For a family of curves $\pi :X\to S$, fix a section $s_0:S\to X$, somehow the sections $s_i$ give a locally analytic map over $S({\mathbb{Z}}_p)$

$$X({\mathbb{Z}}_p)\to (R^1{\pi }_{\ast }{\mathcal{O}}_X)({\mathbb{Q}}_p)$$

which is fibrewise the integration map

$$X({\mathbb{Q}}_p)\to H^0(X,{\mathrm{\Omega }}_X^1{)}^{\ast },x\mapsto {\int }_x^{s_0(\pi (x))}$$

Associated to $s_0$ there is an extension of flat connections (check) in ${\mathrm{Ext}}^1({\mathcal{O}}_X,(R^1{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }{)}^{\ast })$ that corresponds to $1\in {\mathrm{End}}_{{\mathcal{O}}_S}(R^1{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet })$,

$$0\to (R^1{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }{)}^{\ast }\to \mathcal{E}\to {\mathcal{O}}_X\to 0$$

Mysterious Relics

A collection of small pieces of information that I don’t fully understand, but I need to put them here.

Connection Modified De-Rham Cohomology

Let $(\mathcal{W},\mathrm{\nabla })$ be a connection on $X/K$, define

$${\mathrm{\Omega }}_{(\mathcal{W},\mathrm{\nabla })}^{\bullet }={\mathrm{\Omega }}_{X/K}^{\bullet }\otimes \mathcal{W}$$

$$H_{dR}^{\bullet }(X/K,\mathcal{W})=H^{\bullet }({\mathrm{\Omega }}_{(\mathcal{W},\mathrm{\nabla })}^{\bullet })$$

Integration map

${\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^1$ as a sheaf / vector bundle on $S$, is fibrewise $H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1)$. So we want to consider the sheaf $(R^1{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }{)}^{\ast }$ on $S$ whose fibres are $H_{dR}^{\bullet }(X_t/K{)}^{\ast }$.

$$X_t({\mathbb{Q}}_p)\to H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1{)}^{\ast },x\mapsto {\int }_{s_0(\pi (x))}^x$$

Hodge Filtration

We have (where does this come from?)

$$0\to H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1)\to H_{dR}^1(X_t/K)\to H^1(X_t,{\mathcal{O}}_{X_t})\to 0$$

the first group is maximal isotropic with respect to the pairing ($X_t$ is a curve) (not sure if this is correct)

$$H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1)\times H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1)\subset H_{dR}^1(X_t)\times H_{dR}^1(X_t)\to H_{dR}^2(X_t/K)\cong K$$

When $W\subset V$ is maximal isotropic for $V\times V\to K$, then $W^{\ast }\cong V/W$. Here we have

$$H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1{)}^{\ast }\cong H_{dR}^1(X_t)/H^0(X_t,{\mathrm{\Omega }}_{X_t/K}^1)\cong H^1(X_t,{\mathcal{O}}_{X_t}).$$

An Exact Sequence Probably Coming From Some Spectral Sequence

$$0\to H_{dR}^1(S/K,R^0{\pi }_{\ast }{\mathrm{\Omega }}_{(\mathcal{W},\mathrm{\nabla })}^{\bullet })\to H_{dR}^1(X/K,\mathcal{W})\to H_{dR}^0(S/K,R^1{\pi }_{\ast }{\mathrm{\Omega }}_{(\mathcal{W},\mathrm{\nabla })}^{\bullet })$$

This probably comes from a spectral sequence associated to the maps $X\to S\to \{\ast \}$,

$$E_1^{p,q}=H^q(S,R^p{\pi }_{\ast }{\mathrm{\Omega }}_{(\mathcal{W},\mathrm{\nabla })}^{\bullet })\Rightarrow H_{dR}^{p+q}(X,\mathcal{W})$$

Flat Connections in question

Let $\mathcal{V}=(R^1{\pi }_{\ast }{\mathrm{\Omega }}_{X/S}^{\bullet }{)}^{\ast }$ and our connection $\mathcal{E}$ locally splits into ${\mathcal{O}}_X\oplus \mathcal{V}$. A flat section

Connections To Differential Equations

Whenever we are given a connection, we can find out the differential equations their flat sections satisfy.

This can be done via the cyclic vector theorem or a similar process.

Imagine you have a connection $\mathrm{\nabla }$ on ${\mathcal{O}}^n$ which can be locally written as

$$\mathrm{\nabla }=d\cdot I_n-\mathrm{\Lambda }\in \mathrm{Hom}({\mathcal{O}}^n,({\mathrm{\Omega }}^1{)}^n)$$