This is a note for helping me to understand the picture.
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.
Let \(X\) be a curve over \(\mathbb{Q}_p\), we can form the integration map
\[ X(\mathbb{Q}_p)\times X(\mathbb{Q}_p) \to H^0(X,\Omega_X^1)^*, \quad (x,y)\mapsto \int_x^y \]
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 \(S(\mathbb{Z}_p)\)
\[ X(\mathbb{Z}_p)\to (R^1\pi_* \mathcal{O}_X) (\mathbb{Q}_p) \]
which is fibrewise the integration map
\[ X(\mathbb{Q}_p) \to H^0(X,\Omega^1_X)^*, \quad x \mapsto \int_{x}^{s_0(\pi(x))} \]
Associated to \(s_0\) there is an extension of flat connections (check) in \(\mathrm{Ext}^1(\mathcal{O}_X, (R^1\pi_*\Omega^\bullet_{X/S})^*)\) that corresponds to \(1\in \mathrm{End}_{\mathcal{O}_S}(R^1\pi_*\Omega^\bullet_{X/S})\),
\[ 0\to (R^1\pi_*\Omega^\bullet_{X/S})^*\to \mathcal{E}\to \mathcal{O}_X \to 0 \]
A collection of small pieces of information that I don’t fully understand, but I need to put them here.
Let \((\mathcal{W},\nabla)\) be a connection on \(X/K\), define
\[ \Omega_{(\mathcal{W},\nabla)}^\bullet = \Omega_{X/K}^\bullet \otimes \mathcal{W}\]
\[ H^\bullet_{dR}(X/K, \mathcal{W}) = H^\bullet(\Omega_{(\mathcal{W},\nabla)}^\bullet) \]
\(\pi_*\Omega_{X/S}^1\) as a sheaf / vector bundle on \(S\), is fibrewise \(H^0(X_t,\Omega^1_{X_t/K})\). So we want to consider the sheaf \((R^1\pi_*\Omega^\bullet_{X/S})^*\) on \(S\) whose fibres are \(H^\bullet_{dR}(X_t/K)^*\).
\[ X_t(\mathbb{Q}_p) \to H^0(X_t, \Omega^1_{X_t/K})^*, \quad x\mapsto \int^{x}_{s_0(\pi(x))} \]
We have (where does this come from?)
\[ 0\to H^0(X_t, \Omega^1_{X_t/K}) \to H^1_{dR}(X_t/K) \to H^1(X_t, \mathcal{O}_{X_t}) \to 0 \]
the first group is maximal isotropic with respect to the pairing (\(X_t\) is a curve) (not sure if this is correct)
\[ H^0(X_t, \Omega^1_{X_t/K})\times H^0(X_t, \Omega^1_{X_t/K}) \subset H^1_{dR}(X_t)\times H^1_{dR}(X_t) \to H^2_{dR}(X_t/K)\cong K \]
When \(W\subset V\) is maximal isotropic for \(V\times V\to K\), then \(W^*\cong V/W\). Here we have
\[ H^0(X_t, \Omega^1_{X_t/K})^* \cong H^1_{dR}(X_t)/H^0(X_t,\Omega^1_{X_t/K}) \cong H^1(X_t, \mathcal{O}_{X_t}). \]
\[ 0 \to H^1_{dR}(S/K, R^0\pi_* \Omega^\bullet_{(\mathcal{W},\nabla)}) \to H^1_{dR}(X/K, \mathcal{W}) \to H^0_{dR}(S/K, R^1\pi_* \Omega^\bullet_{(\mathcal{W},\nabla)}) \]
This probably comes from a spectral sequence associated to the maps \(X\to S\to \{*\}\),
\[ E_1^{p,q} = H^q(S, R^p\pi_*\Omega^\bullet_{(\mathcal{W},\nabla)}) \Rightarrow H^{p+q}_{dR}(X, \mathcal{W}) \]
Let \(\mathcal{V}= (R^1\pi_*\Omega^\bullet_{X/S})^*\) and our connection \(\mathcal{E}\) locally splits into \(\mathcal{O}_X\oplus \mathcal{V}\). A flat section
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 \(\nabla\) on \(\mathcal{O}^n\) which can be locally written as
\[ \nabla = d\cdot I_n - \Lambda \in \mathrm{Hom}(\mathcal{O}^n, (\Omega^1)^n) \]