Diophantine finiteness results should be a result of the principle rational points are special points. \(\dim_K \mathcal{V}(X)^\nabla \le \mathrm{rank}\mathcal{V}\). There are transecndantal maps \(X(\mathbb{Q}_p) \to V(\mathbb{Q}_p)\cong \mathbb{A}^m\) and a sub-variety \(W(\mathbb{Q}_p) \subset W(\mathbb{Q}_p)\) defined over \(\mathbb{Q}\) such that \(X(\mathbb{Q})\subset W(\mathbb{Q}_p)\).
An idea is that Zilber-pink conjecture would imply Mordell’s conjecture. Suppose \(J=\mathrm{Jac}(X)\) is the Jacobian of a curve, and \(\mathrm{rank}J(\mathbb{Q})=r\), then for \(n>r\) we have that \(P\in J^n(\mathbb{Q})\) is actually contained in a proper subgroup \(A(\mathbb{Q})\).
There is a big difference between vector bundles and vector bundles with connections. Vector bundles are always locally trivial and so they not to be studied locally. However, vector bundles with connections has extra structure and their local study make sense. In fact it is expected that you don’t lose much information by studying them locally.
A horizontan section is a section \(s\in \Gamma(X, \mathcal{V})\) such that \(\nabla s=0\). Equivalently, it can be seen as a morphism \(s: (\mathcal{O}_X,d)\to (\mathcal{V},\nabla)\) from the trivial one-dimensional connection to \(\mathcal{V}\).
The flat sections \(\Gamma(X, \mathcal{V})^\nabla\) form a \(K\)-vector space.
For \((\mathcal{O}_X,d)\) the trivial connection, \(1\in \mathcal{O}(X)\) is a horizontal section and the only one, it forms the basis of \(\Gamma(X, \mathcal{O}_X)^{\nabla}\).
\(\dim_K \mathcal{V}(X)^\nabla \le \mathrm{rank}\mathcal{V}\).
If we think of Coleman integral as a theory of analytic continuation. And think of continuation as giving isomorphism functors between the fibre functors
\[ \pi_1(X,x,y)\to \mathrm{Iso}(x^*\mathcal{V}, y^*\mathcal{V}) \]
We should think of these functors as ‘paths’ in the category of vector bundles with connections. Because real path might not exist, but these functors can see them and is an evidence of some kind of ‘path’.
The simplest example is the tower of logarithm, let’s consider \((\mathcal{O}_X^(\oplus 2), d - \begin{pmatrix} 0 & \frac{dx}{x} \\ 0 & 0 \end{pmatrix})\), and \(\gamma \in \pi_1(\mathbb{C}^*,1)\), then the monodromy is given by