The reference for this post is $11^{th}$ section of the article, N. Katz, Nilpotent connections and the monodromy theorem.
Suppose $k$ is a field of characteristic $0$ and $K/k$ is the function field of a smooth, projective, irreducible curve over $k$, $W$ is a finite-dimensional $K$-vector space. A connection $$\nabla : W \to \Omega^1_{K/k} \otimes W$$ is simply an additive map satisfying the Leibniz rule just almost as same as the cases for connection of vector bundle over smooth manifolds, read $$\nabla(fw) = df \otimes w + f\nabla (w) \ \forall f \in K, w \in W.$$ Equivalently, we can define a connection to be a map $\nabla: \mathrm{Der}_k(K,K) \to \mathrm{End}_k(W)$. Now suppose $\mathfrak{p}$ is a closed point, $\mathcal{O}_{\mathfrak{p}}$ its local ring, $\mathfrak{m}_{\mathfrak{p}}$ its maxiaml ideal, $\mathrm{ord}_{\mathfrak{p}}: K \to \mathbb{Z} \cup \left \{\infty \right\}$ the discrete valuation at $\mathfrak{p}$, so in particular $$\mathcal{O} = \left \{f \in K \mid \mathrm{ord}(f) \geq 0 \right \}$$ $$\mathfrak{m} = \left \{f \in K \mid \mathrm{ord}(f) \geq 1 \right \}.$$ We now have an equality $$\left \{D \in \mathrm{Der}_k(K,K) \mid D(\mathfrak{m}) \subset \mathfrak{m} \right \} = (h\frac{d}{dh})\mathcal{O}.$$ If there is a base $\mathbf{e}$ of $W$ (over K) such that $$\nabla\left(h\frac{d}{dh}\right)\mathbf{e} = B\mathbf{e} \ \text{for some} \ B \in M_{\mathrm{dim}(W)}(\mathcal{O})$$ then $\mathfrak{p}$ is called a regular singular point. A theorem of Turrittin asserts that there is a closed relation between regular singular points and cyclic vectors, i.e. vector $w \in W$ such that there exists a non-zero derivation such that $w,(\nabla(D))(w),...,(\nabla(D))^k(w),...$ span $W$ over $K$. In that case, we say $(W,\nabla)$ is cyclic and $w$ a cyclic vector
Theorem. [Turrittin] Assume that $(W,\nabla)$ is cyclic with a cyclic vector $w$, $\mathfrak{p}$ a closed point and $h$ a uniformizer at $\mathfrak{p}$. Then the following statements are equivalent
- $(W,\nabla)$ does not have a regular singular point at $\mathfrak{p}$.
- In terms of the basis $$\mathbf{e} = \left(w, \nabla\left(h\frac{d}{dh}\right)(w),...\nabla\left(h\frac{d}{dh}\right)^{n-1}(w) \right)^t,$$ the connection is expressed as $$\nabla\left(h\frac{d}{dh}\right)\mathbf{e} = \begin{pmatrix} 0 & 1 & ... & 0 &0 \\ . & &... & & ...\\ 0 & 0 & & 0 & 1\\ -f_0 & -f_1 & . & .& -f_{n-1} \end{pmatrix}\mathbf{e}$$ and for some $i$, $\mathrm{ord}(f_i) < 0$.
From the theorem, we deduce that if $A$ is a $K$-algebra, then any $K$-linear map $W \to A$ compatible with the connection is one-one corresponding to an element $f \in A$ satisfying the differential equation $$\frac{d^n f}{dh^n} + f_{n-1}\frac{d^{(n-1)}f}{dh^{(n-1)}}+...+f_0=0.$$ Here we're back to diffential equation "in the" analytic world, such a equation of the form above is said to have a regular singular point at $a$ if $f_{n-i}$ has a pole of order at most $i$ at $a$. If so, the equation can be solved by the so-called Frobenius method. The matrix appears in the theorem is nothing but making a change of variables, i.e. transforming a $n$-order equation into a system of $n$ 1-order equations and a cyclic basis in this case is just $(f,f',f^{"},...,f^{(n-1)})$. If we view the resulting system of equations as a connection on the trivial bundle then the connection has form $$\nabla_{\frac{d}{dz}} = \frac{d}{dz} + \begin{pmatrix} 0 & 1 & ... & 0 &0 \\ . & &... & & ...\\ 0 & 0 & & 0 & 1\\ -f_0 & -f_1 & . & .& -f_{n-1} \end{pmatrix}dz$$ My question.
- "What" plays the role of $\nabla\left(h\frac{d}{dh}\right)$ (algebraic manner) in the analytical world?
- Maybe I'm confused now but let's pick up a concrete example, say $f^{"}+\frac{p}{z}f'+\frac{q}{z^2}f=0$ ($p,q$ holomorphic), it obviously has a regular singular point at $z=0$ but its matrix is $\begin{pmatrix} 0 & 1\\ \frac{q}{z^2}& \frac{p}{z} \end{pmatrix}$ so by Turrrittin's theorem, it does not have a regular singular at $z=0$? What is the reason for this contradiction? Perhaps I missed some point, any explanation is appreciate.
- Is there any motivation for considering the condition $D(\mathfrak{m})\subset \mathfrak{m}$?
Update. The second question is solved.