Suppose $C/\mathbb{Q}$ is an algebraic curve (not necessarily complete) defined over $\mathbb{Q}$, and $p$ is a $\mathbb{Q}$ valued point of $C$. Suppose there is a algebraic fibration \begin{equation} \pi: Z \rightarrow C \end{equation} such that $Y=\pi^{-1}(p)$ is the only singular fiber, i.e. \begin{equation} \pi: X \rightarrow S \end{equation} where $X=Z \setminus Y$ and $S=C \setminus p$, is a smooth morphism between smooth varieties. Then by theorem 1 of the paper "On the differentiation of de Rham cohomology classes with respect parameters" by Katz and Oda, there is a canonical integrable connetion $\nabla_{GM}$ --Gauss-Manin connection--on the relative de Rham cohomology sheaf $R^q\,\pi_*(\Omega_{X/S}^*)$.

If we take extension of field and then analytification, we got maps \begin{equation} \pi: Z_{\mathbb{C}}^{an} \rightarrow C_{\mathbb{C}}^{an}\\ \pi:X_{\mathbb{C}}^{an} \rightarrow S_{\mathbb{C}}^{an} \end{equation} The Gauss-Manin connection now is the the analytification of $\nabla_{GM}$, and let us denote it by $\nabla_{GM}^{an}$. For the purpose of this question, let me assume that $\nabla_{GM}^{an}$ has regular singularities at $p$. Then from the works of Griffiths and Schmid on variations of Hodge structures, there exists Deligne's canonical extension of the locally free sheaf $R^q\pi_*(\Omega_{X/S}^{*an})$ on $S_{\mathbb{C}}^{an}$ to a locally free sheaf on $C_{\mathbb{C}}^{an}$. The process of extending it over $p$ could be described explicitly as follows.

Choose a local coordinate in a neighbourhood of $p$, say $t$ and $p$ is the point with $t=0$, i.e. origin. Choose muli-valued local sections $e=(e_1,e_2,\cdots,e_n)^{t}$ of $R^q\pi_*(\Omega_{X/S}^{*an})$ in a neighbourhood of $0$ which form a multi-valued local frame. Suppose the monodromy matrix around $t=0$ with respect to thess muli-valued sections is $T$, define $N= \log T$, the vector \begin{equation} \exp (-\frac{\log t}{2 \pi i} \,N)\,e \end{equation} is single valued and we use it as a local frame to do the extension. This is Deligne's canonical extension in analytical case.

My questions are

1, With the assumption of $\nabla_{GM}^{an}$ is regular, is $\nabla_{GM}$ regular. Actually I am not clear with the definition of regularity of $\nabla_{GM}$ when it is defined over $\mathbb{Q}$, could someone explain it?

2, Could $R^q\,\pi_*(\Omega_{X/S}^*)$ be canonically extended to a locally free sheaf on $C$?

3, If such an extension exists, is there an explicit way like the analytical case to describe this extension using monodromy matrix?

4, If such an extension exists, after $\otimes_{\mathbb{Q}} \mathbb{C}$ and analytification, is this extension compatible with the Deligne's canonical extension in the analytical case?

Any comments and references will be appreciated.