Let $k$ be a field of characteristic zero, $X=\mathbb{G}_{m, k}=\mathrm{Spec}\ k[t, t^{-1}]$ the multiplicative group over k and $E=\mathcal{O}_X$ the trivial line bundle.
Consider the connection $\nabla: E \to E \otimes_{\mathcal{O}_X} \Omega^1_X$ defined by $\nabla(1)=\alpha \frac{dt}{t}$ for some $\alpha \in \mathbb{C}$, that is:
$\nabla=d+\alpha \frac{dt}{t}\wedge$
My question is: has $(E, \nabla)$ regular singularities?
Thanks for you help
Dear $p^4$,
I had a chance to glance further at the 4 author paper linked in your comment. Their connection form is $-dt + sdt/t$. This has double pole at $\infty$, so it isn't regular*. However, the connection in your question has logarithmic singularities at $0$ and $\infty$, so it is regular, as I said earlier. There is no contradiction.
*(Afterthought) This is a bit sloppy, since the Fuchs criterion is only a sufficient condition for regularity. But to see the non regularity, observe that the solution they write down $exp(t)t^{-s}$, which is correct (!), has bad singularities at $\infty$.
