$p$-Integrality of canonical coordinates

Abstract

Let $L$ be a differential operator with coefficients in $\mathbb{Q}(z)$ of order $n\geq2$ with maximal unipotent monodromy at zero. In this paper we are interested in determining when the canonical coordinate of $L$ belongs to $\mathbb{Z}_p[[z]]$. For this purpose, motivated by a recent conjecture due to P. Candelas, X. de la Ossa and D. van Straten~\cite{CD}, we study the situation when $L$ has a strong Frobenius structure $\Phi=(\phi_{i,j})_{1\leq i,j\leq n}\in M_n(\mathbb{Z}_p[[z]])$ such that $\phi_{1,1}(0)=1$. We then give a necessary and sufficient condition for the canonical coordinate of $L$ to belong to $\mathbb{Z}_p[[z]]$ when $L$ has such a strong Frobenius structure.