Riccati Differential Equation for Hypergeometric Differential Equation

Abstract

In this paper, we study the solutions of Riccati differential equation corresponding to $p$-adic differential equations which are solvable on the generic disc. As an application, we consider the Grothendieck conjecture for Riccati differential equations. We see that the Riccati differential equations for some globally nilpotent differential equation with coefficients in $\mathbb{Q}(t)$ have, for almost all prime, a solution in rational function field over the finite field ${\mathbb{F}}_p$, but do not have any algebraic solutions.