Fuchsian differential equations from modular arithmetic

Counting combinatorial objects and determining the associated generating functions can be computationally very difficult and expensive when using exact numbers. Doing similar calculations modulo a prime can be orders of magnitude faster. We use two simple polygon models to illustrate this: we study the generating functions of (singly) punctured staircase polygons and imperfect staircase polygons, counted by their extent along the main diagonal. For the former model this is equivalent to counting by the half-perimeter of the outer staircase polygon. We derive long series for these generating functions modulo a single prime, and then proceed to find Fuchsian ODEs satisfied by the generating functions, modulo this prime. Knowledge of a Fuchsian ODE modulo a prime will generally suffice to determine exactly its singular points and the associated characteristic exponents. We also present a procedure for the efficient reconstruction of the exact ODE, using results from multiple modprime calculations. Finally, we demonstrate how modular calculations can be used to factor Fuchsian differential operators.