An upper bound and criteria for the Galois group of weighted walks with rational coefficients in the quarter plane

TL;DR

This paper establishes an upper bound of 24 for the Galois group of weighted quarter-plane walks with rational coefficients, providing explicit criteria for specific group orders using elliptic curve torsion theory.

Contribution

It introduces a new upper bound for the Galois group's order and derives explicit criteria for its specific orders using geometric and division polynomial methods.

Findings

Upper bound of 24 for the Galois group order.

Explicit criteria for orders 4, 6, and 8 of the Galois group.

Simplified method for determining when the Galois group has order 8.

Abstract

Using Mazur's theorem on torsions of elliptic curves, an upper bound 24 for the order of the finite Galois group $\mathcal{H}$ associated with weighted walks in the quarter plane $\mathbb{Z}^2_+$ is obtained. The explicit criterion for $\mathcal{H}$ to have order 4 or 6 is rederived by simple geometric argument. Using division polynomials, a recursive criterion for $\mathcal{H}$ having order $4m$ or $4m+2$ is also obtained. As a corollary, explicit criterion for $\mathcal{H}$ to have order 8 is given and is much simpler than the existing method.