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

Using Mazur’s theorem on torsions of elliptic curves, an upper bound 24 for the order finite Galois group ℋ associated with weighted walks in quarter plane ℤ + 2 is obtained. The explicit criterion to have 4 or 6 rederived by simple geometric arguments. division polynomials, a recursive 4m 4m+2 also As corollary, 8 given through method simpler than existing one.