In the book [FIM], original methods were proposed to determine the invariant
measure of random walks in the quarter plane with small jumps, the general
solution being obtained via reduction to boundary value problems. Among other
things, an important quantity, the so-called group of the walk, allows to
deduce theoretical features about the nature of the solutions. In particular,
when the \emph{order} of the group is finite, necessary and sufficient
conditions have been given in [FIM] for the solution to be rational or
algebraic. In this paper, when the underlying algebraic curve is of genus 1,
we propose a concrete criterion ensuring the finiteness of the group. It turns
out that this criterion can be expressed as the cancellation of a determinant
of a matrix of order 3 or 4, which depends in a polynomial way on the
coefficients of the walk.Comment: 2 figure
