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In the book [FIM], original methods were proposed to determine the invariantmeasure of random walks in the quarter plane with small jumps, the generalsolution being obtained via reduction to boundary value problems. Among otherthings, an important quantity, the so-called group of the walk, allows todeduce theoretical features about the nature of the solutions. In particular,when the \emph{order} of the group is finite, necessary and sufficientconditions have been given in [FIM] for the solution to be rational oralgebraic. In this paper, when the underlying algebraic curve is of genus $1$,we propose a concrete criterion ensuring the finiteness of the group. It turnsout that this criterion can be expressed as the cancellation of a determinantof a matrix of order 3 or 4, which depends in a polynomial way on thecoefficients of the walk.

Random Walks in the Quarter-Plane: Advances in Explicit Criterions for the Finiteness of the Associated Group in the Genus 1 Case