A walk in the quarter plane is a path between integral points of the plane that uses a prescribed set of directions and remains in the first quadrant. In the past years, the enumeration of such walks has attracted the attentionof many authors in combinatorics and probability. The complexity of their enumeration is encoded in the algebraic nature of their associated generatingseries. The main questions are: are these series algebraic, holonomic (solutions of linear differential equations) or differentially algebraic (solutions of algebraicdifferential equations)? In this talk, we will show how this algebraic nature can be understood via the study of a discrete functional equation over a curve E of genus zero or one over a function field . In the genus zero case, the functional equation corresponds to a socalled q-difference equation and the generating series is always differentially transcendental. In genus one, the dynamic of the functional equation is the addition by agiven point P of the elliptic curve E.In that situation, the nature of the generating series is entirely captured by the linear dependence relations of certain prescribed points in the Mordell-Weil lattice of the elliptic surface attached to E. This are joint works with T. Dreyfus, J. Roques and M.F. Singer.