L-functions of exponential sums on curves over rings

Let C be a smooth curve over R=O/plO, O being the valuation ring of an unramified extension of the field Qp of p-adic numbers, with residue field k=Fq. Let f be a function over C, and Ψ be an additive character of order pl over R; in this paper we study the exponential sums associated to f and Ψ over C, and their L-functions. We show the rationality of the L-functions in a more general setting, then in the case of curves we express them as products of L-functions associated to polynomials over the affine line, each factor coming from a singularity of f. Finally we show that in the case of Morse functions (i.e. having only simple singularities), the degree of the L-functions are, up to sign, the same as in the case of finite fields, yielding very similar bounds for exponential sums.