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L-functions of exponential sums on curves over rings

Abstract : Let C be a smooth curve over a Galois ring R. Let f be a function over C, and Ψ be an additive character of order p^l 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.
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Submitted on : Tuesday, January 4, 2011 - 1:40:53 PM
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Régis Blache. L-functions of exponential sums on curves over rings. Finite Fields and Their Applications, Elsevier, 2009, 15 (3), pp.345-359. ⟨10.1016/j.ffa.2009.01.001⟩. ⟨hal-00551470⟩

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