L-functions of exponential sums on curves over rings

Régis Blache (1)

1 AOC - Analyse Optimisation Controle

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.

Mots clés: Exponential sums over p-adic rings; L-functions; Weil numbers; Witt vectors

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Régis Blache. L-functions of exponential sums on curves over rings. Finite Fields and Their Applications, 2009, 15 (3), pp.345-359. ⟨10.1016/j.ffa.2009.01.001⟩. ⟨hal-00551470⟩