Cyclotomie des sommes de Weil binomiales

Aubry, Yves; Katz, Daniel J.; Langevin, Philippe

Comptes Rendus. Mathématique, Tome 352 (2014) no. 5, pp. 373-376.

Les sommes de Weil de la forme ${\mathrm{W}}_{K,d}(a)={\sum }_{x\in K}\psi (x^d+ax)$, où K est un corps fini, ψ un caractère additif de K, d un entier premier à $|K^{\times }|$ et $a\in K^{\times }$, apparaissent naturellement en théorie des nombres ainsi qu'en géométrie finie, en cryptographie, dans l'étude de la corrélation des suites et en théorie des codes. Nous nous intéressons ici au cas où ${\mathrm{W}}_{K,d}(a)$ ne prend que trois valeurs distinctes lorsque a varie dans $K^{\times }$. Via une approche galoisienne, nous donnons plusieurs résultats concernant ces sommes de Weil à trois valeurs, généralisant notamment à toute caractéristique non nulle des résultats de Calderbank–McGuire–Poonen–Rubinstein, de Calderbank–McGuire et de Charpin établis en caractéristique 2.

Weil sums of the form ${\mathrm{W}}_{K,d}(a)={\sum }_{x\in K}\psi (x^d+ax)$, where K is a finite field, ψ is an additive character of K, d is coprime to $|K^{\times }|$, and $a\in K^{\times }$, arise often in number theory, as well as in finite geometry, in cryptography, in the study of the correlation of sequences, and in coding theory. Here we are interested in the case where ${\mathrm{W}}_{K,d}(a)$ takes only three distinct values as a runs through $K^{\times }$. Via a Galois-theoretic approach, we give several results concerning three-valued Weil sums, and, in particular, we generalize to any nonzero characteristic some results of Calderbank–McGuire–Poonen–Rubinstein, of Calderbank–McGuire and of Charpin proved in characteristic 2.

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