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À la recherche de petites sommes d'exponentielles

Étienne Fouvry, Philippe Michel (2002)

Annales de l’institut Fourier

Soit f ( x ) une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers n , ayant exactement deux facteurs premiers, tels que la somme d’exponentielles ∑ x = 1 n exp ( 2 π i f ( x ) / n ) soit en O ( n 1 2 - β f ) , où β f > 0 est une constante ne dépendant que de la géométrie de f . On donne aussi des résultats de répartition du type Sato-Tate, pour certaines sommes de Salié, modulo n , avec n entier comme ci- dessus.

A note on evaluations of some exponential sums

Marko J. Moisio (2000)

Acta Arithmetica

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form S ( a , p α + 1 ) : = ∑ x ∈ q χ ( a x p α + 1 ) where χ is a non-trivial additive character of the finite field q , q = p e odd, and a ∈ * q . In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of p α + 1 . The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain...

Complete arcs arising from a generalization of the Hermitian curve

Herivelto Borges, Beatriz Motta, Fernando Torres (2014)

Acta Arithmetica

We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves, which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.

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