The geometry of the third moment of exponential sums
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 733-760.

Nous donnons une interprétation géométrique à deux types distincts de sommes d’exponentielles. L’une d’elles correspond au moment d’ordre trois des sommes de Kloosterman sur F q de type K(ν 2 ;q). Nous commençons par établir un lien entre les sommes considérées et le nombre de points F q -rationnels sur certaines surfaces projectives lisses : l’une d’entre elles est une surface K3 et l’autre est une surface cubique lisse. Appliquant la théorie de Grothendieck-Lefschetz, on retrouve alors en particulier une formule pour le troisième moment des sommes de Kloosterman obtenue par D. H. et E. Lehmer en 1960.

We give a geometric interpretation (and we deduce an explicit formula) for two types of exponential sums, one of which is the third moment of Kloosterman sums over F q of type K(ν 2 ;q). We establish a connection between the sums considered and the number of F q -rational points on explicit smooth projective surfaces, one of which is a K3 surface, whereas the other is a smooth cubic surface. As a consequence, we obtain, applying Grothendieck-Lefschetz theory, a generalized formula for the third moment of Kloosterman sums first investigated by D. H. and E. Lehmer in the 60’s .

DOI : 10.5802/jtnb.648
Jouve, Florent 1

1 Dept. of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX, 78712, USA.
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Jouve, Florent. The geometry of the third moment of exponential sums. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 733-760. doi : 10.5802/jtnb.648. http://archive.numdam.org/articles/10.5802/jtnb.648/

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