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 .

Reçu le : 2007-11-23
Publié le : 2009-06-04
DOI : https://doi.org/10.5802/jtnb.648
@article{JTNB_2008__20_3_733_0,
     author = {Florent Jouve},
     title = {The geometry of the third moment of exponential sums},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {733--760},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {3},
     year = {2008},
     doi = {10.5802/jtnb.648},
     zbl = {pre05572699},
     mrnumber = {2523315},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2008__20_3_733_0/}
}
Florent Jouve. 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. https://jtnb.centre-mersenne.org/item/JTNB_2008__20_3_733_0/

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