The geometry of the third moment of exponential sums

Florent Jouve (1)

1 IMB - Institut de Mathématiques de Bordeaux

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 certain Kloosterman sums. We establish a connection between the sums considered and the number of rational points over finite fields 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 .

Domaines: Théorie des nombres [math.NT]

Florent Jouve. The geometry of the third moment of exponential sums. Journal de Théorie des Nombres de Bordeaux, 2009, Volume en l'honneur de Henri Cohen. ⟨hal-00387285⟩