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Florent Jouve
The geometry of the third moment of exponential sums
Journal de théorie des nombres de Bordeaux, 20 no. 3 (2008), p. 733-760, doi: 10.5802/jtnb.648
Article PDF | Reviews MR 2523315 | Zbl pre05572699

Résumé - Abstract

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 $\mathbf{F}_q$ of type $K(\nu ^2;q)$. We establish a connection between the sums considered and the number of $\mathbf{F}_q$-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 .

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