The geometry of the third moment of exponential sums

Florent Jouve[1]

  • [1] Dept. of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX, 78712, USA.

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 3, page 733-760
  • ISSN: 1246-7405

Abstract

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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 K 3 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 .

How to cite

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Jouve, Florent. "The geometry of the third moment of exponential sums." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 733-760. <http://eudml.org/doc/10858>.

@article{Jouve2008,
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 .},
affiliation = {Dept. of Mathematics The University of Texas at Austin 1 University Station C1200 Austin, TX, 78712, USA.},
author = {Jouve, Florent},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {third moment of Kloosterman sums; number of rational points on smooth projective surfaces},
language = {eng},
number = {3},
pages = {733-760},
publisher = {Université Bordeaux 1},
title = {The geometry of the third moment of exponential sums},
url = {http://eudml.org/doc/10858},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Jouve, Florent
TI - The geometry of the third moment of exponential sums
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 733
EP - 760
AB - 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 .
LA - eng
KW - third moment of Kloosterman sums; number of rational points on smooth projective surfaces
UR - http://eudml.org/doc/10858
ER -

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