The geometry of the third moment of exponential sums

Jouve, Florent

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 ${\mathbf{F}}_q$ de type $K({\nu }^2;q)$. Nous commençons par établir un lien entre les sommes considérées et le nombre de points ${\mathbf{F}}_q$-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 ${\mathbf{F}}_q$ 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 .

[1] S. Ahlgren, K. Ono, Modularity of a certain Calabi-Yau threefold. Monatsh. Math. 129 (2000), no. 3, 177–190. | MR | Zbl

[2] M. Artin, Supersingular $K3$ surfaces. Ann. Scient. Éc. Norm. Sup., 4e série, 7 (1974), 543–568. | Numdam | MR | Zbl

[3] A. O. L. Atkin, Note on a paper of Birch. J. London Math. Soc. 44 (1969). | MR | Zbl

[4] W. Barth, C. Peters, A. van de Ven, Compact complex surfaces. Springer, Berlin-Heidelberg-New York, 1984. | MR | Zbl

[5] F. Beukers, J. Stienstra, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic $K3$-surfaces. Math. Ann. 271 (1985), 269–304. | MR | Zbl

[6] B. J. Birch, How the number of points of an elliptic curve over a fixed prime field varies. J. London Math. Soc. 43 (1968) 57–60. | MR | Zbl

[7] A. Calabri, R. Ferraro, Explicit resolutions of double point singularities of surfaces. Collect. Math. 53 (2002), no. 2, 99–131. | MR | Zbl

[8] P. Deligne, Cohomologie étale, SGA $4\frac{1}{2}$. Lectures Notes in Math. 569, Springer Verlag 1977. | MR | Zbl

[9] P. Deligne, La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252. | Numdam | MR | Zbl

[10] L. Fu, D. Wan, $L$-functions for symmetric products of Kloosterman sums. J. reine angew. Math. 589 (2005), 79–103. | MR

[11] R. Hartshorne, Algebraic geometry. GTM 52, Springer-Verlag, 1977. | MR | Zbl

[12] B. Hunt, The geometry of some special arithmetic quotients. Lecture Notes in Math. 1637. Springer-Verlag, Berlin, 1996. | MR | Zbl

[13] H. Inose, T. Shioda, On singular $K3$ surfaces. Complex analysis and algebraic geometry (eds Baily, W. and Shioda, T.), Cambridge (1977), 119-136. | MR | Zbl

[14] K. Ireland, M. Rosen, A classical introduction to modern number theory, Second Edition, GTM 84, Springer-Verlag 1990. | MR | Zbl

[15] H. Iwaniec, Topics in classical automorphic forms. Graduate Studies in Mathematics, 17, American Mathematical Society, 1997. | MR | Zbl

[16] A. J. de Jong, N. M. Katz, Monodromy and the Tate conjecture: Picard numbers and Mordell-Weil ranks in families. Israel J. Math. 120 (2000), part A, 47–79. | MR | Zbl

[17] N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies 116. Princeton University Press, Princeton, NJ, 1988 | MR | Zbl

[18] D. H. Lehmer, E. Lehmer, On the cubes of Kloosterman sums. Acta Arith. 6 (1960), 15–22. | MR | Zbl

[19] R. Livné, Cubic exponential sums and Galois representations . Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 247–261, Contemp. Math. 67, Amer. Math. Soc., Providence, RI, (1987). | MR | Zbl

[20] Yu. Manin, Cubic forms. Algebra, geometry, arithmetic. Second edition. North-Holland Mathematical Library, 4. North-Holland Publishing Co., Amsterdam, 1986. | MR | Zbl

[21] L. J. Mordell, On Lehmer’s congruence associated with cubes of Kloosterman’s sums. J. London Math. Soc. 36 (1961), 335–339. | MR | Zbl

[22] H. Salié, Über die Kloostermanschen Summen $S(u,v;q)$. Math. Zeit. 34 (1932), 91–109. | MR | Zbl

[23] T. W. Sederberg, Techniques for cubic algebraic surfaces. IEEE Comp. Graph and Appl., September 1990.

[24] J. Silverman, The arithmetic of elliptic curves. GTM 106, Springer-Verlag, 1986. | MR | Zbl

[25] J. Silverman, Advanced topics in the arithmetic of elliptic curves, Second Edition, GTM 151, Springer-Verlag, 1999. | MR | Zbl

[26] H. P. F. Swinnerton-Dyer, The zeta function of a cubic surface over a finite field. Proc. Camb. Phil. Soc. 63 (1967), 55. | MR | Zbl

[27] H. A. Verril, The $L$-series of certain rigid Calabi-Yau threefolds. J. Number Theory 81 (2000), no. 2, 310–334. | MR | Zbl