Fonctions zêta des hauteurs

de la Bretèche, Régis

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 77-95.

Ce papier présente les récents progrès concernant les fonctions zêta des hauteurs associées à la conjecture de Manin. En particulier, des exemples où on peut prouver un prolongement méromorphe de ces fonctions sont détaillés.

The paper surveys recent progress towards the Height zeta functions related to the Manin’s conjecture. In particular, it details some cases where one can prove meromorphic continuation of these functions.

Mots clés : Height zeta function, Manin’s Conjecture, Cubic surfaces, Natural Boundary

[1] S. J. Arakelov, Theory of intersections on the arithmetic surface. Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 1, 405–408. Canad. Math. Congress, Montreal, Que., 1975. | MR | Zbl

[2] V.V. Batyrev, Y.I. Manin, Sur les points rationnels de hauteur bornée des variétés algébriques. Math. Ann. 286 (1990), 27–43. | MR | Zbl

[3] V.V. Batyrev, Y. Tschinkel, Rational points of bounded height on compactifications of anisotropic toric. Internat. Math. Res. Notices 12 (1995), 591–635. | MR | Zbl

[4] V.V. Batyrev, Y. Tschinkel, Height Zeta functions of Toric Varieties, Algebraic geometry 5, Manin’s Festschrift. J. Math. Sci. 82 (1996), n${}^{\circ }$ 1, 3220–3239. | MR | Zbl

[5] V.V. Batyrev, Y. Tschinkel, Manin’s Conjecture for Toric Varieties. J. of Algebraic Geometry 7 (1998), 15–53. | MR | Zbl

[6] V.V. Batyrev, Y. Tschinkel, Tamagawa numbers of polarized algebraic varieties. Astérisque 251 (1998), 299–340. | MR | Zbl

[7] G. Bhowmik, D. Essouabri, B. Lichtin, Meromorphic Continuation of Multivariable Euler Products and Applications. ArXiv math.NT/0502508, à paraître à Forum Math. | MR

[8] G. Bhowmik, J.-C. Schlage-Puchta, Natural boundaries of Dirichlet series. Functiones et Approximatio 37 (2007), 17–29. | MR | Zbl

[9] R. de la Bretèche, Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière. Astérisque 251 (1998), 51–77. | Numdam | MR | Zbl

[10] R. de la Bretèche, Compter des points sur des variétés toriques. Journal of Number Theory 87 (2001), n${}^{\circ }$ 2, 315–331. | MR | Zbl

[11] R. de la Bretèche, Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5. Duke Math. J. 113 (2002), n${}^{\circ }$ 3, 421–464. | MR | Zbl

[12] R. de la Bretèche, T.D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, I. Michigan Mathematical Journal 55 (2007), 51–80. | MR | Zbl

[13] R. de la Bretèche, T.D. Browning, On Manin’s conjecture for singular del Pezzo surfaces of degree four, II. Math. Proc. Camb. Phil. Soc., à paraître. | Zbl

[14] R. de la Bretèche, T.D. Browning, U. Derenthal, On Manin’s conjecture for a certain singular cubic surface. Annales scientifiques de l’ENS, $4$-ème série, 40 (2007), 1–50. | Numdam | MR | Zbl

[15] R. de la Bretèche, É. Fouvry, L’éclaté du plan projectif en quatre points dont deux conjugués. J. reine angew. Math. 576 (2004), 63–122. | MR | Zbl

[16] R. de la Bretèche, Sir P. Swinnerton-Dyer, Fonction zêta des hauteurs associée à une certaine surface cubique. Bulletin de la SMF 135 (2007), 65–92. | Numdam | MR

[17] T.D. Browning, The density of rational points on a certain singular cubic surface. J. Number Theory 119 (2006), 242–283. | MR | Zbl

[18] T.D. Browning, An overview of Manin’s conjecture for del Pezzo surfaces. Analytic number theory - A tribute to Gauss and Dirichlet (Goettingen, 20th June - 24th June, 2005), Clay Mathematics Proceedings 7 (2007), 39–56. | MR | Zbl

[19] A. Chambert-Loir, Y. Tschinkel, Yuri Points of bounded height on equivariant compactifications of vector groups, I Compositio Math. 124 (2000), no. 1, 65–93. | MR | Zbl

[20] A. Chambert-Loir, Y. Tschinkel, Points of bounded height on equivariant compactifications of vector groups, II. J. Number Theory 85 (2000), no. 2, 172–188. | MR | Zbl

[21] A. Chambert-Loir, Y. Tschinkel, Fonctions zêta des hauteurs des espaces fibrés. Rational points on algebraic varieties, 71–115, Progr. Math. 199, Birkhäuser, Basel, 2001. | MR | Zbl

[22] A. Chambert-Loir, Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452. | MR | Zbl

[23] J.-L. Colliot-Thélène, J.-J. Sansuc, La descente sur les variétés rationnelles. Journée de géométrie algébrique d’Angers (1979) (A. Beauville, ed.) Sijthoff, Noordhoff, Alphen aan den Rijn, 1980, 223–237. | MR | Zbl

[24] J.-L. Colliot-Thélène, J.-J. Sansuc, La descente sur les variétés rationnelles, II. Duke Math. J. 54 (1987), no 2, 375–492. | MR | Zbl

[25] U. Derenthal, Manin’s conjecture for a cubic surface. ArXiv :math.NT/0504016, (2005).

[26] U. Derenthal, Y. Tschinkel, Universal torsors over Del Pezzo surfaces and rational points. « Equidistribution in Number theory, An Introduction », (A. Granville, Z. Rudnick eds.), 169–196, NATO Science Series II 237, Springer, (2007). | MR | Zbl

[27] D. Essouabri, Prolongements analytiques d’une classe de fonction zêta des hauteurs et applications. Bull. Soc. Math. France 133 (2) (2005), 297–329. | Numdam | MR | Zbl

[28] E. Fouvry, Sur la hauteur des points d’une certaine surface cubique singulière. Astérisque 251 (1998), 31–49. | Numdam | MR | Zbl

[29] T. Estermann, On certain functions represented by Dirichlet series. Proc. London Math. Soc. 27 (1928), 433–448.

[30] J. Franke, Y.I. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties. Invent. Math. 95 (1989), 421–435. | MR | Zbl

[31] A. Gorodnik, F. Maucourant, H. Oh, Manin’s conjecture on rational points of bounded height and adelic mixing. Annales scientifiques de l’École Normale Supérieure, série 4, 41, fascicule 3 (2008), 385–437. | Numdam | MR | Zbl

[32] B. Hassett, Y. Tschinkel, Universal torsors and Cox rings. Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 149–173, Progr. Math. 226, Birkhäuser, 2004. | MR | Zbl

[33] R. Heath-Brown, The density of rational points on cubic surfaces. Acta Arith. 19 (1997), 17–30. | MR | Zbl

[34] R. Heath-Brown, B.Z. Moroz, The density of Rational Points on the cubic surface $X_0^3=X_1{X}_2{X}_3$. Math. Proc. Camb. Phil. Soc. 125 (1999), n${}^{\circ }$ 3, 385–395. | MR | Zbl

[35] M. N. Huxley, Exponential sums and the Riemann zeta function, V. Proc. London Math. Soc. (3) 90 (2005), no. 1, 1–41. | MR | Zbl

[36] A. Ivić, The Riemann zeta-function. John Wiley, Sons Inc., New York, (1985). | MR | Zbl

[37] H. Iwaniec, C.J. Mozzochi, On the divisor and circle problems, J. Number Theory 29 (1988), no. 1, 60–93. | MR | Zbl

[38] N. Kurokawa, On the meromorphy of Euler products, I and II. Proc. London Math. Soc. (3) 53 (1986), no. 1, 1–47 and 209–236. | MR | Zbl

[39] Y. Manin, Y. Tschinkel, Points of bounded height on del Pezzo surfaces. Compositio Math. 85 (1993), n${}^{\circ }$ 3, 315–332. | Numdam | MR | Zbl

[40] E. Peyre, Hauteurs et nombres de Tamagawa sur les variétés de Fano. Duke Math. J. 79 (1995), 101–218. | MR | Zbl

[41] E. Peyre, Terme principal de la fonction zêta des hauteurs et torseurs universels. Astérisque 251 (1998), 259–298. | MR | Zbl

[42] E. Peyre, Points de hauteur bornée et géométrie des variétés (d’après Y. Manin et al.). Séminaire Bourbaki, Vol. 2000/2001. Astérisque 282 (2002), Exp. n${}^{\circ }$ 891, ix, 323–344. | Numdam | MR | Zbl

[43] E. Peyre, Points de hauteur bornée, topologie adélique et mesures de Tamagawa. Les XXIIèmes Journées Arithmetiques (Lille, 2001), J. Théor. Nombres Bordeaux 15 (2003), n${}^{\circ }$ 1, 319–349. | Numdam | MR | Zbl

[44] E. Peyre, Counting points on varieties using universal torsors. Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 61–81, Progr. Math. 226, Birkhäuser Boston, Boston, MA, 2004. | MR

[45] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque 251 (1998), 91–258. | Numdam | MR | Zbl

[46] M. du Sautoy, Zeta Functions of Groups and Natural Boundaries. Preprint 2000, 79 pages. | MR

[47] J.A. Shalika, Y. Tschinkel, Height zeta functions of equivariant compactifications of the Heisenberg group. Contributions to automorphic forms, geometry, and number theory, 743–771, Johns Hopkins Univ. Press, Baltimore, MD, 2004. | MR

[48] J.A. Shalika, R. Takloo-Bighash, Y. Tschinkel, Rational points on compactifications of semi-simple groups of rank 1. Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 205–233, Progr. Math. 226, Birkhüser Boston, Boston, MA, 2004. | MR

[49] M. Strauch, Y. Tschinkel, Height zeta functions of twisted products. Math. Res. Lett. 4 (1997), n${}^{\circ }$ 2-3, 273–282. | MR | Zbl

[50] M. Strauch, Y. Tschinkel, Height zeta functions of toric bundles over flag varieties. Selecta Math. (N.S.) 5 (1999), n${}^{\circ }$ 3, 325–396. | MR | Zbl

[51] Sir P. Swinnerton-Dyer, Counting points on cubic surfaces II. Geometric methods in algebra and number theory, (Eds. F. Bogomolov and Y. Tschinkel), Progress in Mathematics 235, Birkhäuser, 2004. | MR | Zbl

[52] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres. Cours spécialisés, n${}^{\circ }$ 1, Société Mathématique de France (1995), xv + 457 pp. | MR | Zbl

[53] E.C. Titchmarsh, The theory of the Riemann zeta-function. 2nd ed., revised by D.R. Heath-Brown. Oxford University Press, 1986. | MR | Zbl

[54] Y. Tschinkel, Lectures on height zeta functions of toric varieties. Geometry of toric varieties, 227–247, Sémin. Congr. 6, Soc. Math. France, Paris, 2002. | MR | Zbl

[55] Y. Tschinkel, Fujita’s program and rational points. “Higher Dimensional Varieties and Rational Points”, (K. J. Böröczky, J. Kollár, T. Szamuely eds.), Bolyai Society Mathematical Studies 12, Springer Verl., 2003, 283–310. | MR | Zbl