Imaginary quadratic points on toric varieties via universal torsors

Abstract

Inspired by a paper of Salberger we give a new proof of Manin’s conjecture for toric varieties over imaginary quadratic number fields by means of universal torsor parameterizations and elementary lattice point counting.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Arzhantsev I., Derenthal U., Hausen J., Laface A.: Cox Rings, Volume 144 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2015)

    MATH  Google Scholar 

  2. 2

    Baier S., Browning T.D.: Inhomogeneous cubic congruences and rational points on del Pezzo surfaces. J. reine angew. Math. 680, 69–151 (2013)

    MathSciNet  MATH  Google Scholar 

  3. 3

    Batyrev V.V., Tschinkel Y.: Rational points on some Fano cubic bundles. C. R. Acad. Sci. Paris Sér. I Math. 323(1), 41–46 (1996)

    MathSciNet  MATH  Google Scholar 

  4. 4

    Batyrev V.V., Tschinkel Y.: Manin’s conjecture for toric varieties. J. Algebraic Geom. 7(1), 15–53 (1998)

    MathSciNet  MATH  Google Scholar 

  5. 5

    Birch, B.J.: Forms in many variables. Proc. R. Soc. Ser. A 265, 245–263 (1961/1962)

  6. 6

    Bourqui, D.: Fonction zêta des hauteurs des variétés toriques non déployées. Mem. Am. Math. Soc. 211(994), viii+151 (2011)

  7. 7

    Cassels J.W.S.: An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd. 99. Springer, Berlin (1959)

    Google Scholar 

  8. 8

    Chambert-Loir A., Tschinkel Y.: On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148(2), 421–452 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Cox D.A.: The homogeneous coordinate ring of a toric variety. J. Algebraic Geom. 4(1), 17–50 (1995)

    MathSciNet  MATH  Google Scholar 

  10. 10

    Colliot-Thélène, J.-L., Sansuc, J.-J.: Torseurs sous des groupes de type multiplicatif; applications à l’étude des points rationnels de certaines variétés algébriques. C. R. Acad. Sci. Paris Sér. A-B 282(18), Aii, A1113–Aii, A1116 (1976)

  11. 11

    Colliot-Thélène J.-L., Sansuc J.-J.: La descente sur les variétés rationnelles. II. Duke Math. J. 54(2), 375–492 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

    de la Bretèche R.: Compter des points d’une variété torique. J. Number Theory 87(2), 315–331 (2001)

    MathSciNet  Article  Google Scholar 

  13. 13

    de la Bretèche R.: Estimation de sommes multiples de fonctions arithmétiques. Compositio Math. 128(3), 261–298 (2001)

    MathSciNet  Article  Google Scholar 

  14. 14

    de la Bretèche R.: Nombre de points de hauteur bornée sur les surfaces de del Pezzo de degré 5. Duke Math. J. 113(3), 421–464 (2002)

    MathSciNet  Article  Google Scholar 

  15. 15

    de la Bretèche R., Browning T.D.: On Manin’s conjecture for singular del Pezzo surfaces of degree four. II. Math. Proc. Cambridge Philos. Soc. 143(3), 579–605 (2007)

    MathSciNet  MATH  Google Scholar 

  16. 16

    de la Bretèche R., Browning T.D., Derenthal U.: On Manin’s conjecture for a certain singular cubic surface. Ann. Sci. École Norm. Sup. (4) 40(1), 1–50 (2007)

    MATH  Google Scholar 

  17. 17

    de la Bretèche R., Browning T., Peyre E.: On Manin’s conjecture for a family of Châtelet surfaces. Ann. Math. (2) 175(1), 297–343 (2012)

    Article  MATH  Google Scholar 

  18. 18

    Derenthal U., Frei C.: Counting imaginary quadratic points via universal torsors. Compos. Math. 150(10), 1631–1678 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Derenthal U., Frei C.: Counting imaginary quadratic points via universal torsors, II. Math. Proc. Cambridge Philos. Soc. 156(3), 383–407 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20

    Derenthal U., Frei C.: On Manin’s conjecture for a certain singular cubic surface over imaginary quadratic fields. Int. Math. Res. Not. IMRN 10, 2728–2750 (2015)

    MathSciNet  MATH  Google Scholar 

  21. 21

    Derenthal, U., Janda, F.: Gaussian rational points on a singular cubic surface. In: Torsors, étale homotopy and applications to rational points, volume 405 of London Math. Soc. Lecture Note Ser., pp. 210–230. Cambridge Univ. Press, Cambridge (2013)

  22. 22

    Derenthal, U., Pieropan, M.: Cox rings over nonclosed fields (2014) arXiv:1408.5358v1

  23. 23

    Franke J., Manin Y.I., Tschinkel Y.: Rational points of bounded height on Fano varieties. Invent. Math. 95(2), 421–435 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Frei, C., Pieropan, M.: O-minimality on twisted universal torsors and Manin’s conjecture over number fields. Ann. Sci. Éc. Norm. Supér., to appear (2015). arXiv:1312.6603v3

  25. 25

    Frei C.: Counting rational points over number fields on a singular cubic surface. Algebra Number Theory 7(6), 1451–1479 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Lang S.: Algebraic Number Theory. Addison-Wesley Publishing Co. Inc., Reading (1970)

    MATH  Google Scholar 

  27. 27

    Oda, T.: Torus embeddings and applications, Volume 57 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; by Springer, Berlin (1978). Based on joint work with Katsuya Miyake

  28. 28

    Ono T.: Arithmetic of algebraic tori. Ann. Math. (2) 74, 101–139 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Peyre E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J. 79(1), 101–218 (1995)

    MathSciNet  Article  Google Scholar 

  30. 30

    Peyre, E.: Points de hauteur bornée, topologie adélique et mesures de Tamagawa. J. Théor. Nombres Bordeaux, 15(1):319–349, (2003). Les XXIIèmes Journées Arithmetiques (Lille, 2001).

  31. 31

    Robbiani M.: On the arithmetic of isotropic del Pezzo surfaces of degree six. J. Reine Angew. Math. 503, 1–45 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32

    Salberger, P.: Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Astérisque (251):91–258 (1998). Nombre et répartition de points de hauteur bornée (Paris, 1996)

  33. 33

    Schanuel S.H.: Heights in number fields. Bull. Soc. Math. France 107(4), 433–449 (1979)

    MathSciNet  MATH  Google Scholar 

  34. 34

    Skorobogatov A.: Torsors and Rational Points, Volume 144 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  35. 35

    Weil A.: Basic Number Theory, Die Grundlehren der mathematischen Wissenschaften, Band 144. Springer, New York (1967)

    Google Scholar 

  36. 36

    Weil, A.: Adeles and Algebraic Groups, Volume 23 of Progress in Mathematics. Birkhäuser, Boston, Mass. With appendices by M. Demazure and Takashi Ono (1982)

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Marta Pieropan.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pieropan, M. Imaginary quadratic points on toric varieties via universal torsors. manuscripta math. 150, 415–439 (2016). https://doi.org/10.1007/s00229-015-0817-8

Download citation

Mathematics Subject Classification

  • 11G35 (14G05, 14M25)