Keller, Timo

On an Analogue of the Conjecture of Birch and Swinnerton-Dyer for Abelian Schemes over Higher Dimensional Bases over Finite Fields

Doc. Math. 24, 915-993 (2019)
DOI: 10.25537/dm.2019v24.915-993
Communicated by Otmar Venjakob

Summary

We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic \(p\). We prove the prime-to-\(p\) part conditionally on the finiteness of the \(p\)-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-\(p\) part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-\(p\) part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of \(\mathscr{A}[p]\) or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.

Mathematics Subject Classification

11G40, 11G50, 19F27, 11G10, 14F20, 14K15

Keywords/Phrases

\(L\)-functions of varieties over global fields, Birch-Swinnerton-Dyer conjecture, heights, étale cohomology, higher regulators, zeta and \(L\)-functions, abelian varieties of dimension \(> 1\), étale and other Grothendieck topologies and cohomologies, arithmetic ground fields

References

  • 1. Altman, Allen and Kleiman, Steven: Introduction to Grothendieck duality theory. Lecture Notes in Mathematics 146, Berlin-Heidelberg-New York: Springer-Verlag, 1970. DOI 10.1007/bfb0060932; zbl 0215.37201; MR0274461.
  • 2. Bauer, Werner: On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic $p > 0$. Invent. Math., 108(2) (1992), 263-287. DOI 10.1007/BF02100606; zbl 0807.14014; MR1161093.
  • 3. Beilinson, Alexander; Bernstein, J. and Deligne, Pierre: Faisceaux pervers. Astérisque, 100 (1982). zbl 0536.14011; MR0751966.
  • 4. Bombieri, Enrico and Gubler, Walter: Heights in Diophantine geometry. New Mathematical Monographs 4. Cambridge: Cambridge University Press. xvi + 652 p., 2006. DOI 10.2277/0511138091; zbl 1115.11034; MR2216774.
  • 5. Bloch, Spencer: A note on height pairings, Tamagawa numbers, and the Birch and Swinnerton-Dyer conjecture. Invent. Math., 58 (1980), 65-76. DOI 10.1007/BF01402274; zbl 0444.14015; MR0570874.
  • 6. Bosch, Siegfried; Lütkebohmert, Werner and Raynaud, Michel: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21. Berlin etc.: Springer-Verlag. x + 325 p., 1990. zbl 0705.14001; MR1045822.
  • 7. Bayer, Pilar and Neukirch, Jürgen: On values of zeta functions and $\ell$-adic Euler characteristics. Invent. Math., 50 (1978), 35-64. DOI 10.1007/BF01406467; zbl 0409.12018; MR0516603.
  • 8. Conrad, Brian: Chow's $K/k$-image and $K/k$-trace, and the Lang-Néron theorem. Enseign. Math., (2), 52(1-2) (2006), 37-108. zbl 1133.14028; MR2255529.
  • 9. Conrad, Brian: Weil and Grothendieck approaches to adelic points. Enseign. Math., (2), 58(1-2) (2012), 61-97. DOI 10.4171/lem/58-1-3; zbl 1316.14002; MR2985010.
  • 10. Deligne, Pierre: La conjecture de Weil. II. Publ. Math. IHÉS, 52(2) (1980), 137-252. DOI 10.1007/bf02684780; zbl 0456.14014; MR0601520.
  • 11. Demazure, Michel and Grothendieck, Alexandre: Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes. Number 151-153 in Lecture Notes in Mathematics, Springer, Berlin, 1970. SGA3. vol. 151: DOI 10.1007/bfb0058993; zbl 0207.51401 vol. 152: DOI 10.1007/bfb0059005; zbl 0209.24201; vol. 153: DOI 10.1007/bfb0059027; zbl 0212.52810; MR0274460.
  • 12. Deninger, Christopher and Murre, Jacob: Motivic decomposition of abelian schemes and the Fourier transform. J. Reine Angew. Math., 422 (1991), 201-219. zbl 0745.14003; MR1133323.
  • 13. Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc et al.: Fundamental algebraic geometry: Grothendieck's FGA explained. Mathematical Surveys and Monographs 123. Providence, RI: American Mathematical Society (AMS). x + 339 p., 2005. zbl 1085.14001; MR2222646.
  • 14. Freitag, Eberhard and Kiehl, Reinhardt: Etale cohomology and the Weil conjecture. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, 1988. zbl 0643.14012; MR0926276.
  • 15. Fu, Lei: Étale cohomology theory. Nankai Tracts in Mathematics 13. Hackensack, NJ: World Scientific. ix + 611 p., 2011. zbl 1228.14001; MR2791606.
  • 16. Grothendieck, Alexandre and Dieudonné, Jean: Éléments de géométrie algébrique: I. Le langage des schémas. Math. IHÉS, 4 (1960) 5-228. DOI 10.1007/bf02684778; zbl 0118.36206; MR0217083.
  • 17. Grothendieck, Alexandre and Dieudonné, Jean: Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie. Publ. Math. IHÉS, 32 (1967), 5-361. zbl 0153.22301; MR0238860.
  • 18. Gelfand, Sergei I. and Manin, Yuri I.: Methods of homological algebra. Transl. from the Russian. 2nd ed., Berlin: Springer, 2003. zbl 1006.18001; MR1950475.
  • 19. Grothendieck, Alexandre: Séminaire de Géométrie Algébrique du Bois Marie - 1960-61 - Revêtements étales et groupe fondamental - (SGA 1). Lecture Notes in Mathematics 224, Berlin; New York: Springer-Verlag. xxii + 447 p., 1971. DOI 10.1007/bfb0058656; zbl 0234.14002.
  • 20. Grothendieck, Alexandre: Standard conjectures on algebraic cycles. Algebr. Geom., Bombay Colloq. 1968, 193-199 (1969). zbl 0201.23301; MR0268189.
  • 21. Graber, Tom and Starr, Jason Michael: Restriction of sections for families of abelian varieties. In: A celebration of algebraic geometry. A conference in honor of Joe Harris' 60th birthday, Harvard University, Cambridge, MA, USA, August 25-28, 2011, pp. 311-327, Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute, 2013. zbl 1327.14201; MR3114946.
  • 22. Görtz, Ulrich and Wedhorn, Torsten: Algebraic geometry I. Schemes. With examples and exercises. Advanced Lectures in Mathematics. Wiesbaden: Vieweg+Teubner. vii + 615 p., 2010. zbl 1213.14001; MR2675155.
  • 23. Hartshorne, Robin: Algebraic geometry. Corr. 3rd printing. Graduate Texts in Mathematics 52. New York-Heidelberg-Berlin: Springer-Verlag. xvi + 496 p., 1983. zbl 0531.14001.
  • 24. Hindry, Marc and Silverman, Joseph H.: Diophantine geometry. An introduction. New York, NY: Springer, 2000. zbl 0948.11023; MR1745599.
  • 25. Huybrechts, Daniel: Lectures on $K$3 surfaces, volume 158. Cambridge: Cambridge University Press, 2016. DOI 10.1017/cbo9781316594193; zbl 1360.14099; MR3586372.
  • 26. Ito, Kazuhiro: Finiteness of Brauer groups of $K 3$ surfaces in characteristic 2. Int. J. Number Theory, 14(6) (2018), 1813-1825. DOI 10.1142/S1793042118501087; zbl 1398.14044; MR3827960; arxiv 1704.06232.
  • 27. Jannsen, Uwe: Continuous étale cohomology. Math. Ann., 280(2) (1988), 207-245. DOI 10.1007/bf01456052; zbl 0649.14011; MR0929536.
  • 28. Jannsen, Uwe: Weights in arithmetic geometry. Jpn. J. Math., (3), 5(1) (2010), 73-102. DOI 10.1007/s11537-010-0947-4; zbl 1204.14011; MR2609323; arxiv 1003.0927.
  • 29. Jossen, Peter: On the arithmetic of $1-$motives. PhD thesis, Central European University Budapest, 2009. http://www.jossenpeter.ch/PdfDvi/Dissertation.pdf.
  • 30. Keller, Timo: On the Tate-Shafarevich group of Abelian schemes over higher dimensional bases over finite fields. Manuscripta Math., 150(1-2) (2016), 211-245. https://www.timokeller.name/TateShafarevich.pdf. DOI 10.1007/s00229-015-0803-1; zbl 1342.19005; MR3483177; arxiv 1410.5293.
  • 31. Keller, Timo: Finiteness properties for flat cohomology of varieties over finite fields. Preprint, 2018. https://www.timokeller.name/Shap.pdf.
  • 32. Katz, Nicholas M. and Mazur, Barry: Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108. Princeton, New Jersey: Princeton University Press. xiv + 514 p., 1985. DOI 10.1515/9781400881710; zbl 0576.14026; MR0772569.
  • 33. Kato, Kazuya and Trihan, Fabien: On the conjectures of Birch and Swinnerton-Dyer in characteristic $p > 0$. In: Invent. Math., 153(3) (2003), 537-592. DOI 10.1007/s00222-003-0299-2; zbl 1046.11047; MR2000469.
  • 34. Kiehl, Reinhardt and Weissauer, Rainer: Weil conjectures, perverse sheaves and $l$-adic Fourier transform. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 42. Berlin: Springer. xii + 375 p., 2001. zbl 0988.14009; MR1855066.
  • 35. Lang, Serge: Abelian varieties. Interscience Tracts in Pure and Applied Mathematics 7. New York-London: Interscience Publishers. xii + 256 p., 1958. zbl 0098.13201; MR0106225.
  • 36. Lang, Serge: Algebra. Graduate Texts in Mathematics 211, Revised Third Edition, Springer-Verlag, 2002. zbl 0984.00001; MR1878556.
  • 37. Liedtke, Christian: A note on non-reduced Picard schemes. J. Pure Appl. Algebra, 213(5) (2009), 737-741. DOI 10.1016/j.jpaa.2008.09.001; zbl 1156.14031; MR2494366; arxiv 0805.0848.
  • 38. Liu, Qing: Algebraic geometry and arithmetic curves. Transl. by Reinie Erné. Oxford Graduate Texts in Mathematics 6. Oxford: Oxford University Press. xv + 577 p., 2006. zbl 1103.14001; MR1917232.
  • 39. Liu, Qing; Lorenzini, Dino and Raynaud, Michel: On the Brauer group of a surface. Invent. Math., 159(3) (2005), 673-676. DOI 10.1007/s00222-004-0403-2; zbl 1077.14023; MR2125738.
  • 40. Moret-Bailly, Laurent: Métriques permises. Szpiro, Lucien (ed.), Séminaire sur les pinceaux arithmétiques: La conjecture de Mordell, Paris, 1983-84, Astérisque 127 (1985), 29-87. zbl 1182.11028; MR0801918.
  • 41. Mumford, David and Fogarty, John: Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 34. Berlin-Heidelberg-New York: Springer-Verlag. xii + 220 p., 1982. zbl 0504.14008; MR0719371.
  • 42. Milne, James S.: The Tate-Safarevic Group of a Constant Abelian Variety. Invent. Math., 6 (1968), 91-105. DOI 10.1007/bf01389836; zbl 0159.22402; MR0244264.
  • 43. Milne, James S.: Étale cohomology. Princeton Mathematical Series 33. Princeton, New Jersey: Princeton University Press. xiii + 323 p., 1980. zbl 0433.14012; MR0559531.
  • 44. Milne, James S.: Abelian varieties. Arithmetic Geometry, Pap. Conf., Storrs/Conn. 1984, 103-150 (1986). zbl 0604.14028; MR0861974.
  • 45. Milne, James S.: Arithmetic duality theorems. Perspectives in Mathematics, Vol. 1. Boston etc.: Academic Press. Inc. Harcourt Brace Jovanovich Publishers. x + 421 p., 1986. zbl 0613.14019.
  • 46. Milne, James S.: Motivic cohomology and values of zeta functions. Compos. Math., 68(1) (1988), 59-102. zbl 0681.14007; MR0962505.
  • 47. Mochizuki, Shinichi: Topics in absolute anabelian geometry I: Generalities. J. Math. Sci., Tokyo, 19(2) (2012), 139-242. zbl 1267.14039; MR2987306.
  • 48. Mumford, David: Abelian varieties. Oxford University Press. viii + 242 p., 1970. zbl 0223.14022; MR0282985.
  • 49. Neukirch, Jürgen; Schmidt, Alexander and Wingberg, Kay: Cohomology of Number Fields. Grundlehren der mathematischen Wissenschaften, Vol. 323, Springer-Verlag, 2000. zbl 0948.11001; MR1737196.
  • 50. Oda, Tadao: The first de Rham cohomology group and Dieudonné modules. Ann. Sci. Éc. Norm. Supér. (4), 2 (1969), 63-135. DOI 10.24033/asens.1175; zbl 0175.47901; MR0241435.
  • 51. Oort, Frans: Commutative group schemes. Lecture Notes in Mathematics 15. Berlin-Heidelberg-New York: Springer-Verlag vi + 133 p., 1966. DOI 10.1007/bfb0097479; zbl 0216.05603; MR0213365.
  • 52. Oort, Frans: Subvarieties of moduli spaces. Invent. Math., 24 (1974), 95-119. DOI 10.1007/bf01404301; zbl 0259.14011; MR0424813.
  • 53. Oort, Frans: A stratification of a moduli space of abelian varieties. In: Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999, pp. 345-416, Basel: Birkhäuser, 2001. zbl 1052.14047; MR1827027.
  • 54. Poonen, Bjorn: Bertini theorems over finite fields. Ann. Math., 160(3) (2005), 1099-1127. DOI 10.4007/annals.2004.160.1099; zbl 1084.14026; MR2144974; arxiv math/0204002.
  • 55. Raynaud, Michel: Faisceaux Amples sur les Schémas en Groupes et les Espaces Homogènes. Lecture Notes in Mathematics 119. Berlin-Heidelberg-New York: Springer-Verlag, 1970. DOI 10.1007/bfb0059504; zbl 0195.22701; MR0260758.
  • 56. Schneider, Peter: On the values of the zeta function of a variety over a finite field. Compos. Math., 46 (1982), 133-143. zbl 0505.14020; MR0659920.
  • 57. Schneider, Peter: Zur Vermutung von Birch und Swinnerton-Dyer über globalen Funktionenkörpern. Math. Ann., 260 (1982), 495-510. DOI 10.1007/BF01457028; zbl 0509.14022; MR0670197.
  • 58. Silverman, Joseph H.: The arithmetic of elliptic curves. 2nd ed., New York, NY: Springer, 2009. DOI 10.1007/978-0-387-09494-6; zbl 1194.11005; MR2514094.
  • 59. Stacks Project Authors, The: Stacks Project. http://stacks.math.columbia.edu, 2018.
  • 60. Skorobogatov, Alexei N. and Zarhin, Yuri G.: A finiteness theorem for the Brauer group of $K3$ surfaces in odd characteristic. Int. Math. Res. Not., 2015(21) (2015), 11,404-11,418. DOI 10.1093/imrn/rnv030; zbl 1347.14003; MR3456048; arxiv 1403.0849.
  • 61. Tate, John T.: Endomorphisms of Abelian varieties over finite fields. Invent. Math., 2 (1966), 134-144. DOI 10.1007/bf01404549; zbl 0147.20303; MR0206004.
  • 62. Tate, John T.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3 (1968), 189-214; Sém. Bourbaki 1965/66, Exp. No. 306 (1966), 415-440. zbl 0199.55604; MR3202555.
  • 63. Tate, John T.: $p$-divisible groups. Proc. Conf. Local Fields, NUFFIC Summer School Driebergen 1966, 158-183, 1967. zbl 0157.27601; MR0231827.
  • 64. Zeta function of Abelian variety over finite field. MathOverflow 2017. Version: 2017-09-29, https://mathoverflow.net/q/282315.
  • 65. Tate, John T. and Shafarevich, Igor R.: The rank of elliptic curves. Sov. Math., Dokl., 8 (1967), 917-920. zbl 0168.42201.
  • 66. Ulmer, Douglas: Elliptic curves with large rank over function fields. Ann. Math. (2), 155(1) (2002), 295-315. DOI 10.2307/3062158; zbl 1109.11314; MR1888802; arxiv math/0109163.
  • 67. Picard of the product of two curves. MathOverflow 2014. Version: 2014-11-18, https://mathoverflow.net/q/187445.
  • 68. Weibel, Charles A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994. zbl 0797.18001; MR1269324.
  • 69. Zarkhin, Yuri G.: The Brauer group of an abelian variety over a finite field. Math. USSR, Izv., 20 (1983), 203-234. DOI 10.1070/im1983v020n02abeh001348; zbl 0514.14016.

Affiliation

Keller, Timo
Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

Downloads