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Hirzebruch–Zagier cycles and twisted triple product Selmer groups

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Let E be an elliptic curve over \({{\mathbb {Q}}}\) and A another elliptic curve over a real quadratic number field. We construct a \({{\mathbb {Q}}}\)-motive of rank 8, together with a distinguished class in the associated Bloch–Kato Selmer group, using Hirzebruch–Zagier cycles, that is, graphs of Hirzebruch–Zagier morphisms. We show that, under certain assumptions on E and A, the non-vanishing of the central critical value of the (twisted) triple product L-function attached to (E, A) implies that the dimension of the associated Bloch–Kato Selmer group of the motive is 0; and the non-vanishing of the distinguished class implies that the dimension of the associated Bloch–Kato Selmer group of the motive is 1. This can be viewed as the triple product version of Kolyvagin’s work on bounding Selmer groups of a single elliptic curve using Heegner points.

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References

  1. Bachmat, E., Goren, E.Z.: On the non-ordinary locus in Hilbert–Blumenthal surfaces. Math. Ann. 313(3), 475–506 (1999). doi:10.1007/s002080050270

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertolini, M., Darmon, H.: Iwasawa’s main conjecture for elliptic curves over anticyclotomic \(\mathbb{Z}_{p}\)-extensions. Ann. Math. (2) 162(1), 1–64 (2005). doi:10.4007/annals.2005.162.1

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertolini, M., Darmon, H.: Hida families and rational points on elliptic curves. Invent. Math. 168(2), 371–431 (2007). doi:10.1007/s00222-007-0035-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Bloch, S., Kato, K.: \(L\)-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift, vol. I. Progr. Math., vol. 86, pp. 333–400. Birkhäuser, Boston (1990)

  5. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21. Springer, Berlin (1990)

    Google Scholar 

  6. Boutot, J.-F., Carayol, H.: Uniformisation \(p\)-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel’d. Astérisque, pp. 196–197 (1991), vol. 7, pp. 45.158 (1992). (French, with English summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988)

  7. Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001). (electronic). doi:10.1090/S0894-0347-01-00370-8

  8. Breuil, C., Messing, W.: Torsion étale and crystalline cohomologies. Cohomologies \(p\)-adiques et applications arithmétiques, II. Astérisque 279, 81–124 (2002)

  9. Brylinski, J.-L., Labesse, J.-P.: Cohomologie d’intersection et fonctions \(L\) de certaines variétés de Shimura. Ann. Sci. École Norm. Sup. (4) 17(3), 361–412 (1984). (French)

    MathSciNet  Google Scholar 

  10. Bump, D., Friedberg, S., Hoffstein, J.: Nonvanishing theorems for \(L\)-functions of modular forms and their derivatives. Invent. Math. 102(3), 543–618 (1990). doi:10.1007/BF01233440

    Article  MathSciNet  MATH  Google Scholar 

  11. Darmon, H., Rotger, V.: Diagonal cycles and Euler systems I: a \(p\)-adic Gross–Zagier formula. Ann. Sci. Éc. Norm. Supér. (4) 47(4), 779–832 (2014). (English, with English and French summaries)

    MathSciNet  MATH  Google Scholar 

  12. Darmon, H., Rotger, V.: Diagonal cycles and Euler systems II: the Birch and Swinnerton–Dyer conjecture for Hasse–Weil–Artin \(L\)-series. (preprint). http://www.math.mcgill.ca/darmon/pub/pub.html

  13. Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980). (French)

    Article  MathSciNet  MATH  Google Scholar 

  14. Freitas, N., Le Hung, B.V., Siksek, S.: Elliptic curves over real quadratic fields are modular. Invent. Math. 201(1), 159–206 (2015). doi:10.1007/s00222-014-0550-z

    Article  MathSciNet  MATH  Google Scholar 

  15. Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber), Algebraic geometry: Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo 2002, 153–183 (2000)

  16. Gan, W.T., Gross, B.H., Prasad, D.: Symplectic local root numbers, central critical \(L\) values, and restriction problems in the representation theory of classical groups. Astérisque 346, 1–109 (2012). (English, with English and French summaries). Sur les conjectures de Gross et Prasad. I

    MathSciNet  MATH  Google Scholar 

  17. Garrett, P.B.: Decomposition of Eisenstein series: Rankin triple products. Ann. Math. (2) 125(2), 209–235 (1987). doi:10.2307/1971310

    Article  MathSciNet  MATH  Google Scholar 

  18. Goren, E.Z.: Lectures on Hilbert modular varieties and modular forms. In: CRM Monograph Series. With the Assistance of Marc–Hubert Nicole, vol. 14. American Mathematical Society, Providence (2002)

  19. Goren, E.Z., Oort, F.: Stratifications of Hilbert modular varieties. J. Algebraic Geom. 9(1), 111–154 (2000)

    MathSciNet  MATH  Google Scholar 

  20. Gross, B.H.: Kolyvagin’s work on modular elliptic curves, \(L\)-functions and arithmetic (Durham, 1989). London Math. Soc. Lecture Note Ser., vol. 153, pp. 235–256. Cambridge Univ. Press, Cambridge (1991). doi:10.1017/CBO9780511526053.009

  21. Gross, B.H., Keating, K.: On the intersection of modular correspondences. Invent. Math. 112(2), 225–245 (1993). doi:10.1007/BF01232433

    Article  MathSciNet  MATH  Google Scholar 

  22. Gross, B.H., Kudla, S.S.: Heights and the central critical values of triple product \(L\)-functions. Compositio Math. 81(2), 143–209 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Gross, B.H., Parson, J.A.: On the Local Divisibility of Heegner Points, Number Theory, Analysis and Geometry. Springer, New York (2012). doi:10.1007/978-1-4614-1260-1

    MATH  Google Scholar 

  24. Gross, B.H., Prasad, D.: On the decomposition of a representation of \({{\rm SO}}_{n}\) when restricted to \({{\rm SO}}_{n-1}\). Can. J. Math. 44(5), 974–1002 (1992). doi:10.4153/CJM-1992-060-8

    Article  MathSciNet  MATH  Google Scholar 

  25. Gross, B.H., Schoen, C.: The modified diagonal cycle on the triple product of a pointed curve. Ann. Inst. Fourier (Grenoble) 45(3), 649–679 (1995). (English, with English and French summaries)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gross, B.H., Zagier, D.B.: Heegner points and derivatives of \(L\)-series. Invent. Math. 84(2), 225–320 (1986). doi:10.1007/BF01388809

    Article  MathSciNet  MATH  Google Scholar 

  27. Grothendieck, D.A.: Avec la collaboration de M. Raynaud et D. S. Rim Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, vol. 288. Springer, Berlin (1972) (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I)

  28. Harder, G., Langlands, R.P., Rapoport, M.: Algebraische Zyklen auf Hilbert–Blumenthal–Flächen. J. Reine Angew. Math. 366, 53–120 (1986). (German)

    MathSciNet  MATH  Google Scholar 

  29. Helm, D.: On maps between modular Jacobians and Jacobians of Shimura curves. Israel J. Math. 160, 61–117 (2007). doi:10.1007/s11856-007-0056-0

    Article  MathSciNet  MATH  Google Scholar 

  30. Ichino, A.: Trilinear forms and the central values of triple product \(L\)-functions. Duke Math. J. 145(2), 281–307 (2008). doi:10.1215/00127094-2008-052

    Article  MathSciNet  MATH  Google Scholar 

  31. Jannsen, U.: Continuous étale cohomology. Math. Ann. 280(2), 207–245 (1988). doi:10.1007/BF01456052

    Article  MathSciNet  MATH  Google Scholar 

  32. Jannsen, U.: Mixed motives and algebraic \(K\)-theory. In: Lecture Notes in Mathematics, vol. 1400. Springer, Berlin (1990)

  33. de Jong, A.J.: Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math. 83, 51–93 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kida, M.: Potential good reduction of elliptic curves. J. Symbol. Comput. 34(3), 173–180 (2002). doi:10.1006/jsco.2002.0555

    Article  MathSciNet  MATH  Google Scholar 

  35. Kolyvagin, V.A.: Euler systems, The Grothendieck Festschrift, vol. II, Progr. Math., vol. 87, pp. 435-483. Birkhäuser, Boston (1990)

  36. Lapid, E., Rogawski, J.: On twists of cuspidal representations of GL(2). Forum Math. 10(2), 175–197 (1998). doi:10.1515/form.10.2.175

    Article  MathSciNet  MATH  Google Scholar 

  37. Le Hung Viet, B.: Modularity of some elliptic curves over totally real fields. Thesis (Ph.D.)-Harvard University, ProQuest LLC, Ann Arbor (2014)

  38. Milne, J.S.: Étale Cohomology. Princeton Mathematical Series, vol. 33. Princeton University Press, Princeton (1980)

    Google Scholar 

  39. Milne, J.S.: Arithmetic Duality Theorems, 2nd edn. BookSurge, LLC, Charleston (2006)

  40. Murre, J.P.: On the motive of an algebraic surface. J. Reine Angew. Math. 409, 190–204 (1990). doi:10.1515/crll.1990.409.190

    MathSciNet  MATH  Google Scholar 

  41. Nekovář, J.: \(p\)-adic Abel–Jacobi maps and p-adic heights. The arithmetic and geometry of algebraic cycles (Banff, AB, 1998). In: CRM Proc. Lecture Notes, vol. 24, pp. 367–379. Amer. Math. Soc. Providence (2000)

  42. Nicole, M.-H.: The supersingular locus of Hilbert modular surfaces modulo \(p\), 2000. Thesis (Ph.D.), McGill University

  43. Nizioł, W.: Cohomology of crystalline representations. Duke Math. J. 71(3), 747–791 (1993). doi:10.1215/S0012-7094-93-07128-1

    Article  MathSciNet  MATH  Google Scholar 

  44. Pollack, R., Weston, T.: On anticyclotomic \(\mu \)-invariants of modular forms. Compos. Math. 147(5), 1353–1381 (2011). doi:10.1112/S0010437X11005318

    Article  MathSciNet  MATH  Google Scholar 

  45. Prasad, D.: Invariant forms for representations of \(GL_{2}\) over a local field. Am. J. Math. 114(6), 1317–1363 (1992). doi:10.2307/2374764

    Article  MATH  Google Scholar 

  46. Piatetski-Shapiro, I., Rallis, S.: Rankin triple \(L\) functions. Compositio Math. 64(1), 31–115 (1987)

    MathSciNet  MATH  Google Scholar 

  47. Rapoport, M.: Compactifications de l’espace de modules de Hilbert–Blumenthal. Compositio Math. 36(3), 255–335 (1978). (French)

    MathSciNet  MATH  Google Scholar 

  48. Rapoport, M., Zink, T.: Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik. Invent. Math. 68(1), 21–101 (1982). doi:10.1007/BF01394268. (German)

    Article  MathSciNet  MATH  Google Scholar 

  49. Ribet, K.A.: Bimodules and abelian surfaces. Algebraic number theory. In: Adv. Stud. Pure Math., vol. 17, pp. 359-407. Academic Press, Boston (1989)

  50. Ribet, K.A.: On modular representations of Gal \((\overline{{\bf Q}}/{{\bf Q}})\) arising from modular forms. Invent. Math. 100(2), 431–476 (1990). doi:10.1007/BF01231195

    Article  MathSciNet  MATH  Google Scholar 

  51. Saito, S., Sato, K.: A finiteness theorem for zero-cycles over \(p\)-adic fields. Ann. Math. (2) 172(3), 1593–1639 (2010). doi:10.4007/annals.2010.172.1593. With an appendix by Uwe Jannsen

    Article  MathSciNet  MATH  Google Scholar 

  52. Scholl, A.J.: Classical motives, motives (Seattle, WA, 1991). In: Proc. Sympos. Pure Math., vol. 55, pp. 163–187. Amer. Math. Soc., Providence (1994)

  53. Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259–331 (1972). (French)

    Article  MathSciNet  Google Scholar 

  54. Skinner, C.: A converse to a theorem of Gross, Zagier, and Kolyvagin (2014). arXiv:0808.1725 [math]

  55. Tate, J.: Relations between \(K_{2}\) and Galois cohomology. Invent. Math. 36, 257–274 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  56. Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141(3), 553–572 (1995). doi:10.2307/2118560

    Article  MathSciNet  MATH  Google Scholar 

  57. Tian, Y., Xiao, L.: On Goren–Oort stratification for quaternionic Shimura varieties (2013). arXiv:1308.0790 [math]

  58. Tian, Y., Xiao, L.: Tate cycles on some quaternionic Shimura varieties mod \(p\) (2014). arXiv:1410.2321 [math]

  59. van der Geer, G.: Hilbert Modular Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16. Springer, Berlin (1988)

    Google Scholar 

  60. Vignéras, M.-F.: Arithmétique des algèbres de quaternions. In: Lecture Notes in Mathematics, vol. 800. Springer, Berlin (1980)

  61. Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995). doi:10.2307/2118559

    Article  MathSciNet  MATH  Google Scholar 

  62. Yuan, X., Zhang, S., Zhang, W.: Triple product \(L\)-series and Hirzebruch–Zagier cycles. https://web.math.princeton.edu/~shouwu/publications/triple2012. (preprint)

  63. Zhang, W.: On arithmetic fundamental lemmas. Invent. Math. 188(1), 197–252 (2012). doi:10.1007/s00222-011-0348-1

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhang, W.: Selmer groups and the indivisibility of Heegner points. Cambr. J. Math. 2(2), 191–253 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zink, T.: The display of a formal \(p\)-divisible group. Cohomologies p-adiques et applications arithmétiques, I. Astérisque 278, 127–248 (2002)

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Acknowledgments

The author would like to thank Henri Darmon, Wee Tech Gan, Benedict Gross, Atsushi Ichino, Ye Tian, Yichao Tian, Shouwu Zhang, and Wei Zhang for helpful discussions and comments. He appreciates Yichao Tian and Liang Xiao for sharing their preprint [58] at the early stage. He also thanks the anonymous referees for very careful reading and useful comments. The author is partially supported by NSF Grant DMS-1302000.

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    Yifeng Liu

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Liu, Y. Hirzebruch–Zagier cycles and twisted triple product Selmer groups. Invent. math. 205, 693–780 (2016). https://doi.org/10.1007/s00222-016-0645-9

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