Abstract
In this paper, we deduce the vanishing of Selmer groups for the Rankin–Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated L-value, thus establishing the rank 0 case of the Bloch–Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and p-adic L-functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin’s method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur’s conjecture (as proven in weight 2 by Cornut–Vatsal), and as a consequence of our results, we deduce from Nekovář’s work a proof of the parity conjecture in this setting.
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Notes
Here our convention is that p-adic cyclotomic character has Hodge–Tate weight \(+1\).
As explained in [30, Example (5.3.4)(5)], this follows from properties [loc.cit.,(2)-(3)] for \(\mathcal T_\phi \), whose verification is immediate. Indeed, \((\mathcal T_\phi ,\mathcal T^+_{p,\phi })\) satisfies the Panchishkin condition of [30, Def. (3.3.1)] by construction, and \(\mathcal T_\phi \) is pure of weight 1 at all finite places, since Ramanujan��s conjecture holds for f; and anticyclotomic Hecke characters are pure of weight 0.
References
Bertolini, M., Darmon, H.: Kolyvagin’s descent and Mordell–Weil groups over ring class fields. J. Reine Angew. Math. 412, 63–74 (1990)
Bertolini, M., Darmon, H., Prasanna, K.: Generalized Heegner cycles and \(p\)-adic Rankin \(L\)-series. Duke Math. J. 162(6), 1033–1148 (2013)
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, Boston (1990)
Brakočević, M.: Anticyclotomic \(p\)-adic \(L\)-function of central critical Rankin–Selberg \(L\)-value. Int. Math. Res. Not. 2011(21), 4967–5018 (2011)
Castella, F.: Heegner cycles and higher weight specializations of big Heegner points. Math. Ann. 356(4), 1247–1282 (2013)
Castella, F.: On the \(p\)-adic Variation of Heegner Points. Preprint. arXiv:1410.6591 (2014)
Deligne, P.: Formes modulaires et représentations \(l\)-adiques. In: Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363. Lecture Notes in Mathematics, vol. 175, Exp. No. 355, pp. 139–172. Springer, Berlin (1971)
Deligne, P.: La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980)
de Shalit, E.: Iwasawa theory of elliptic curves with complex multiplication: \(p\)-adic \(L\) functions. In: Perspectives in Mathematics, vol. 3. Academic Press, Inc., Boston (1987)
Faltings, G., Chai, C.-L.: Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Springer, Berlin. (1990). With an appendix by David Mumford
Fontaine, J.-M.: Représentations \(p\)-adiques semi-stables. Astérisque, 223, 113–184 (1994). Périodes \(p\)-adiques (Bures-sur-Yvette, 1988). With an appendix by Pierre Colmez
Fontaine, J.-M., Perrin-Riou, B.: Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions \(L\). In: Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math, vol. 55, pp. 599–706. American Mathematical Society, Providence (1994)
Hida, H.: Elementary theory of \(L\)-functions and Eisenstein series. In: London Mathematical Society Student Texts, vol. 26. Cambridge University Press, Cambridge (1993)
Hida, H.: \(p\)-adic automorphic forms on Shimura varieties. In: Springer Monographs in Mathematics. Springer, New York (2004)
Howard, B.: Variation of Heegner points in Hida families. Invent. Math. 167(1), 91–128 (2007)
Hsieh, M.-L.: Special values of anticyclotomic Rankin–Selberg \(L\)-functions. Doc. Math. 19, 709–767 (2014)
Jacquet, H.: Automorphic forms on \({\rm GL}(2)\). Part II. In: Lecture Notes in Mathematics, vol. 278, Springer, Berlin (1972)
Jacquet, H., Langlands, R.P.: Automorphic forms on \({\rm GL}(2)\). In: Lecture Notes in Mathematics, vol. 114. Springer, Berlin (1970)
Katz, N.M.: \(p\)-adic properties of modular schemes and modular forms. Modular functions of one variable, III. In: Proceedings of the International Summer School, University of Antwerp, Antwerp, 1972. Lecture Notes in Mathematics, vol. 350, pp. 69–190. Springer, Berlin (1973)
Katz, N.: \(p\)-adic \(L\)-functions for CM fields. Invent. Math. 49(3), 199–297 (1978)
Katz, N.: Serre–Tate local moduli, Algebraic surfaces (Orsay, 1976–78). In: Lecture Notes in Math, vol. 868, pp. 138–202. Springer, Berlin (1981)
Loeffler, D., Zerbes, S.L.: Iwasawa theory and \(p\)-adic \(L\)-functions over \(\mathbb{Z}_p^2\)-extensions. Int. J. Number Theory 10(8), 2045–2095 (2014)
Loeffler, D., Zerbes, S.L.: Rankin–Eisenstein classes in Coleman families. Res. Math. Sci. 3(29), 53 (2016)
Mazur, B.: Modular curves and arithmetic. In: Proceedings of the International Congress of Mathematicians (Warsaw, 1983), vol. 1, no 2, pp. 185–211. PWN, Warsaw (1984)
Mumford, D.: Abelian varieties. In: Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay, 2008. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second edition (1974)
Nekovář, J.: Kolyvagin’s method for Chow groups of Kuga–Sato varieties. Invent. Math. 107(1), 99–125 (1992)
Nekovář, J.: On the \(p\)-adic height of Heegner cycles. Math. Ann. 302(4), 609–686 (1995)
Nekovář, J.: \(p\)-adic Abel–Jacobi maps and \(p\)-adic heights. In: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998). CRM Proceedings and Lecture Notes, vol. 24, American Mathematical Society, Providence, pp. 367–379 (2000)
Nekovář, J.: Selmer complexes, Astérisque, no. 310, viii+559 (2006)
Nekovář, J.: On the parity of ranks of Selmer groups. III. Doc. Math. 12, 243–274 (2007)
Nekovář, J.: Erratum for “On the parity of ranks of Selmer groups. III” cf. Documenta Math. 12: 243–274 [mr2350290]. Doc. Math. 14(2009), 191–194 (2007)
Niziol, W.: On the image of \(p\)-adic regulators. Invent. Math. 127(2), 375–400 (1997)
Perrin-Riou, B.: Théorie d’Iwasawa des représentations \(p\)-adiques sur un corps local. Invent. Math. 115(1), 81–161 (1994). With an appendix by Jean-Marc Fontaine
Rubin, K.: Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. Invent. Math. 64(3), 455–470 (1981)
Rubin, K.: Euler systems. In: Annals of Mathematics Studies, Hermann Weyl Lectures, vol. 147. The Institute for Advanced Study, Princeton University Press, Princeton (2000)
Scholl, A.J.: Motives for modular forms. Invent. Math. 100(2), 419–430 (1990)
Schmidt, R.: Some remarks on local newforms for \({\rm GL}(2)\). J. Ramanujan Math. Soc. 17(2), 115–147 (2002)
Shimura, G.: Abelian Varieties with Complex Multiplication and Modular Functions, Princeton Mathematical Series, vol. 46. Princeton University Press, Princeton, NJ (1998)
Shnidman, A.: \(p\)-adic heights of generalized Heegner cycles. Ann. Inst. Fourier (Grenoble) 66(3), 1117–1174 (2016)
Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. (2) 88, 492–517 (1968)
Tate, J.: Number theoretic background, Automorphic forms, representations and \(L\)-functions. In: Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII. American Mathamatical Society, Providence, pp. 3–26 (1979)
Tsuji, T.: \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2), 233–411 (1999)
Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math. 94(3), 529–573 (1988)
Yang, T.: On CM abelian varieties over imaginary quadratic fields. Math. Ann. 329(1), 87–117 (2004)
Zhang, S.: Heights of Heegner cycles and derivatives of \(L\)-series. Invent. Math. 130(1), 99–152 (1997)
Acknowledgements
Fundamental parts of this paper were written during the visits of the first-named author to the second-named author in Taipei during February 2014 and August 2014; it is a pleasure to thank NCTS and the National Taiwan University for their hospitality and financial support. We would also like to thank Ben Howard, Shinichi Kobayashi and David Loeffler for their comments and enlightening conversations related to this work.
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Communicated by Toby Gee.
During the preparation of this paper, F. Castella was partially supported by Grant MTM2012-34611 and by Prof. Hida’s NSF Grant DMS-0753991. M.-L. Hsieh was partially supported by a MOST Grant 103-2115-M-002-012-MY5.
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Castella, F., Hsieh, ML. Heegner cycles and p-adic L-functions. Math. Ann. 370, 567–628 (2018). https://doi.org/10.1007/s00208-017-1517-3
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DOI: https://doi.org/10.1007/s00208-017-1517-3