On the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$, I: Modular curves
Author:
Ashay A. Burungale
Journal:
J. Algebraic Geom. 29 (2020), 329-371
DOI:
https://doi.org/10.1090/jag/748
Published electronically:
January 8, 2020
MathSciNet review:
4069652
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Abstract |
References |
Additional Information
Abstract: Generalised Heegner cycles are associated with a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf {Q}$. Let $p$ be an odd prime split in $K/\mathbf {Q}$ and let $\ell \neq p$ be an odd unramified prime. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf {Z}_\ell$-anticylotomic extension of $K$. The result is evidence for the refined Bloch–Beilinson and the Bloch–Kato conjecture. In the case of weight two and $\ell$ an ordinary prime, it provides a non-trivial refinement of the results of Cornut and Vatsal on Mazur’s conjecture regarding the non-triviality of Heegner points over the $\mathbf {Z}_\ell$-anticylotomic extension of $K$. In the case of weight two and $\ell$ a supersingular prime, it settles Mazur’s conjecture earlier known just for $\ell$ ordinary.
References
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- A. Burungale and D. Disegni, On the non-vanishing of $p$-adic heights for CM abelian varieties, and the arithmetic of Katz p-adic L-functions, arXiv:1803.09268, 2018, Ann. Inst. Fourier (Grenoble) (to appear).
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- Haruzo Hida, $p$-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004. MR 2055355
- Haruzo Hida, Non-vanishing modulo $p$ of Hecke $L$-values and application, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 207–269. MR 2392356, DOI https://doi.org/10.1017/CBO9780511721267.007
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- Haruzo Hida, Elliptic curves and arithmetic invariants, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3098991
- Benjamin Howard, Special cohomology classes for modular Galois representations, J. Number Theory 117 (2006), no. 2, 406–438. MR 2213774, DOI https://doi.org/10.1016/j.jnt.2005.07.002
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- Ming-Lun Hsieh, Special values of anticyclotomic Rankin-Selberg $L$-functions, Doc. Math. 19 (2014), 709–767. MR 3247801, DOI https://doi.org/10.1016/j.cnsns.2013.07.005
- H. Jacquet and R. P. Langlands, Automorphic forms on ${\rm GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
- Nicholas M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. MR 513095, DOI https://doi.org/10.1007/BF01390187
- Yifeng Liu, Shouwu Zhang, and Wei Zhang, A $p$-adic Waldspurger formula, Duke Math. J. 167 (2018), no. 4, 743–833. MR 3769677, DOI https://doi.org/10.1215/00127094-2017-0045
- B. Mazur, Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 185–211. MR 804682
- Jan Nekovář, Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992), no. 1, 99–125. MR 1135466, DOI https://doi.org/10.1007/BF01231883
- Jan Nekovář, $p$-adic Abel-Jacobi maps and $p$-adic heights, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 367–379. MR 1738867, DOI https://doi.org/10.1090/crmp/024/18
- Takeshi Saito, Weight-monodromy conjecture for $l$-adic representations associated to modular forms. A supplement to: “Modular forms and $p$-adic Hodge theory” [Invent. Math. 129 (1997), no. 3, 607–620; MR1465337 (98g:11060)], The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 427–431. MR 1744955
- Chad Schoen, Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 (1986), no. 3, 771–794. MR 860672, DOI https://doi.org/10.1215/S0012-7094-86-05343-3
- A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. MR 1047142, DOI https://doi.org/10.1007/BF01231194
- Goro Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149–192. MR 156001, DOI https://doi.org/10.2307/1970507
- Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449
- Eric Urban, Nearly overconvergent modular forms, Iwasawa theory 2012, Contrib. Math. Comput. Sci., vol. 7, Springer, Heidelberg, 2014, pp. 401–441. MR 3586822
- V. Vatsal, Uniform distribution of Heegner points, Invent. Math. 148 (2002), no. 1, 1–46. MR 1892842, DOI https://doi.org/10.1007/s002220100183
- V. Vatsal, Special values of anticyclotomic $L$-functions, Duke Math. J. 116 (2003), no. 2, 219–261. MR 1953292, DOI https://doi.org/10.1215/S0012-7094-03-11622-1
- Vinayak Vatsal, Special values of $L$-functions modulo $p$, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 501–514. MR 2275607
- J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242 (French). MR 783511
References
- Esther Aflalo and Jan Nekovář, Non-triviality of CM points in ring class field towers, with with an appendix by Christophe Cornut, Israel J. Math. 175 (2010), 225–284. MR 2607546, DOI https://doi.org/10.1007/s11856-010-0011-3
- Massimo Bertolini, Henri Darmon, and Kartik Prasanna, Generalized Heegner cycles and $p$-adic Rankin $L$-series, with an appendix by Brian Conrad, Duke Math. J. 162 (2013), no. 6, 1033–1148. MR 3053566, DOI https://doi.org/10.1215/00127094-2142056
- Massimo Bertolini, Henri Darmon, and Kartik Prasanna, $p$-adic Rankin $L$-series and rational points on CM elliptic curves, Pacific J. Math. 260 (2012), no. 2, 261–303. MR 3001796, DOI https://doi.org/10.2140/pjm.2012.260.261
- Massimo Bertolini, Henri Darmon, and Kartik Prasanna, $p$-adic $L$-functions and the coniveau filtration on Chow groups, with an appendix by Brian Conrad, J. Reine Angew. Math. 731 (2017), 21–86. MR 3709060, DOI https://doi.org/10.1515/crelle-2014-0150
- Spencer Bloch and Kazuya Kato, $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400. MR 1086888
- Miljan Brakočević, Anticyclotomic $p$-adic $L$-function of central critical Rankin-Selberg $L$-value, Int. Math. Res. Not. IMRN 21 (2011), 4967–5018. MR 2852303, DOI https://doi.org/10.1093/imrn/rnq275
- M. Brakocevic, Non-vanishing modulo $p$ of central critical Rankin-Selberg L-values with anticyclotomic twists, https://arxiv.org/abs/1010.6066, 2010.
- Ashay A. Burungale, An $l\ne p$-interpolation of genuine $p$-adic L-functions, Res. Math. Sci. 3 (2016), Paper No. 16, 26 pp. MR 3518730, DOI https://doi.org/10.1186/s40687-016-0060-2
- Ashay A. Burungale and Haruzo Hida, André-Oort conjecture and nonvanishing of central $L$-values over Hilbert class fields, Forum Math. Sigma 4 (2016), e20, 26. MR 3523339, DOI https://doi.org/10.1017/fms.2015.30
- Ashay A. Burungale, Non-triviality of generalised Heegner cycles over anticyclotomic towers: a survey, $p$-adic aspects of modular forms, World Sci. Publ., Hackensack, NJ, 2016, pp. 279–306. MR 3587960
- Ashay A. Burungale, On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves, J. Inst. Math. Jussieu 16 (2017), no. 1, 189–222. MR 3591965, DOI https://doi.org/10.1017/S147474801500016X
- Ashay A. Burungale and Haruzo Hida, $\mathfrak {p}$-rigidity and Iwasawa $\mu$-invariants, Algebra Number Theory 11 (2017), no. 8, 1921–1951. MR 3720935, DOI https://doi.org/10.2140/ant.2017.11.1921
- A. Burungale and Y. Tian, Horizontal non-vanishing of Heegner points and toric periods, Adv. Math. (to appear).
- A. Burungale and D. Disegni, On the non-vanishing of $p$-adic heights for CM abelian varieties, and the arithmetic of Katz p-adic L-functions, arXiv:1803.09268, 2018, Ann. Inst. Fourier (Grenoble) (to appear).
- Christophe Cornut, Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), no. 3, 495–523. MR 1908058, DOI https://doi.org/10.1007/s002220100199
- Christophe Cornut and Vinayak Vatsal, Nontriviality of Rankin-Selberg $L$-functions and CM points, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186. MR 2392354, DOI https://doi.org/10.1017/CBO9780511721267.005
- Ching-Li Chai, Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Invent. Math. 121 (1995), no. 3, 439–479. MR 1353306, DOI https://doi.org/10.1007/BF01884309
- C.-L. Chai, Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear subvarieties and Hecke orbits, preprint, 2003.
- Ching-Li Chai, Hecke orbits as Shimura varieties in positive characteristic, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 295–312. MR 2275599
- Benedict H. Gross and James A. Parson, On the local divisibility of Heegner points, Number theory, analysis and geometry, Springer, New York, 2012, pp. 215–241. MR 2867919, DOI https://doi.org/10.1007/978-1-4614-1260-1_11
- Haruzo Hida, Non-vanishing modulo $p$ of Hecke $L$-values, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 735–784. MR 2099085
- Haruzo Hida, $p$-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer-Verlag, New York, 2004. MR 2055355
- Haruzo Hida, Non-vanishing modulo $p$ of Hecke $L$-values and application, $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser., vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 207–269. MR 2392356, DOI https://doi.org/10.1017/CBO9780511721267.007
- Haruzo Hida, Irreducibility of the Igusa tower, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 1, 1–20. MR 2465518, DOI https://doi.org/10.1007/s10114-008-6490-z
- Haruzo Hida, Elliptic curves and arithmetic invariants, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3098991
- Benjamin Howard, Special cohomology classes for modular Galois representations, J. Number Theory 117 (2006), no. 2, 406–438. MR 2213774, DOI https://doi.org/10.1016/j.jnt.2005.07.002
- Ming-Lun Hsieh, On the non-vanishing of Hecke $L$-values modulo $p$, Amer. J. Math. 134 (2012), no. 6, 1503–1539. MR 2999287, DOI https://doi.org/10.1353/ajm.2012.0049
- Ming-Lun Hsieh, Special values of anticyclotomic Rankin-Selberg $L$-functions, Doc. Math. 19 (2014), 709–767. MR 3247801
- H. Jacquet and R. P. Langlands, Automorphic forms on $\mathrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
- Nicholas M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297. MR 513095, DOI https://doi.org/10.1007/BF01390187
- Yifeng Liu, Shouwu Zhang, and Wei Zhang, A $p$-adic Waldspurger formula, Duke Math. J. 167 (2018), no. 4, 743–833. MR 3769677, DOI https://doi.org/10.1215/00127094-2017-0045
- B. Mazur, Modular curves and arithmetic, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 185–211. MR 804682
- Jan Nekovář, Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992), no. 1, 99–125. MR 1135466, DOI https://doi.org/10.1007/BF01231883
- Jan Nekovář, $p$-adic Abel-Jacobi maps and $p$-adic heights, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) CRM Proc. Lecture Notes, vol. 24, Amer. Math. Soc., Providence, RI, 2000, pp. 367–379. MR 1738867
- Takeshi Saito, Weight-monodromy conjecture for $l$-adic representations associated to modular forms. A supplement to: “Modular forms and $p$-adic Hodge theory” [Invent. Math. 129 (1997), no. 3, 607–620; MR1465337 (98g:11060)], The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., Dordrecht, 2000, pp. 427–431. MR 1744955
- Chad Schoen, Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 (1986), no. 3, 771–794. MR 860672, DOI https://doi.org/10.1215/S0012-7094-86-05343-3
- A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430. MR 1047142, DOI https://doi.org/10.1007/BF01231194
- Goro Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149–192. MR 156001, DOI https://doi.org/10.2307/1970507
- Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46, Princeton University Press, Princeton, NJ, 1998. MR 1492449
- Eric Urban, Nearly overconvergent modular forms, Iwasawa theory 2012, Contrib. Math. Comput. Sci., vol. 7, Springer, Heidelberg, 2014, pp. 401–441. MR 3586822
- V. Vatsal, Uniform distribution of Heegner points, Invent. Math. 148 (2002), no. 1, 1–46. MR 1892842, DOI https://doi.org/10.1007/s002220100183
- V. Vatsal, Special values of anticyclotomic $L$-functions, Duke Math. J. 116 (2003), no. 2, 219–261. MR 1953292, DOI https://doi.org/10.1215/S0012-7094-03-11622-1
- Vinayak Vatsal, Special values of $L$-functions modulo $p$, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 501–514. MR 2275607
- J.-L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math. 54 (1985), no. 2, 173–242 (French). MR 783511
Additional Information
Ashay A. Burungale
Affiliation:
Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540; and California Institute of Technolology, 1200 East California Boulevard, Pasadena California 91125
MR Author ID:
890463
Email:
ashayburungale@gmail.com
Received by editor(s):
July 9, 2017
Received by editor(s) in revised form:
July 29, 2018, and June 25, 2019
Published electronically:
January 8, 2020
Article copyright:
© Copyright 2020
University Press, Inc.