p-adic heights of generalized Heegner cycles
Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1117-1174.

We relate the p-adic heights of generalized Heegner cycles to the derivative of a p-adic L-function attached to a pair (f,χ), where f is an ordinary weight 2r newform and χ is an unramified imaginary quadratic Hecke character of infinity type (,0), with 0<<2r. This generalizes the p-adic Gross-Zagier formula in the case =0 due to Perrin-Riou (in weight two) and Nekovář (in higher weight).

Nous relions les hauteurs p-adiques des cycles de Heegner généralisés à la dérivée d’une fonction L p-adique attachée à une paire (f,χ), où f est une forme modulaire ordinaire de poids 2r et χ est un caractère de Hecke non-ramifé de type (,0), pour 0<<2r. Ceci généralise la formule de Perrin-Riou (en poids deux) and Nekovář (poids plus élevé).

Published online:
DOI: 10.5802/aif.3033
Classification: 11G40, 11G18
Keywords: algebraic cycles, modular forms, $p$-adic $L$-functions
Mot clés : cycles algébriques, formes modulaires, fonctions $L$ $p$-adiques
Shnidman, Ariel 1

1 Department of Mathematics Boston College, Chestnut Hill MA 02467 (U.S.A.)
     author = {Shnidman, Ariel},
     title = {$p$-adic heights of generalized {Heegner} cycles},
     journal = {Annales de l'Institut Fourier},
     pages = {1117--1174},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {66},
     number = {3},
     year = {2016},
     doi = {10.5802/aif.3033},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3033/}
AU  - Shnidman, Ariel
TI  - $p$-adic heights of generalized Heegner cycles
JO  - Annales de l'Institut Fourier
PY  - 2016
SP  - 1117
EP  - 1174
VL  - 66
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3033/
DO  - 10.5802/aif.3033
LA  - en
ID  - AIF_2016__66_3_1117_0
ER  - 
%0 Journal Article
%A Shnidman, Ariel
%T $p$-adic heights of generalized Heegner cycles
%J Annales de l'Institut Fourier
%D 2016
%P 1117-1174
%V 66
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3033/
%R 10.5802/aif.3033
%G en
%F AIF_2016__66_3_1117_0
Shnidman, Ariel. $p$-adic heights of generalized Heegner cycles. Annales de l'Institut Fourier, Volume 66 (2016) no. 3, pp. 1117-1174. doi : 10.5802/aif.3033. https://aif.centre-mersenne.org/articles/10.5802/aif.3033/

[1] Bertolini, Massimo; Darmon, Henri; Prasanna, Kartik Generalized Heegner cycles and p-adic Rankin L-series, Duke Math. J., Volume 162 (2013) no. 6, pp. 1033-1148 (With an appendix by Brian Conrad) | DOI

[2] Bertolini, Massimo; Darmon, Henri; Prasanna, Kartik Chow-Heegner points on CM elliptic curves and values of p-adic L-functions, Int. Math. Res. Not. IMRN (2014) no. 3, pp. 745-793

[3] Bloch, Spencer; Kato, Kazuya L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progr. Math.), Volume 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400

[4] Castella, F; Hsieh, M Heegner cycles and p-adic L-functions (preprint)

[5] Castella, Francesc Heegner cycles and higher weight specializations of big Heegner points, Math. Ann., Volume 356 (2013) no. 4, pp. 1247-1282 | DOI

[6] Colmez, Pierre Fonctions L p-adiques, Astérisque (2000) no. 266, pp. Exp. No. 851, 3, 21-58 (Séminaire Bourbaki, Vol. 1998/99)

[7] Conrad, Brian Lifting global representations with local properties (preprint)

[8] Conrad, Brian Gross-Zagier revisited, Heegner points and Rankin L-series (Math. Sci. Res. Inst. Publ.), Volume 49, Cambridge Univ. Press, Cambridge, 2004, pp. 67-163 (With an appendix by W. R. Mann) | DOI

[9] Déglise, F; Niziol, W On p-adic absolute hodge cohomology and syntomic coefficients, I (preprint)

[10] Disegni, D p-adic heights of Heegner points on Shimura curves (preprint)

[11] Elias, Y On the Selmer group attached to a modular form and an algebraic Hecke character (preprint)

[12] Faltings, Gerd Crystalline cohomology and p-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80

[13] Gross, Benedict H. Arithmetic on elliptic curves with complex multiplication, Lecture Notes in Mathematics, 776, Springer, Berlin, 1980, iii+95 pages (With an appendix by B. Mazur)

[14] Gross, Benedict H.; Zagier, Don B. Heegner points and derivatives of L-series, Invent. Math., Volume 84 (1986) no. 2, pp. 225-320 | DOI

[15] Hida, Haruzo A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math., Volume 79 (1985) no. 1, pp. 159-195 | DOI

[16] Howard, Benjamin The Iwasawa theoretic Gross-Zagier theorem, Compos. Math., Volume 141 (2005) no. 4, pp. 811-846 | DOI

[17] Hunter Brooks, Ernest Shimura curves and special values of p-adic L-functions, Int. Math. Res. Not. IMRN (2015) no. 12, pp. 4177-4241 | DOI

[18] Katz, Nicholas M.; Mazur, Barry Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108, Princeton University Press, Princeton, NJ, 1985, xiv+514 pages

[19] Kobayashi, Shinichi The p-adic Gross-Zagier formula for elliptic curves at supersingular primes, Invent. Math., Volume 191 (2013) no. 3, pp. 527-629 | DOI

[20] Liu, Y; Zhang, S; Zhang, W On p-adic Waldspurger formula (preprint)

[21] Miyake, Toshitsune Modular forms, Springer-Verlag, Berlin, 1989, x+335 pages (Translated from the Japanese by Yoshitaka Maeda) | DOI

[22] Nekovář, Jan Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math., Volume 107 (1992) no. 1, pp. 99-125 | DOI

[23] Nekovář, Jan On p-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91 (Progr. Math.), Volume 108, Birkhäuser Boston, Boston, MA, 1993, pp. 127-202 | DOI

[24] Nekovář, Jan On the p-adic height of Heegner cycles, Math. Ann., Volume 302 (1995) no. 4, pp. 609-686 | DOI

[25] Nekovář, Jan p-adic Abel-Jacobi maps and p-adic heights, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) (CRM Proc. Lecture Notes), Volume 24, Amer. Math. Soc., Providence, RI, 2000, pp. 367-379

[26] Ogg, Andrew Modular forms and Dirichlet series, W. A. Benjamin, Inc., New York-Amsterdam, 1969, xvi+173 pp. (not consecutively paged) paperbound pages

[27] Olsson, Martin C. On Faltings’ method of almost étale extensions, Algebraic geometry—Seattle 2005. Part 2 (Proc. Sympos. Pure Math.), Volume 80, Amer. Math. Soc., Providence, RI, 2009, pp. 811-936 | DOI

[28] Perrin-Riou, Bernadette Points de Heegner et dérivées de fonctions L p-adiques, Invent. Math., Volume 89 (1987) no. 3, pp. 455-510 | DOI

[29] Perrin-Riou, Bernadette Fonctions L p-adiques associées à une forme modulaire et à un corps quadratique imaginaire, J. London Math. Soc. (2), Volume 38 (1988) no. 1, pp. 1-32 | DOI

[30] Perrin-Riou, Bernadette p-adic L-functions and p-adic representations, SMF/AMS Texts and Monographs, 3, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2000, xx+150 pages (Translated from the 1995 French original by Leila Schneps and revised by the author)

[31] Petrequin, Denis Classes de Chern et classes de cycles en cohomologie rigide, Bull. Soc. Math. France, Volume 131 (2003) no. 1, pp. 59-121

[32] Rohrlich, David E. Root numbers of Hecke L-functions of CM fields, Amer. J. Math., Volume 104 (1982) no. 3, pp. 517-543 | DOI

[33] Scholl, A. J. Motives for modular forms, Invent. Math., Volume 100 (1990) no. 2, pp. 419-430 | DOI

[34] Serre, Jean-Pierre; Tate, John Good reduction of abelian varieties, Ann. of Math. (2), Volume 88 (1968), pp. 492-517

[35] de Shalit, Ehud Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics, 3, Academic Press, Inc., Boston, MA, 1987, x+154 pages (p-adic L functions)

[36] Shiho, Atsushi Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo, Volume 9 (2002) no. 1, pp. 1-163

[37] Walling, Lynne H. The Eichler commutation relation for theta series with spherical harmonics, Acta Arith., Volume 63 (1993) no. 3, pp. 233-254

[38] Wiles, A. On ordinary λ-adic representations associated to modular forms, Invent. Math., Volume 94 (1988) no. 3, pp. 529-573 | DOI

[39] Zhang, Shouwu Heights of Heegner cycles and derivatives of L-series, Invent. Math., Volume 130 (1997) no. 1, pp. 99-152 | DOI

Cited by Sources: