no. 5
On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions
[Sur la non-annulation des hauteurs p-adiques sur les variétés abéliennes CM, et l’arithmétique des fonctions L p-adiques de Katz]
Burungale, Ashay A.1 ; Disegni, Daniel2
1 California Institute of Technology, 1200 E California Blvd, Pasadena CA 91125, (USA)
2 Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, (Israel)
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2077-2101.

Soient B une variété abélienne CM simple sur un corps CM E, p un premier rationnel. On suppose que B a une réduction potentiellement ordinaire au dessus de p et est auto-duale avec signe -1. Sous quelques hypothèses supplementaires, on montre la non-annulation générique des hauteurs p-adiques (cyclotomiques) sur B le long de Z p -extensions anticyclotomiques de E. Cela confirme partiellement la conjecture de Schneider sur la non-annulation des hauteurs p-adiques. Pour les courbes elliptiques CM sur Q, le résultat était déjà connu comme conséquence de travaux de Bertrand, Gross–Zagier et Rohrlich dans les années 80. Notre preuve est basée sur des résultats de non-annulation pour les fonctions L p-adiques de Katz, et sur une formule de Gross–Zagier qui les relie à des familles de points rationnels sur B.

Let B be a simple CM abelian variety over a CM field E, p a rational prime. Suppose that B has potentially ordinary reduction above p and is self-dual with root number -1. Under some further conditions, we prove the generic non-vanishing of (cyclotomic) p-adic heights on B along anticyclotomic ℤ p -extensions of E. This provides evidence towards Schneider’s conjecture on the non-vanishing of p-adic heights. For CM elliptic curves over ℚ, the result was previously known as a consequence of works of Bertrand, Gross–Zagier and Rohrlich in the 1980s. Our proof is based on non-vanishing results for Katz p-adic L-functions and a Gross–Zagier formula relating the latter to families of rational points on B.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3381
Classification : 11G50, 11G10, 11G40
Keywords: Keywords: $p$-adic heights, Katz $p$-adic $L$-functions, CM abelian varieties
Mot clés : hauteurs $p$-adiques, fonctions $L$ $p$-adiques de Katz, variétés abéliennes CM

Burungale, Ashay A. 1 ; Disegni, Daniel 2

1 California Institute of Technology, 1200 E California Blvd, Pasadena CA 91125, (USA)
2 Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, (Israel)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2020__70_5_2077_0,
     author = {Burungale, Ashay A. and Disegni, Daniel},
     title = {On the non-vanishing of $p$-adic heights on {CM} abelian varieties, and the arithmetic of {Katz} $p$-adic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     pages = {2077--2101},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3381},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3381/}
}
TY  - JOUR
AU  - Burungale, Ashay A.
AU  - Disegni, Daniel
TI  - On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 2077
EP  - 2101
VL  - 70
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3381/
DO  - 10.5802/aif.3381
LA  - en
ID  - AIF_2020__70_5_2077_0
ER  - 
%0 Journal Article
%A Burungale, Ashay A.
%A Disegni, Daniel
%T On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions
%J Annales de l'Institut Fourier
%D 2020
%P 2077-2101
%V 70
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3381/
%R 10.5802/aif.3381
%G en
%F AIF_2020__70_5_2077_0
Burungale, Ashay A.; Disegni, Daniel. On the non-vanishing of $p$-adic heights on CM abelian varieties, and the arithmetic of Katz $p$-adic $L$-functions. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2077-2101. doi : 10.5802/aif.3381. https://aif.centre-mersenne.org/articles/10.5802/aif.3381/

[1] Aflalo, Esther; Nekovář, Jan Non-triviality of CM points in ring class field towers, Isr. J. Math., Volume 175 (2010), pp. 225-284 (With an appendix by Christophe Cornut) | DOI | MR | Zbl

[2] Agboola, Adebisi; Burns, David J. On twisted forms and relative algebraic K-theory, Proc. Lond. Math. Soc., Volume 92 (2006) no. 1, pp. 1-28 | DOI | MR | Zbl

[3] Agboola, Adebisi; Howard, Benjamin Anticyclotomic Iwasawa theory of CM elliptic curves, Ann. Inst. Fourier, Volume 56 (2006) no. 6, pp. 1001-1048 | DOI | Numdam | MR | Zbl

[4] Bernardi, Dominique; Perrin-Riou, Bernadette Variante p-adique de la conjecture de Birch et Swinnerton–Dyer (le cas supersingulier), C. R. Math. Acad. Sci. Paris, Volume 317 (1993) no. 3, pp. 227-232 | MR | Zbl

[5] Bertolini, Massimo; Darmon, Henri; Prasanna, Kartik p-adic Rankin L-series and rational points on CM elliptic curves, Pac. J. Math., Volume 260 (2012) no. 2, pp. 261-303 | DOI | MR | Zbl

[6] Bertrand, Daniel Propriétés arithmétiques de fonctions thêta à plusieurs variables, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 17-22 | DOI | Zbl

[7] Burungale, Ashay A. On the μ-invariant of the cyclotomic derivative of a Katz p-adic L-function, J. Inst. Math. Jussieu, Volume 14 (2015) no. 1, pp. 131-148 | DOI | MR | Zbl

[8] Burungale, Ashay A. Non-triviality of generalised Heegner cycles over anticyclotomic towers: a survey, p-adic aspects of modular forms, World Scientific, 2016, pp. 279-306 | DOI | MR | Zbl

[9] Burungale, Ashay A. On the non-triviality of the p-adic Abel-Jacobi image of generalised Heegner cycles modulo p, II: Shimura curves, J. Inst. Math. Jussieu, Volume 16 (2017) no. 1, pp. 189-222 | DOI | MR | Zbl

[10] Burungale, Ashay A. On the non-triviality of generalised Heegner cycles modulo p, I: modular curves, J. Algebr. Geom., Volume 29 (2020) no. 2, pp. 329-371 | DOI | MR

[11] Burungale, Ashay A.; Hida, Haruzo 𝔭-rigidity and Iwasawa μ-invariants, Algebra Number Theory, Volume 11 (2017) no. 8, pp. 1921-1951 | DOI | MR | Zbl

[12] Burungale, Ashay A.; Tian, Ye Horizontal non-vanishing of Heegner points and toric periods, Adv. Math., Volume 362 (2020), 106938 | DOI | MR

[13] Burungale, Ashay A.; Tian, Ye p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin, Invent. Math., Volume 220 (2020) no. 1, pp. 211-253 | DOI | MR

[14] Bushnell, Colin J.; Henniart, Guy The local Langlands conjecture for GL (2), Grundlehren der Mathematischen Wissenschaften, 335, Springer, 2006 | MR | Zbl

[15] Chai, Ching-Li Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli, Invent. Math., Volume 121 (1995) no. 3, pp. 439-479 | DOI | MR | Zbl

[16] Chai, Ching-Li Families of ordinary abelian varieties: canonical coordinates, p-adic monodromy, Tate-linear subvarieties and Hecke orbits (2003) (https://www.math.upenn.edu/~chai/papers_pdf/fam_ord_av.pdf)

[17] Chai, Ching-Li Hecke orbits as Shimura varieties in positive characteristic, International Congress of Mathematicians. Vol. II (2006), pp. 295-312 | Zbl

[18] Chai, Ching-Li; Conrad, Brian; Oort, Frans Complex multiplication and lifting problems, Mathematical Surveys and Monographs, 195, American Mathematical Society, 2014 | MR | Zbl

[19] Darmon, Henri René; Rotger, Victor Diagonal cycles and Euler systems II: The Birch and Swinnerton-Dyer conjecture for Hasse–Weil–Artin L-functions, J. Am. Math. Soc., Volume 30 (2017) no. 3, pp. 601-672 | DOI | MR | Zbl

[20] Disegni, Daniel p-adic heights of Heegner points on Shimura curves, Algebra Number Theory, Volume 9 (2015) no. 7, pp. 1571-1646 | DOI | MR | Zbl

[21] Disegni, Daniel The p-adic Gross–Zagier formula on Shimura curves, Compos. Math., Volume 153 (2017) no. 10, pp. 1987-2074 | DOI | MR | Zbl

[22] Disegni, Daniel The universal p-adic Gross–Zagier formula (2019) (https://arxiv.org/abs/2001.00045)

[23] Hida, Haruzo On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves, Am. J. Math., Volume 103 (1981) no. 4, pp. 727-776 | DOI | MR | Zbl

[24] Hida, Haruzo Hilbert modular forms and Iwasawa theory, Oxford Mathematical Monographs; Oxford Science Publications, Oxford Mathematical Monographs; Clarendon Press, 2006 | Zbl

[25] Hida, Haruzo The Iwasawa μ-invariant of p-adic Hecke L-functions, Ann. Math., Volume 172 (2010) no. 1, pp. 41-137 | DOI | MR | Zbl

[26] Hida, Haruzo Hecke fields of Hilbert modular analytic families, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski–Shapiro (Contemporary Mathematics), Volume 614, American Mathematical Society, 2014, pp. 97-137 | DOI | MR | Zbl

[27] Hida, Haruzo; Tilouine, Jacques Anti-cyclotomic Katz p-adic L-functions and congruence modules, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 2, pp. 189-259 | DOI | Numdam | MR | Zbl

[28] Hsieh, Ming-Lun On the μ-invariant of anticyclotomic p-adic L-functions for CM fields, J. Reine Angew. Math., Volume 688 (2014), pp. 67-100 | MR | Zbl

[29] Kashio, Tomokazu; Yoshida, Hiroyuki On p-adic absolute CM-periods. II, Publ. Res. Inst. Math. Sci., Volume 45 (2009) no. 1, pp. 187-225 | DOI | MR | Zbl

[30] Katz, Nicholas M. p-adic L-functions for CM fields, Invent. Math., Volume 49 (1978) no. 3, pp. 199-297 | DOI | MR | Zbl

[31] 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 | MR | Zbl

[32] Liu, Yifeng; Zhang, Shou-Wu; Zhang, Wei A p-adic Waldspurger formula, Duke Math. J., Volume 167 (2018) no. 4, pp. 743-833 | MR | Zbl

[33] Mazur, Barry; Tate, John T. JUN. Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I (Progress in Mathematics), Volume 35, Birkhäuser, 1983, pp. 195-237 | DOI | MR | Zbl

[34] Nekovář, Jan On p-adic height pairings, Séminaire de Théorie des Nombres, Paris, 1990–91 (Progress in Mathematics), Volume 108 (1993), pp. 127-202 | DOI | MR | Zbl

[35] Nekovář, Jan Selmer complexes, Astérisque, 310, Société Mathématique de France, 2006 | Numdam | MR | Zbl

[36] Perrin-Riou, Bernadette Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. Fr., Volume 115 (1987) no. 4, pp. 399-456 | DOI | Numdam

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

[38] Rubin, Karl p-adic variants of the Birch and Swinnerton–Dyer conjecture for elliptic curves with complex multiplication, p-adic monodromy and the Birch and Swinnerton–Dyer conjecture (Boston, MA, 1991) (Contemporary Mathematics), Volume 165, American Mathematical Society, 1994, pp. 71-80 | DOI | MR | Zbl

[39] Schneider, Peter p-adic height pairings. II, Invent. Math., Volume 79 (1985) no. 2, pp. 329-374 | DOI | MR | Zbl

[40] Shimura, Goro On some arithmetic properties of modular forms of one and several variables, Ann. Math., Volume 102 (1975) no. 3, pp. 491-515 | DOI | MR | Zbl

[41] Shimura, Goro Automorphic forms and the periods of abelian varieties, J. Math. Soc. Japan, Volume 31 (1979) no. 3, pp. 561-592 | MR | Zbl

[42] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, 1998 | MR | Zbl

[43] Skinner, Christopher M. A converse to a theorem of Gross, Zagier and Kolyvagin, Ann. Math., Volume 191 (2020) no. 2, pp. 329-354 | DOI | MR

[44] Tate, John T. JUN. Number theoretic background, Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proceedings of Symposia Pure Mathematics), Volume 33 (1979), pp. 3-26 | Zbl

[45] Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei The Gross–Zagier formula on Shimura curves, Annals of Mathematics Studies, 184, Princeton University Press, 2013 | MR | Zbl

[46] Zhang, Wei Selmer groups and the indivisibility of Heegner points, Camb. J. Math., Volume 2 (2014) no. 2, pp. 191-253 | DOI | MR | Zbl

Cité par Sources :