On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves

[Sur les réalisations de de Rham et $p$-adiques du polylogarithme elliptique des courbes elliptiques à multiplication complexe]

Bannai, Kenichi ; Kobayashi, Shinichi ; Tsuji, Takeshi

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 185-234 / Harvested from Numdam

Résumé

Dans cet article, nous donnons une description explicite des réalisations de de Rham et $p$-adiques des polylogarithmes elliptiques en utilisant la fonction thêta de Kronecker. Considérons en particulier une courbe elliptique $E$ définie sur un corps quadratique imaginaire $\mathbb {K}$, à multiplication complexe par l’anneau des entiers $\mathcal {O}_\mathbb {K}$ de $\mathbb {K}$, et ayant bonne réduction en chaque place au-dessus d’un nombre premier $p \ge 5$ non ramifié dans $\mathbb {K}$. On notera que le nombre de classe de $\mathbb {K}$ est nécessairement égal à un. Nous montrons alors que les spécialisations des polylogarithmes $p$-adiques aux points de torsion de $E$ d’ordre premier à $p$ sont reliées aux nombres d’Eisenstein-Kronecker $p$-adiques. Ce résultat est valable même si $E$ a une réduction supersingulière en $p$. C’est un analogue $p$-adique d’un cas spécial du résultat de Beilinson et Levin exprimant la réalisation de Hodge du polylogarithme elliptique en utilisant les séries d’Eisenstein-Kronecker-Lerch. Si $p$ est quelconque, nous établissons un lien entre les nombres d’Eisenstein-Kronecker $p$-adiques et les valeurs spéciales des fonctions $L$ associées aux caractères de Hecke de $\mathbb {K}$.

Abstract

In this paper, we give an explicit description of the de Rham and $p$-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $\mathbb {K}$ with complex multiplication by the full ring of integers $\mathcal {O}_\mathbb {K}$ of $\mathbb {K}$. Note that our condition implies that $\mathbb {K}$ has class number one. Assume in addition that $E$ has good reduction above a prime $p \ge 5$ unramified in $\mathcal {O}_\mathbb {K}$. In this case, we prove that the specializations of the $p$-adic elliptic polylogarithm to torsion points of $E$ of order prime to $p$ are related to $p$-adic Eisenstein-Kronecker numbers. Our result is valid even if $E$ has supersingular reduction at $p$. This is a $p$-adic analogue in a special case of the result of Beilinson and Levin, expressing the Hodge realization of the elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. When $p$ is ordinary, then we relate the $p$-adic Eisenstein-Kronecker numbers to special values of $p$-adic $L$-functions associated to certain Hecke characters of $\mathbb {K}$.

Détail

Publié le : 2010-01-01

MR 2662664 | Zbl 1197.11073

DOI : https://doi.org/10.24033/asens.2119
Classification:  11G55,  11G07,  11G15,  14F30,  14G10
Mots clés: courbes elliptiques, multiplication complexe, polylogarithmes elliptiques, fonctions $L$ $p$-adiques

Bibliographie

[1] M. Abramowitz & I. A. Stegun (éds.), Weierstrass elliptic and related functions, Ch. 18, in Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., 1992, p. 627-671. | MR 1225604 | Zbl 0643.33001

[2] F. Baldassarri & B. Chiarellotto, Algebraic versus rigid cohomology with logarithmic coefficients, in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, 1994, 11-50. | MR 1307391 | Zbl 0833.14010

[3] K. Bannai, Rigid syntomic cohomology and $p$-adic polylogarithms, J. reine angew. Math. 529 (2000), 205-237. | MR 1799937 | Zbl 1006.19002

[4] K. Bannai, On the $p$-adic realization of elliptic polylogarithms for CM-elliptic curves, Duke Math. J. 113 (2002), 193-236. | MR 1909217 | Zbl 1019.11018

[5] K. Bannai & G. Kings, $p$-adic elliptic polylogarithm, $p$-adic Eisenstein series and Katz measure, preprint arXiv:0707.3747, to appear in Amer. J. Math. | MR 2766179 | Zbl 1225.11075

[6] K. Bannai & S. Kobayashi, Algebraic theta functions and $p$-adic interpolation of Eisenstein-Kronecker numbers, preprint arXiv:math.NT/0610163, to appear in Duke Math. J. | MR 2667134 | Zbl 1205.11076

[7] A. Beĭlinson & A. Levin, The elliptic polylogarithm, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., 1994, 123-190. | MR 1265553 | Zbl 0817.14014

[8] P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique $p$, Mém. Soc. Math. France (N.S.) 23 (1986), 7-32. | Numdam | MR 865810 | Zbl 0606.14017

[9] P. Berthelot, Cohomologie rigide et cohomologie rigide à support propre, première partie, preprint IRMAR 96-03, 1996.

[10] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), 329-377. | MR 1440308 | Zbl 0908.14005

[11] J. L. Boxall, A new construction of ${\mathfrak {p}}$-adic $L$-functions attached to certain elliptic curves with complex multiplication, Ann. Inst. Fourier (Grenoble) 36 (1986), 31-68. | Numdam | MR 867915 | Zbl 0608.14015

[12] J. L. Boxall, $p$-adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups, Ann. Inst. Fourier (Grenoble) 36 (1986), 1-27. | Numdam | MR 865657 | Zbl 0587.12007

[13] P. Colmez, Fonctions $L$ $p$-adiques, Séminaire Bourbaki, vol. 1998/99, exposé no 851, Astérisque 266 (2000), 21-58. | Numdam | MR 1772669 | Zbl 0964.11055

[14] R. M. Damerell, $L$-functions of elliptic curves with complex multiplication. I, Acta Arith. 17 (1970), 287-301. | Zbl 0209.24603

[15] R. M. Damerell, $L$-functions of elliptic curves with complex multiplication. II, Acta Arith. 19 (1971), 311-317. | Zbl 0229.12015

[16] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III, Astérisque 65, Soc. Math. France, 1979, 3-80. | Zbl 0429.14016

[17] A. Huber & G. Kings, Degeneration of $l$-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), 545-594. | Zbl 0955.11027

[18] A. Huber & J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27-133; correction: idem, 297-299. | Zbl 0906.19004

[19] N. M. Katz, $p$-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104 (1976), 459-571. | Zbl 0354.14007

[20] A. Levin, Elliptic polylogarithms: an analytic theory, Compositio Math. 106 (1997), 267-282. | Zbl 0905.11028

[21] A. Levin & G. Racinet, Towards multiple elliptic polylogarithm, preprint arXiv:math/0703237.

[22] J. I. Manin & M. M. Višik, $p$-adic Hecke series of imaginary quadratic fields, Mat. Sb. (N.S.) 95 (1974), 357-383. | Zbl 0352.12013

[23] P. Monsky & G. Washnitzer, Formal cohomology. I, Ann. of Math. 88 (1968), 181-217. | Zbl 0162.52504

[24] B. Perrin-Riou, Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque 229 (1995). | Zbl 0845.11040

[25] P. Schneider & J. Teitelbaum, $p$-adic Fourier theory, Doc. Math. 6 (2001), 447-481. | MR 1871671 | Zbl 1028.11069

[26] E. De Shalit, Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics 3, Academic Press Inc., 1987. | MR 917944 | Zbl 0674.12004

[27] A. Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), 1-163. | MR 1889223 | Zbl 1057.14025

[28] N. Solomon, $p$-adic elliptic polylogarithms and arithmetic applications, Thèse, Ben-Gurion University, 2009.

[29] N. Tsuzuki, On base change theorem and coherence in rigid cohomology, Doc. Math. extra vol. (2003), 891-918. | MR 2046617 | Zbl 1093.14503

[30] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebn. Math. Grenzg. 88, Springer, 1976. | MR 562289 | Zbl 0318.33004

[31] E. Weisstein, Weierstrass sigma function, http://mathworld.wolfram.com/WeierstrassSigmaFunction.html.

[32] J. Wildeshaus, Realizations of polylogarithms, Lecture Notes in Math. 1650, Springer, 1997. | MR 1482233 | Zbl 0877.11001

[33] S. Yamamoto, On $p$-adic $L$-functions for CM elliptic curves at supersingular primes, Mémoire, University of Tokyo, 2002.