Anticyclotomic <i>p</i>-adic <i>L</i>-functions for elliptic curves at some additive reduction primes

Daniel Kohen; Ariel Pacetti

doi:10.1016/j.crma.2018.09.005

Number theory

Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes
[Fonctions L p-adiques anti-cyclotomiques pour les courbes elliptiques en des premiers de réduction additive]

Présenté par : the Editorial Board

Daniel Kohen ¹ ; Ariel Pacetti ²

¹ Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS, Conicet, Argentina
² FAMAF-CIEM, Universidad Nacional de Córdoba, C.P:5000, Córdoba, Argentina

Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 973-983.

Résumé
Abstract

Soit E une courbe elliptique rationnelle et p un premier impair de réduction additive. Soit K un corps quadratique imaginaire et c un entier positif, premier au conducteur de E. Le but de cette Note est de définir une fonction L p-adique, anti-cyclotomique, notée $L$ , attachée à $E / K$ lorsque $E / Q_{p}$ atteint la réduction semi-stable sur une extension abélienne. Nous montrons que $L$ satisfait les propriétés d'interpolation escomptées. Précisément, nous montrons que, si χ est un caractère anti-cyclotomique de conducteur $c p^{n}$ , alors $χ (L)$ est égal (à des constantes explicites près) à $L (E, χ, 1)$ ou $L^{'} (E, χ, 1)$ .

Let E be a rational elliptic curve and let p be an odd prime of additive reduction. Let K be an imaginary quadratic field and fix a positive integer c prime to the conductor of E. The main goal of the present article is to define an anticyclotomic p-adic L-function $L$ attached to $E / K$ when $E / Q_{p}$ attains semistable reduction over an abelian extension. We prove that $L$ satisfies the expected interpolation properties; namely, we show that if χ is an anticyclotomic character of conductor $c p^{n}$ , then $χ (L)$ is equal (up to explicit constants) to $L (E, χ, 1)$ or $L^{'} (E, χ, 1)$ .

Reçu le : 2018-04-27
Accepté le : 2018-09-11
Publié le : 2018-09-27

DOI : 10.1016/j.crma.2018.09.005

Affiliations des auteurs :

Daniel Kohen ¹ ; Ariel Pacetti ²

@article{CRMATH_2018__356_10_973_0,
     author = {Daniel Kohen and Ariel Pacetti},
     title = {Anticyclotomic \protect\emph{p}-adic {\protect\emph{L}-functions} for elliptic curves at some additive reduction primes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {973--983},
     publisher = {Elsevier},
     volume = {356},
     number = {10},
     year = {2018},
     doi = {10.1016/j.crma.2018.09.005},
     language = {en},
}

TY  - JOUR
AU  - Daniel Kohen
AU  - Ariel Pacetti
TI  - Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 973
EP  - 983
VL  - 356
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2018.09.005
LA  - en
ID  - CRMATH_2018__356_10_973_0
ER  -

%0 Journal Article
%A Daniel Kohen
%A Ariel Pacetti
%T Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes
%J Comptes Rendus. Mathématique
%D 2018
%P 973-983
%V 356
%N 10
%I Elsevier
%R 10.1016/j.crma.2018.09.005
%G en
%F CRMATH_2018__356_10_973_0

Daniel Kohen; Ariel Pacetti. Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes. Comptes Rendus. Mathématique, Volume 356 (2018) no. 10, pp. 973-983. doi : 10.1016/j.crma.2018.09.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.09.005/

Bibliographie
Cité par

[1] Y. Amice; J. Vélu Distributions p-adiques associées aux séries de Hecke, Conférence, Université de Bordeaux, Bordeaux, France, 1974 (Astérisque), Volume vol. 24–25, Soc. Math. France, Paris (1975), pp. 119-131

[2] M. Bertolini; H. Darmon Heegner points on Mumford–Tate curves, Invent. Math., Volume 126 (1996) no. 3, pp. 413-456

[3] M. Bertolini; H. Darmon Iwasawa's main conjecture for elliptic curves over anticyclotomic $Z_{p}$ -extensions, Ann. of Math. (2), Volume 162 (2005) no. 1, pp. 1-64

[4] C. Breuil; B. Conrad; F. Diamond; R. Taylor On the modularity of elliptic curves over $Q$ : wild 3-adic exercises, J. Amer. Math. Soc., Volume 14 (2001) no. 4, pp. 843-939

[5] L. Cai; J. Shu; Y. Tian Explicit Gross–Zagier and Waldspurger formulae, Algebra Number Theory, Volume 8 (2014) no. 10, pp. 2523-2572

[6] M. Chida; M.-L. Hsieh Special values of anticyclotomic L-functions for modular forms, J. Reine Angew. Math. (Crelles J.), Volume 741 (2018), pp. 87-131

[7] C. Cornut; V. Vatsal Nontriviality of Rankin–Selberg L-functions and CM points, L-Functions and Galois Representations, Lond. Math. Soc. Lect. Note Ser., vol. 320, Cambridge University Press, Cambridge, UK, 2007, pp. 121-186

[8] H. Darmon; A. Iovita The anticyclotomic main conjecture for elliptic curves at supersingular primes, J. Inst. Math. Jussieu, Volume 7 (2008) no. 2, pp. 291-325

[9] D. Delbourgo Iwasawa theory for elliptic curves at unstable primes, Compos. Math., Volume 113 (1998) no. 2, pp. 123-153

[10] P. Deligne Les constantes des équations fonctionnelles des fonctions L, Lect. Notes Math., Volume 349 (1973), pp. 501-597

[11] F. Diamond; J. Shurman A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005

[12] D. Disegni The p-adic Gross–Zagier formula on Shimura curves, Compos. Math., Volume 153 (2017) no. 10, pp. 1987-2074

[13] Be.H. Gross Local orders, root numbers, and modular curves, Amer. J. Math., Volume 110 (1988) no. 6, pp. 1153-1182

[14] B.H. Gross; D. Prasad Test vectors for linear forms, Math. Ann., Volume 291 (1991) no. 2, pp. 343-355

[15] H. Jacquet Automorphic Forms on $GL (2)$ . Part II, Lecture Notes in Mathematics, vol. 278, Springer-Verlag, Berlin–New York, 1972

[16] K. Kato p-Adic Hodge theory and values of zeta functions of modular forms, Astérisque, Volume 295:ix (2004), pp. 117-290

[17] D. Kohen Heegner Point Constructions, University of Buenos Aires, 2017 (PhD thesis)

[18] D. Kohen; A. Pacetti On Heegner points for primes of additive reduction ramifying in the base field, Trans. Amer. Math. Soc., Volume 370 (2018) no. 2, pp. 911-926

[19] M. Longo; M.R. Pati Exceptional zero formulae for anticyclotomic p-adic L-functions of elliptic curves in the ramified case, J. Number Theory, Volume 190 (2018), pp. 187-211

[20] M. Longo; S. Vigni Quaternion algebras, Heegner points and the arithmetic of Hida families, Manuscr. Math., Volume 135 (2011) no. 3–4, pp. 273-328

[21] B. Mazur Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972), pp. 183-266

[22] B. Mazur; P. Swinnerton-Dyer Arithmetic of Weil curves, Invent. Math., Volume 25 (1974), pp. 1-61

[23] B. Mazur; J. Tate; J. Teitelbaum On p-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math., Volume 84 (1986) no. 1, pp. 1-48

[24] B. Perrin-Riou Théorie d'Iwasawa des représentations p-adiques sur un corps local, Invent. Math., Volume 115 (1994) no. 1, pp. 81-161 (With an appendix by Jean–Marc Fontaine)

[25] R. Pollack On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J., Volume 118 (2003) no. 3, pp. 523-558

[26] H. Saito On Tunnell's formula for characters of $GL (2)$ , Compos. Math., Volume 85 (1993) no. 1, pp. 99-108

[27] J.B. Tunnell Local ϵ-factors and characters of $GL (2)$ , Amer. J. Math., Volume 105 (1983) no. 6, pp. 1277-1307

[28] J. Van Order p-Adic interpolation of automorphic periods for $GL (2)$ , Doc. Math., Volume 22 (2017), pp. 1467-1499

[29] M.-F. Vignéras Arithmétique des Algèbres de Quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980

[30] M.M. Višik Nonarchimedean measures associated with Dirichlet series, Mat. Sb. (N.S.), Volume 99(141) (1976) no. 2, pp. 248-260 (296)

[31] A. Wiles Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 443-551

[32] X. Yuan; S.-W. Zhang; W. Zhang The Gross–Zagier Formula on Shimura Curves, Annals of Mathematics Studies, vol. 184, Princeton University Press, Princeton, NJ, USA, 2013

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

The vanishing of anticyclotomic $μ$ -invariants for non-ordinary modular forms

Jeffrey Hatley; Antonio Lei

C. R. Math (2023)

Plus and minus logarithms and Amice transform

Cédric Dion; Antonio Lei

C. R. Math (2017)

Period relations for automorphic induction and applications, I

Jie Lin

C. R. Math (2015)