Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes

Kohen, Daniel; Pacetti, Ariel

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

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/Qp 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 cpn, 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/Qp 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 cpn, then χ(L) is equal (up to explicit constants) to L(E,χ,1) or L′(E,χ,1).

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

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

[4] Breuil, C.; Conrad, B.; Diamond, F.; Taylor, R. 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] Cai, L.; Shu, J.; Tian, Y. Explicit Gross–Zagier and Waldspurger formulae, Algebra Number Theory, Volume 8 (2014) no. 10, pp. 2523-2572

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[24] Perrin-Riou, B. 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] Pollack, R. 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] Saito, H. On Tunnell's formula for characters of GL(2), Compos. Math., Volume 85 (1993) no. 1, pp. 99-108

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

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

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

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

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

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