Hida families, p-adic heights, and derivatives
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, p. 2275-2299
This paper concerns the arithmetic of certain p-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a p-adic regulator, and the derivative of a p-adic L-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first conjecture implies the equivalence of the last two.
Cet article concerne l’arithmétique de certaines familles p-adiques de formes modulaires elliptiques. En utilisant une formule de Rubin, on examine quelques aspects de la théorie d’Iwasawa pour les objets du titre, dont trois affirment la non-trivialité d’un système d’Euler, d’un régulateur p-adique, et de la dérivée d’une fonction L p-adique. En particulier, on étudie des conditions suffisantes pour que la première conjecture soit vraie et on démontre que, sous des hypothèses supplémentaires, la première conjecture implique que les deux dernières conjectures sont équivalentes.
DOI : https://doi.org/10.5802/aif.2584
Classification:  11R23,  11G40,  11F11,  11S25
Keywords: Iwasawa theory, Hida family, p-adic height, p-adic L-function
@article{AIF_2010__60_6_2275_0,
     author = {Arnold, Trevor},
     title = {Hida families, $p$-adic heights, and derivatives},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     pages = {2275-2299},
     doi = {10.5802/aif.2584},
     mrnumber = {2791658},
     zbl = {1259.11099},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_6_2275_0}
}
Arnold, Trevor. Hida families, $p$-adic heights, and derivatives. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2275-2299. doi : 10.5802/aif.2584. http://www.numdam.org/item/AIF_2010__60_6_2275_0/

[1] Deligne, P. Formes modulaires et représentations -adiques, Séminaire Bourbaki 1968/1969, exp. 355, Springer, Berlin (Lecture Notes in Math.) Tome 179 (1971), pp. 139-172 | Numdam | Zbl 0206.49901

[2] Greenberg, R. Iwasawa theory for motives, L -functions and arithmetic (Durham, 1989), Cambridge Univ. Press (London Math. Soc. Lecture Note Ser.) Tome 153 (1991), pp. 211-233 | MR 1110394 | Zbl 0727.11043

[3] Greenberg, R. Elliptic curves and p-adic deformations, Elliptic curves and related topics, AMS (CRM Proc. Lecture Notes) Tome 4 (1994), pp. 101-110 | MR 1260957 | Zbl 0821.14021

[4] Greenberg, R. On the structure of certain Galois cohomology groups, Doc. Math. (2006) (Extra Vol., p. 335–391 (electronic)) | MR 2290593 | Zbl 1138.11048

[5] Hida, H. Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4), Tome 19 (1986) no. 2, pp. 231-273 | Numdam | MR 868300 | Zbl 0607.10022

[6] Kato, K. p-adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004) no. 295, pp. 117-290 | MR 2104361 | Zbl 1142.11336

[7] Kitagawa, K. On standard p-adic L-functions of families of elliptic cusp forms, p -adic monodromy and the Birch and Swinnerton-Dyer conjecture, AMS (Contemporary Mathematics) Tome 165 (1994), pp. 81-110 | MR 1279604 | Zbl 0841.11028

[8] Mazur, B.; Tilouine, J. Représentations galoisiennes, différentielles de Kähler et ‘conjectures principales’, Inst. Hautes Études Sci. Publ. Math. (1990) no. 71, pp. 65-103 | Article | Numdam | MR 1079644 | Zbl 0744.11053

[9] Milne, J. Arithmetic duality theorems, Academic Press, Perspectives in Mathematics, Tome 1 (1986) | MR 881804 | Zbl 0613.14019

[10] Nekovář, J. Selmer complexes, Astérisque (2006) no. 310, pp. viii+559 | MR 2333680 | Zbl pre05161833

[11] Ochiai, T. A generalization of the Coleman map for Hida deformations, Amer. J. Math., Tome 125 (2003), pp. 849-892 | Article | MR 1993743 | Zbl 1057.11048

[12] Ochiai, T. Euler system for Galois deformation, Ann. Inst. Fourier (Grenoble), Tome 55 (2005), pp. 113-146 | Article | Numdam | MR 2141691 | Zbl 1112.11031

[13] Ochiai, T. On the two-variable Iwasawa main conjecture, Compositio Math., Tome 142 (2006), pp. 1157-1200 | Article | MR 2264660 | Zbl 1112.11051

[14] Perrin-Riou, B. Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math., Tome 109 (1992), pp. 137-185 | Article | MR 1168369 | Zbl 0781.14013

[15] Plater, A. Height pairings in families of deformations, J. reine angew. Math., Tome 486 (1997), pp. 97-127 | Article | MR 1450752 | Zbl 0872.14018

[16] Rubin, K. Abelian varieties, p-adic heights and derivatives, Algebra and number theory, Walter de Gruyter and Co. (1994), pp. 247-266 | MR 1285370 | Zbl 0829.11034

[17] Rubin, K. Euler Systems, Princeton University Press (2000) | MR 1749177 | Zbl 0977.11001

[18] Wiles, A. On λ-adic representations associated to modular forms, Invent. Math., Tome 94 (1988), pp. 529-573 | Article | MR 969243 | Zbl 0664.10013