Hida families, $p$-adic heights, and derivatives

Arnold, Trevor

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2275-2299.

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.

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.

Classification : 11R23, 11G40, 11F11, 11S25
Keywords: Iwasawa theory, Hida family, $p$-adic height, $p$-adic $L$-function
Mot clés : théorie d’Iwasawa, famille de Hida, hauteur $p$-adique, fonction $L$ $p$-adique

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

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

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

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

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

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

[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 (Contemporary Mathematics), Volume 165, AMS, 1994, pp. 81-110 | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

[18] Wiles, A. On $\lambda$-adic representations associated to modular forms, Invent. Math., Volume 94 (1988), pp. 529-573 | DOI | MR | Zbl