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On the two-variable Iwasawa main conjecture. (English) Zbl 1112.11051
This article is a continuation of the author’s work on the Iwasawa theory for the two-variable nearly ordinary modular Galois deformations [J. Number Theory, 88, No.1, 59–85 (2001; Zbl 1090.11034); Am. J. Math. 125, No. 4, 849–892 (2003; Zbl 1057.11048); Ann. Inst. Fourier 55, No.1, 113–146 (2005; Zbl 1112.11031)].
Fix a prime number $$p$$. Consider a pair $$({\mathcal T}, P)$$, where $${\mathcal R}$$ is a local Noetherian algebra which is finite flat over the Iwasawa algebra $$\mathbb{Z}_p[[X_1,\dots, X_g]]$$, $${\mathcal T}$$ is a free $${\mathcal R}$$-module of finite rank with $${\mathcal R}$$-linear action of $$\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})$$ unramified outside a finite set of primes containing $$\{p,\infty\}$$, and $$P$$ is a dense subset of $$\text{Spec}({\mathcal R})$$ which consists of the kernels of various specializations $${\mathcal R}\to\overline\mathbb{Q}$$ such that the specialization of $${\mathcal T}$$ at $$\kappa\in P$$ is a $$\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})$$-stable lattice of the $$p$$-adic realization of a critical motive.
Given such a pair $$({\mathcal T}, P)$$, we would like to study analytic and algebraic $$p$$-adic $$L$$-functions $$L^{\text{anal}}_p({\mathcal T})$$, $$L^{\text{alg}}_p({\mathcal T})$$. Their existence as non-zero elements in $$\text{Frac}({\mathcal R})$$ is expected only when $$({\mathcal T}, P)$$ satisfies a certain local condition (“nearly ordinary” or “Dąbrowski-Panchishkin” or “admissible”). In this case one expects equality $$(L^{\text{anal}}_p({\mathcal T}))= (L^{\text{alg}}_p({\mathcal T}))$$ as non-zero ideals in $${\mathcal R}$$ (the Iwasawa main conjecture).
The author recalls, in section 2, the results for the Iwasawa theory of Hida deformations obtained in his previous articles cited above. In section 4 he reviews the theory of Selmer groups for a two-variable nearly ordinary deformations $${\mathcal T}$$ and its various specializations.
The author discuses the two-variable $$p$$-adic $$L$$-function for a nearly ordinary deformation $${\mathcal T}$$ through Beilinson-Kato elements in section 6. In section 7 he formulates and discusses the Iwasawa main conjecture for various specializations of $${\mathcal T}$$.
In section 9 he studies examples of two-variable nearly ordinary deformations where one can determine the structure of the Selmer group or one can prove the equality in addition to the inequality result (2) on p. 1160 by using Beilinson-Kato elements.

##### MSC:
 11R23 Iwasawa theory 11F33 Congruences for modular and $$p$$-adic modular forms 11F80 Galois representations 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R34 Galois cohomology
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