## Galois representations, Kähler differentials and “main conjectures”. (Représentations galoisiennes, différentielles de Kähler et “conjectures principales”.)(French)Zbl 0744.11053

A “main conjecture” typically relates an algebraic object, usually an Iwasawa module, to an analytic object, for example a $$p$$-adic $$L$$- function. For example, see the recent papers of B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)], A. Wiles [Ann. Math., II. Ser. 123, 407-456 (1986; Zbl 0613.12013)], and K. Rubin [Invent. Math. 103, 25-68 (1991; Zbl 0737.11030)]. The present article gives a fairly general setting for this type of result.
Let $$\bar\rho: \hbox{Gal}(\bar\mathbb{Q}/\mathbb{Q})\to GL_ 2(\kappa)$$ be a representation over a finite field $$\kappa$$ associated to a $$p$$-adic $$p$$- ordinary Hecke eigenform $$f$$. Let $$Np^ r$$ be the exact conductor of $$f$$, with $$(N,p)=1$$. Let $$R^ 0$$ be the ring of universal co-ordinary (i.e., the coinvariants of inertia at $$p$$ are free of rank 1) deformations of $$\bar\rho$$. Let $$h_ \infty^{ord}$$ be the ordinary Hecke algebra of level $$Np^ \infty$$ constructed by H. Hida [Ann. Sci. Ec. Norm. Super., IV. Ser. 19, 231-273 (1986; Zbl 0607.10022)] and let $$R$$ be the localization of $$h_ \infty^{ord}$$ at the maximal ideal whose residue field is $$\kappa$$. There is a natural homomorphism (of algebras over the Iwasawa algebra $$\Lambda$$) $$\phi: R^ 0\to R$$. Under the hypotheses that $$\bar\rho$$ is absolutely irreducible and all the $$\Lambda$$-adic forms occurring in $$R$$ are of level divisible by $$N$$, the authors conjecture that $$\phi$$ is an isomorphism. The main purpose of the present paper is to prove the surjectivity of $$\phi$$ under these hypotheses plus the assumptions that $$p\geq 5$$ and the character $$\det(\bar\rho)$$ is of exact conductor $$Np$$.
A consequence of this surjectivity is that the map on differentials $$d\phi:R\otimes_ \Lambda\Omega_{R^ 0/\Lambda}\to\Omega_{R/\Lambda}$$ is surjective. The module on the left can be viewed as an algebraically defined Iwasawa module and can be described in terms of Galois cohomology. The module on the right is a more analytic object. The surjectivity of $$d\phi$$ implies that the characteristic ideal of $$\Omega_{R/\Lambda}$$ divides that of $$R\otimes_ \Lambda\Omega_{R^ 0/\Lambda}$$.
The authors use their results to deduce certain divisibilities of one- variable power series predicted by the two-variable main conjecture for imaginary quadratic fields. For related results in this direction, see J. Tilouine [Duke Math. J. 59, 629-673 (1989; Zbl 0707.11079)].

### MSC:

 11R23 Iwasawa theory 11F85 $$p$$-adic theory, local fields 11S40 Zeta functions and $$L$$-functions 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 11R34 Galois cohomology
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### References:

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