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)].


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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