zbMATH — the first resource for mathematics

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
Full Text: DOI Numdam EuDML
[1] Bourbaki N.,Algèbre Commutative, chapitre 7: Diviseurs, Paris, Hermann, 1965. · Zbl 0141.03501
[2] Carayol H., Sur les représentations-adiques attachées aux formes modulaires de Hilbert,C.R. Acad. Sc. Paris, série I, 296 (1985), 629–632.
[3] Coates J., Schmidt K., Iwasawa theory for the symmetric square of an elliptic curve,J. reine angew. Math.,375 (1987), 104–156. · Zbl 0609.14013 · doi:10.1515/crll.1987.375-376.104
[4] Gillard R., Fonctions Lp-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes,J. reine angew. Math.,358 (1985), 76–91. · Zbl 0551.12011 · doi:10.1515/crll.1985.358.76
[5] Greenberg R., On the conjecture of Birch and Swinnerton-Dyer,Inv. Math.,72 (1983), 241–265. · Zbl 0546.14015 · doi:10.1007/BF01389322
[6] Greenberg R., Iwasawa theory forp-adic representations,Advanced Studies in Pure Mathematics,17 (1989), 97–137. · Zbl 0739.11045
[7] Hida H., Iwasawa modules attached to congruences of cusp forms,Ann. Scient. Ec. Norm. Sup., 4e série,19 (1986), 231–273. · Zbl 0607.10022
[8] Hida H., Galois representations into GL2(Z p [[X]]) attached to ordinary cusp forms,Inv. Math.,85 (1986), 545–577. · Zbl 0612.10021 · doi:10.1007/BF01390329
[9] Hida H., Ap-adic measure attached to the zeta functions associated with two elliptic modular forms, I,Inv. Math.,79 (1985), 159–195. · Zbl 0573.10020 · doi:10.1007/BF01388661
[10] Hida H., Hecke algebras for GL1 and GL2,Sém. Th. N. Paris, 1985–1986, 131–163, Birkhäuser Verlag, 1986.
[11] Katz N.,p-adic interpolation of real analytic Eisenstein series,Ann. of Math.,104 (1976), 459–571. · Zbl 0354.14007 · doi:10.2307/1970966
[12] Katz N., Mazur B.,Arithmetic Moduli of Elliptic Curves, Ann. of Math. Studies, number 108, Princeton Univ. Press, 1985. · Zbl 0576.14026
[13] Langlands R. P., Automorphic forms and-adic representations, inProc. Int. Summer School on Modular Functions of One Variable II, Antwerp, 1972, Lecture Notes in Math.,349, 361–500, Springer-Verlag, 1973. · Zbl 0279.14007 · doi:10.1007/978-3-540-37855-6_6
[14] Mazur B., Modular Curves and the Eisenstein Ideal,Publ. Math. I.H.E.S.,47 (1977), 33–186. · Zbl 0394.14008
[15] Mazur B., Wiles A., Class fields of abelian extensions ofQ,Inv. Math.,76 (1984), 179–330. · Zbl 0545.12005 · doi:10.1007/BF01388599
[16] Mazur B., Wiles A., Onp-adic families of Galois representations,Comp. Math.,59 (1986), 231–264. · Zbl 0654.12008
[17] Mazur B., Deforming Galois Representations, inGalois Groups over Q, 385–438, Springer-Verlag, 1989.
[18] Mazur B., Ribet K., Two-dimensional representations in the arithmetic of modular curves, à paraître dansSém. Orsay, 1987–1988, Ed. G. Henniart,Astérisque. · Zbl 0780.14015
[19] Rubin K.,The “ main conjectures ” in Iwasawa theory for imaginary quadratic fields, Preprint, Ohio State U., Columbus, Ohio, 1990. · Zbl 0737.11030
[20] deShalit E.,Iwasawa Theory of Elliptic Curves with Complex Multiplication, Persp. in Math., vol. 3, Academic Press, 1987.
[21] Shimura G.,Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton Univ. Press, 1972. · Zbl 0243.14012
[22] Tilouine J., Un sous-groupep-divisible de la jacobienne de X1(Np r) comme module sur l’algèbre de Hecke,Bull. Soc. Math. Fr.,115 (1987), 329–360. · Zbl 0677.14006
[23] Tilouine J.,Kummer’s Criterion over \(\Lambda\) and Hida’s Congruence Module, Hokkaido University Technical Report Series in Mathematics, vol. 4, 1987.
[24] Tilouine J., Théorie d’Iwasawa classique et de l’algèbre de Hecke ordinaire,Comp. Math.,65 (1988), 265–320. · Zbl 0663.12008
[25] Tilouine J., Une conséquence de la conjecture principale dans la théorie d’Iwasawa d’un corps quadratique imaginaire,C.R. Acad. Sci. Paris, série 1,306 (1988), 217–221. · Zbl 0674.12002
[26] Tilouine J., Sur la conjecture principale anticyclotomique,Duke Math. J.,59 (1989), 629–673. · Zbl 0707.11079 · doi:10.1215/S0012-7094-89-05929-2
[27] Wiles A., Onp-adic representations over a totally real field,Ann. of Math.,123 (1986), 407–456. · Zbl 0613.12013 · doi:10.2307/1971332
[28] Yager R.,p-adic measures on Galois groups,Inv. Math.,76 (1984), 331–343. · Zbl 0555.12006 · doi:10.1007/BF01388600
[29] Gelbart S.,Automorphic Forms on Adele Groups, Ann. of Math. Studies, Princeton Univ. Press, 1975. · Zbl 0329.10018
[30] Weil A., On a certain type of characters of idèle-class group of an algebraic number-field, inOEuvres scientifiques, vol. 2, [1955c], 255–261, Springer-Verlag, 1980.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.