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  • Print publication year: 2011
  • Online publication date: January 2012

On the MH(G)-conjecture

Summary

Introduction

Let p be any prime number, and let G be a compact p-adic Lie group with a closed normal subgroup H such that G/H is isomorphic to the additive subgroup of p-adic integers ℤp. Write ∧(G) (respectively, ∧(H)) for the Iwasawa algebra of G (respectively, H) with coefficients in ℤp. As was shown in [5], there exists an Ore set in ∧(G) which enables one to define a characteristic element, with all the desirable properties, for a special class of torsion ∧(G)-modules, namely those finitely generated left ∧(G)-modules W such that W/W(p) is finitely generated over ∧(H); here W(p) denotes the p-primary submodule of W. This simple piece of pure algebra leads to a class of deep arithmetic problems, which will be the main concern of this paper. We shall loosely call these problems the MH(G)-conjectures, and it should be stressed that their validity is essential even for the formulation of the main conjectures of non-commutative Iwasawa theory.

Let F be a finite extension of ℚ, and F a Galois extension of F satisying (i) G = Gal(F/F) is a p-adic Lie group, (ii) F/F is unramified outside a finite set of primes of F, and (iii) F contains the cyclotomic ℤp-extension of F, which we denote by Fcyc.

References
[1] A. C., Sharma, Iwasawa invariants for the False-Tate extension and congruences between modular forms, Jour. Number Theory 129, (2009), 1893–1911.
[2] N., Bourbaki, Elements of Mathematics, Commutative Algebra, Chapters 1–7, Springer (1989).
[3] J., CoatesFragments of the Iwasawa theory of elliptic curves without complex multiplication. Arithmetic theory of elliptic curves, (Cetraro, 1997), Lecture Notes in Math., 1716, Springer, Berlin (1999), 1–50.
[4] J., Coates, R., Greenberg, Kummer theory for abelian varieties over local fields, Invent. Math. 124 (1996), 129–174.
[5] J., Coates, T., Fukaya, K., Kato, R., Sujatha, O., Venjakob, The GL2 main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 163–208.
[6] J., Coates, P., Schneider, R., Sujatha, Links between cyclotomic and GL2 Iwasawa theory, Kazuya Kato's fiftieth birthday. Doc. Math. 2003, Extra Vol., 187–215.
[7] A., Cuoco, P., Monsky, Class numbers in -extensions, Math. Ann. 255 (1981), 235–258.
[8] M., Emerton, R., Pollack, T., Weston, Variation of Iwasawa invariants in Hida families, Invent. Math. 163 (2006), 523–580.
[9] R., Greenberg, Iwasawa theory for p-adic representations, Algebraic number theory, 97–137, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989.
[10] R., Greenberg, Iwasawa theory and p-adic deformations of motives. Motives (Seattle, WA, 1991), 193–223, Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, (1994).
[11] R., Gross, On the integrality of some Galois representations, Proc. Amer. Math. Soc. 123 (1995), 299–301.
[12] K., Kato, p-adic Hodge theory and values of zeta functions of modular forms, Cohomologies p-adiques et applications arithmétiques. III. Astérisque 295 (2004), 117–290.
[13] Y., Hachimori, O., Venjakob, Completely faithful Selmer groups over Kummer extensions, Kazuya Kato's fiftieth birthday. Doc. Math. 2003, Extra Vol., 443–478.
[14] H., Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. 19 (1986), 231–273.
[15] H., Hida, Galois representations into GL2 (Zpp[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), 545–613.
[16] T., Levasseur, Some properties of non-commutative regular rings, Glasgow Journal of Math. 34 (1992), 277–300.
[17] B., Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.
[18] B., Mazur, A., Wiles, On p-adic analytic families of Galois representations, Compositio Math. 59 (1986), 231–264.
[19] T., Ochiai, On the two-variable Iwasawa main conjecture, Compositio Math. 142 (2006), 1157–1200.
[20] S., Sudhanshu, R., Sujatha, On the structure of Selmer groups of ∧-adic deformations over p-adic Lie extensions, preprint.
[21] O., Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc. 4 (2002), 271–311.
[22] A., Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math. 94 (1988), 529–573.