Abstract
The central result of this paper is a refinement of Hida’s duality theorem between ordinary \({\varLambda }\)-adic modular forms and the universal ordinary Hecke algebra. In particular, we give a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary \({\varLambda }\)-adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt p-adic L-function. The motivation behind these results stems from a proof of the Iwasawa main conjecture over \({\mathbb {Q}}\) by Ohta. While simple and elegant, Ohta’s proof requires some restrictive hypotheses which we are able to remove using our results.
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References
Diamond, F., Shurman, J.: A First Course in Modular Forms, Graduate Texts in Mathematics, 1st edn, p. 228. Springer, Berlin (2015)
Emerton, M.: The Eisenstein ideal in Hida’s ordinary Hecke algebra. Int Math Res Notices 1999(2), 793–802 (1999)
Ferrero, B., Greenberg, R.: On the behavior of \(p\)-adic \(L\)-functions at \(s=0\). Invent. Math. No. 50, 91–102 (1978)
Fukaya, T., Kato, K., Sharifi, R.: Modular Symbols in Iwasawa Theory, Iwasawa Theory 2012: State of the Art and Recent Advances. Springer, Berlin (2015)
Harder, G., Pink, R.: Modular konstruierte unverzweigte abelsche \(p\)-Erweiterungen von \({\mathbb{Q}}(\zeta _p)\) und die Struktur ihrer Galoisgruppen. Mathematische Nachrichten 159, 83–99 (1992)
Hida, H.: Iwasawa modules attached to congruences of cusp forms. Annales Scientifiques de l’É.N.S., 4 série, tome 19(2), 231–273 (1986)
Hida, H.: Galois representations into \({\text{ GL }}_2 ({\mathbb{Z}}_p \llbracket {X} \rrbracket )\) attached to ordinary cusp forms. Invent. Math. 85, 545–613 (1986)
Hida, H.: Elementary Theory of \(L\)-Functions and Eisenstein Series, London Mathematical Society Student Texts (85), 2nd edn. Cambridge University Press, Cambridge (1993)
Kurihara, M.: Ideal class groups of cyclotomic fields and modular forms of level 1. J. Num. Theory 45, 281–294 (1993)
Mazur, B., Wiles, A.: Class fields of abelian extensions of \({\mathbb{Q}}\). Invent. Math. 76, 179–330 (1984)
Ohta, M.: On the \(p\)-adic Eichler-Shimura isomorphism for \(\Lambda \)-adic cusp forms. Journal für die reine und angewandte Mathematik 463, 49–98 (1995)
Ohta, M.: Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves. Compositio Mathematica 115, 241–301 (1999)
Ohta, M.: Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves II. Mathematische Annalen 318, 557–583 (2000)
Ohta, M.: Congruence modules related to Eisenstein series. Annales scientifiquesde l’École Normale Supérieure, 4 e série 36, 225–269 (2003)
Ohta, M.: Companion forms and the structure of \(p\)-adic Hecke algebras. Journal für die reine und angewandte Mathematik 585, 141–172 (2005)
Ohta, M.: Companion forms and the structure of \(p\)-adic Hecke algebras II. J. Math. Soc. Jpn. 59(4), 913–951 (2007)
Rubin, K.: The “main conjectures” of Iwasawa theory for imaginary quadratic fields. Invent. Math. 103(1), 25–68 (1991)
Rubin, K.: Euler Systems. Annals of Mathematics Studies. Princeton University Press, Princeton (2000)
Sharifi, R.: A reciprocity map and the two variable \(p\)-adic \(L\)-function. Ann. Math. (1) 173, 251–300 (2011)
Washington, L.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83, 2nd edn. Springer, Berlin (1996)
Wiles, A.: The Iwasawa conjecture for totally real fields. Ann. Math. (3) 131, 493–540 (1990)
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Communicated by Toby Gee.
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