## Duke Mathematical Journal

### Iwasawa theory and the Eisenstein ideal

Romyar T. Sharifi

#### Abstract

We verify, for each odd prime $p \le 1000$, a conjecture of W. G. McCallum and R. T. Sharifi on the surjectivity of pairings on $p$-units constructed out of the cup product on the first Galois cohomology group of the maximal unramified outside $p$ extension of ${\bf Q}(\mu_p)$ with $\mu_p$-coefficients. In the course of the proof, we relate several Iwasawa-theoretic and Hida-theoretic objects. In particular, we construct a canonical isomorphism between an Eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical Iwasawa module over an abelian pro-$p$ Kummer extension of the cyclotomic ${\bf Z}_p$-extension of an abelian field. This Kummer extension arises from the Galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves which was considered by M. Ohta in order to give another proof of the Iwasawa main conjecture in the spirit of that of B. Mazur and A. Wiles. In turn, we relate the Iwasawa module over the Kummer extension to the quotient of the tensor product of the classical cyclotomic Iwasawa module and the Galois group of the Kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower. We give an application to the structure of the Selmer groups of Ohta's modular representation taken modulo the Eisenstein ideal

#### Article information

Source
Duke Math. J., Volume 137, Number 1 (2007), 63-101.

Dates
First available in Project Euclid: 8 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1173373451

Digital Object Identifier
doi:10.1215/S0012-7094-07-13713-X

Mathematical Reviews number (MathSciNet)
MR2309144

Zentralblatt MATH identifier
1131.11068

#### Citation

Sharifi, Romyar T. Iwasawa theory and the Eisenstein ideal. Duke Math. J. 137 (2007), no. 1, 63--101. doi:10.1215/S0012-7094-07-13713-X. https://projecteuclid.org/euclid.dmj/1173373451

#### References

• L. J. Federer and B. H. Gross, Regulators and Iwasawa modules, with an appendix by W. Sinnott, Invent. Math. 62 (1981), 443--457.
• B. Ferrero and L. C. Washington, The Iwasawa invariant $\mu_p$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), 377--395.
• R. Greenberg, Iwasawa theory and $p$-adic deformations of motives'' in Motives (Seattle, 1991), Proc. Sympos. Pure Math. 55, Part 2, Amer. Math. Soc., Providence, 1994, 193--223.
• Y. Hachimori and R. Sharifi, On the failure of pseudo-nullity of Iwasawa modules, with an appendix by R. T. Sharifi, J. Algebraic Geom. 14 (2005), 567--591.
• G. Harder and R. Pink, Modular konstruierte unverzweigte abelsche $p$-Erweiterungen von $\Q(\zeta_p)$ und die Struktur ihrer Galoisgruppen, Math. Nachr. 159 (1992), 83--99.
• H. Hida, Galois representations into $_2(\zp[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), 545--613.
• —, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), 231--273.
• Y. Ihara, Some arithmetic aspects of Galois actions in the pro-$p$ fundamental group of $\mathbbP^1 - \0,1,\infty\$'' in Arithmetic Fundamental Groups and Noncommutative Algebra (Berkeley, 1999), Proc. Sympos. Pure Math. 70, Amer. Math. Soc., Providence, 2002, 247--273.
• K. Kitagawa, On standard $p$-adic $L$-functions of families of elliptic cusp forms'' in $p$-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, 1991), Contemp. Math. 165, Amer. Math. Soc., Providence, 1994, 81--110.
• M. Kurihara, Ideal class groups of cyclotomic fields and modular forms of level $1$, J. Number Theory 45 (1993), 281--294.
• S. Lang, Cyclotomic Fields I and II, combined 2nd ed., with an appendix by Karl Rubin, Grad. Texts in Math. 121, Springer, New York, 1990.
• J.-C. Lario and R. Schoof, Some computations with Hecke rings and deformation rings, with an appendix by A. Agashe and W. Stein, Experiment. Math. 11 (2002), 303--311.
• B. Mazur and A. Wiles, Class fields of abelian extensions of $\Q$, Invent. Math. 76 (1984), 179--330.
• W. G. Mccallum and R. T. Sharifi, A cup product in the Galois cohomology of number fields, Duke Math. J. 120 (2003), 269--310.
• M. Ohta, On the $p$-adic Eichler-Shimura isomorphism for $\Lambda$-adic cusp forms, J. Reine Angew. Math. 463 (1995), 49--98.
• —, Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves, Compositio Math. 115 (1999), 241--301.
• —, Ordinary $p$-adic étale cohomology groups attached to towers of elliptic modular curves, II, Math. Ann. 318 (2000), 557--583.
• —, Congruence modules related to Eisenstein series, Ann. Sci. École Norm. Sup. (4) 36 (2003), 225--269.
• —, Companion forms and the structure of $p$-adic Hecke algebras, J. Reine Angew. Math. 585 (2005), 141--172.
• K. Ribet, A modular construction of unramified $p$-extensions of $\bf Q(\mu_p)$, Invent. Math. 34 (1976), 151--162.
• R. T. Sharifi, Determination of conductors from Galois module structure, Math. Z. 241 (2002), 227--245.
• —, Massey products and ideal class groups, to appear in J. Reine Angew. Math., preprint.
• C. M. Skinner and A. J. Wiles, Ordinary representations and modular forms, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), 10520--10527.
• C. Soulé, On higher $p$-adic regulators'' in Algebraic $K$-theory (Evanston, Ill., 1980), Lecture Notes in Math. 854, Springer, Berlin, 1981, 372--401.
• W. Stein, Modular Forms: A Computational Approach, with an appendix by P. E. Gunnells, Grad. Stud. in Math. 79, Amer. Math. Soc., Providence, 2007.
• J. Sturm, On the congruence of modular forms'' in Number Theory (New York, 1984--1985.), Lecture Notes in Math. 1240, Springer, Berlin, 1987, 275--280.
• L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, New York, 1997.
• A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493--540.