## Modularity of residual Galois extensions and the Eisenstein ideal

HTML articles powered by AMS MathViewer

- by Tobias Berger and Krzysztof Klosin PDF
- Trans. Amer. Math. Soc.
**372**(2019), 8043-8065 Request permission

## Abstract:

For a totally real field $F$, a finite extension $\mathbf {F}$ of $\mathbf {F}_p$, and a Galois character $\chi : G_F \to \mathbf {F}^{\times }$ unramified away from a finite set of places $\Sigma \supset \{\mathfrak {p} \mid p\}$, consider the Bloch–Kato Selmer group $H:=H^1_{\Sigma }(F, \chi ^{-1})$. The authors previously proved that the number $d$ of isomorphism classes of (nonsemisimple, reducible) residual representations ${\overline \rho }$ giving rise to lines in $H$ which are modular by some $\rho _f$ (also unramified outside $\Sigma$) satisfies $d \geq n:= \dim _{\mathbf {F}} H$. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal $J$ is nonprincipal, then $d >n$. When $F=\mathbf {Q}$ we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the nonprincipality of $J$ that can be checked in practice, allowing us to furnish examples where $d>n$.## References

- Joël Bellaïche and Gaëtan Chenevier,
*Families of Galois representations and Selmer groups*, Astérisque**324**(2009), xii+314 (English, with English and French summaries). MR**2656025** - Spencer Bloch and Kazuya Kato,
*$L$-functions and Tamagawa numbers of motives*, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400. MR**1086888** - Tobias Berger and Krzysztof Klosin,
*A deformation problem for Galois representations over imaginary quadratic fields*, J. Inst. Math. Jussieu**8**(2009), no. 4, 669–692. MR**2540877**, DOI 10.1017/S1474748009000036 - Tobias Berger and Krzysztof Klosin,
*On deformation rings of residually reducible Galois representations and $R=T$ theorems*, Math. Ann.**355**(2013), no. 2, 481–518. MR**3010137**, DOI 10.1007/s00208-012-0793-1 - T. Berger and K. Klosin,
*On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields*, Int. Math. Res. Not. IMRN**20**(2015), no. 20, 10525–10562. - Tobias Berger, Krzysztof Klosin, and Kenneth Kramer,
*On higher congruences between automorphic forms*, Math. Res. Lett.**21**(2014), no. 1, 71–82. MR**3247039**, DOI 10.4310/MRL.2014.v21.n1.a5 - Nicolas Billerey and Ricardo Menares,
*On the modularity of reducible $\textrm {mod}\, l$ Galois representations*, Math. Res. Lett.**23**(2016), no. 1, 15–41. MR**3512875**, DOI 10.4310/MRL.2016.v23.n1.a2 - Nicolas Billerey and Ricardo Menares,
*Strong modularity of reducible Galois representations*, Trans. Amer. Math. Soc.**370**(2018), no. 2, 967–986. MR**3729493**, DOI 10.1090/tran/6979 - C. Breuil,
*$p$-adic Hodge theory, deformations and local Langlands*, 2001, http://www.ihes.fr/~breuil/PUBLICATIONS/Barcelone.pdf. - Laurent Clozel, Michael Harris, and Richard Taylor,
*Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations*, Publ. Math. Inst. Hautes Études Sci.**108**(2008), 1–181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras. MR**2470687**, DOI 10.1007/s10240-008-0016-1 - Henri Darmon, Fred Diamond, and Richard Taylor,
*Fermat’s last theorem*, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) Int. Press, Cambridge, MA, 1997, pp. 2–140. MR**1605752** - Neil Dummigan and Daniel Fretwell,
*Ramanujan-style congruences of local origin*, J. Number Theory**143**(2014), 248–261. MR**3227346**, DOI 10.1016/j.jnt.2014.04.008 - David Eisenbud,
*Commutative algebra*, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR**1322960**, DOI 10.1007/978-1-4612-5350-1 - Spencer Hamblen and Ravi Ramakrishna,
*Deformations of certain reducible Galois representations. II*, Amer. J. Math.**130**(2008), no. 4, 913–944. MR**2427004**, DOI 10.1353/ajm.0.0008 - Krzysztof Klosin,
*Congruences among modular forms on $\textrm {U}(2,2)$ and the Bloch-Kato conjecture*, Ann. Inst. Fourier (Grenoble)**59**(2009), no. 1, 81–166 (English, with English and French summaries). MR**2514862**, DOI 10.5802/aif.2427 - B. Mazur,
*Modular curves and the Eisenstein ideal*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR**488287**, DOI 10.1007/BF02684339 - Toshitsune Miyake,
*Modular forms*, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR**1021004**, DOI 10.1007/3-540-29593-3 - B. Mazur and A. Wiles,
*Class fields of abelian extensions of $\textbf {Q}$*, Invent. Math.**76**(1984), no. 2, 179–330. MR**742853**, DOI 10.1007/BF01388599 - Tadashi Ochiai,
*Control theorem for Bloch-Kato’s Selmer groups of $p$-adic representations*, J. Number Theory**82**(2000), no. 1, 69–90. MR**1755154**, DOI 10.1006/jnth.1999.2483 - Masami Ohta,
*Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II*, Tokyo J. Math.**37**(2014), no. 2, 273–318. MR**3304683**, DOI 10.3836/tjm/1422452795 - Tomomi Ozawa,
*Constant terms of Eisenstein series over a totally real field*, Int. J. Number Theory**13**(2017), no. 2, 309–324. MR**3606625**, DOI 10.1142/S1793042117500208 - Karl Rubin,
*Euler systems*, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study. MR**1749177**, DOI 10.1515/9781400865208 - Christopher Skinner,
*Main conjectures and modular forms*, Current developments in mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 141–161. MR**2459294** - D. Spencer,
*Congruences of local origin for higher levels*, 2018. Thesis (Ph.D.)–University of Sheffield, http://etheses.whiterose.ac.uk/id/eprint/21423. - Hae-Sang Sun,
*Cuspidal class number of the tower of modular curves $X_1(Np^n)$*, Math. Ann.**348**(2010), no. 4, 909–927. MR**2721646**, DOI 10.1007/s00208-010-0505-7 - C. M. Skinner and A. J. Wiles,
*Ordinary representations and modular forms*, Proc. Nat. Acad. Sci. U.S.A.**94**(1997), no. 20, 10520–10527. MR**1471466**, DOI 10.1073/pnas.94.20.10520 - C. M. Skinner and A. J. Wiles,
*Residually reducible representations and modular forms*, Inst. Hautes Études Sci. Publ. Math.**89**(1999), no. 1, 5–126. - Sage Developers,
*Sagemath, the Sage Mathematics Software System $($version $8.3)$*, 2018, http://www.sagemath.org. - Eric Urban,
*Selmer groups and the Eisenstein-Klingen ideal*, Duke Math. J.**106**(2001), no. 3, 485–525. MR**1813234**, DOI 10.1215/S0012-7094-01-10633-9 - Lawrence C. Washington,
*Introduction to cyclotomic fields*, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR**1421575**, DOI 10.1007/978-1-4612-1934-7 - P. Wake and C. Wang-Erickson,
*The Eistenstein ideal with squarefree level*, arXiv:1804.06400 (2018). - Hwajong Yoo,
*The index of an Eisenstein ideal and multiplicity one*, Math. Z.**282**(2016), no. 3-4, 1097–1116. MR**3473658**, DOI 10.1007/s00209-015-1579-4

## Additional Information

**Tobias Berger**- Affiliation: School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, United Kingdom
- MR Author ID: 830077
**Krzysztof Klosin**- Affiliation: Queens College, City University of New York, Queens, New York 11367
- MR Author ID: 842947
- Email: krzysztof.klosin@yahoo.com; kklosin@qc.cuny.edu
- Received by editor(s): October 17, 2018
- Received by editor(s) in revised form: March 3, 2019, and March 6, 2019
- Published electronically: June 3, 2019
- Additional Notes: The first author’s research was supported by EPSRC Grant #EP/R006563/1.

The second author was supported by Young Investigator Grant #H98230-16-1-0129 from the National Security Agency, by Collaboration for Mathematicians Grant #578231 from the Simons Foundation, and by a PSC–CUNY award jointly funded by the Professional Staff Congress and the City University of New York. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 8043-8065 - MSC (2010): Primary 11F80; Secondary 11F33, 11R34
- DOI: https://doi.org/10.1090/tran/7851
- MathSciNet review: 4029689