Higher congruences between newforms and Eisenstein series of squarefree level
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 503-525.

Let $p\ge 5$ be prime. For elliptic modular forms of weight 2 and level ${\Gamma }_{0}\left(N\right)$ where $N>6$ is squarefree, we bound the depth of Eisenstein congruences modulo $p$ (from below) by a generalized Bernoulli number with correction factors and show how this depth detects the local non-principality of the Eisenstein ideal. We then use admissibility results of Ribet and Yoo to give an infinite class of examples where the Eisenstein ideal is not locally principal. Lastly, we illustrate these results with explicit computations and give an interesting commutative algebra application related to Hilbert–Samuel multiplicities.

Soit $p\ge 5$ un nombre premier. Pour les formes modulaires elliptiques de poids $2$ et de niveau ${\Gamma }_{0}\left(N\right),$$N>6$ est sans facteurs carrés, nous donnons une minoration de la profondeur des congruences d’Eisenstein modulo $p$ en fonction d’un nombre de Bernoulli généralisé et de certains facteurs de correction, et montrons que cette profondeur détecte la non principalité locale de l’idéal d’Eisenstein. Nous utilisons ensuite les résultats d’admissibilité de Ribet et Yoo pour donner une infinité d’exemples où l’idéal d’Eisenstein n’est pas localement principal. Finalement, nous illustrons ces résultats par des calculs explicites et en donnons une application intéressante aux multiplicités de Hilbert–Samuel.

Accepted:
Published online:
DOI: 10.5802/jtnb.1092
Classification: 11F33
Keywords: Congruences between modular forms, Eisentein ideal
Catherine M. Hsu 1

1 School of Mathematics University of Bristol Bristol, BS8 1TH, UK and the Heilbronn Institute for Mathematical Research Bristol, UK
@article{JTNB_2019__31_2_503_0,
author = {Catherine M. Hsu},
title = {Higher congruences between newforms and {Eisenstein} series of squarefree level},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {503--525},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {31},
number = {2},
year = {2019},
doi = {10.5802/jtnb.1092},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1092/}
}
TY  - JOUR
TI  - Higher congruences between newforms and Eisenstein series of squarefree level
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2019
DA  - 2019///
SP  - 503
EP  - 525
VL  - 31
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1092/
UR  - https://doi.org/10.5802/jtnb.1092
DO  - 10.5802/jtnb.1092
LA  - en
ID  - JTNB_2019__31_2_503_0
ER  -
%0 Journal Article
%T Higher congruences between newforms and Eisenstein series of squarefree level
%J Journal de Théorie des Nombres de Bordeaux
%D 2019
%P 503-525
%V 31
%N 2
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.1092
%R 10.5802/jtnb.1092
%G en
%F JTNB_2019__31_2_503_0
Catherine M. Hsu. Higher congruences between newforms and Eisenstein series of squarefree level. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 503-525. doi : 10.5802/jtnb.1092. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1092/

[1] Tobias Berger; Krzysztof Klosin Modularity of residual Galois extensions and the Eisenstein ideal (to appear in Trans. Am. Math. Soc.) | Article

[2] Tobias Berger; Krzysztof Klosin On deformation rings of residually reducible Galois representations and $R=T$ theorems, Math. Ann., Volume 355 (2013) no. 2, pp. 481-518 | Article | MR: 3010137 | Zbl: 1292.11065

[3] Tobias Berger; Krzysztof Klosin; Kenneth Kramer On higher congruences between automorphic forms, Math. Res. Lett., Volume 21 (2014) no. 1, pp. 71-82 | Article | MR: 3247039 | Zbl: 1367.11050

[4] Wieb Bosma; John Cannon; Catherine Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | Article | MR: 1484478 | Zbl: 0898.68039

[5] Frank Calegari; Matthew Emerton On the ramification of Hecke algebras at Eisenstein primes, Invent. Math., Volume 160 (2005) no. 1, pp. 97-144 | Article | MR: 2129709 | Zbl: 1145.11314

[6] Imin Chen; Ian Kiming; Jonas B. Rasmussen On congruences $\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}{𝔭}^{m}$ between eigenforms and their attached Galois representations, J. Number Theory, Volume 130 (2010), pp. 608-619 | Article | MR: 2584844 | Zbl: 1129.11072

[7] Henri Darmon; Fred Diamond; Richard Taylor Fermat’s last theorem, Current developments in mathematics (1995), International Press., 1995, pp. 1-107 | Zbl: 0877.11035

[8] Fred Diamond; John Im Modular forms and modular curves, Seminar on Fermat’s last theorem (Toronto, 1993–1994) (CMS Conference Proceedings), Volume 17, American Mathematical Society, 1993, pp. 1993-1994 | Zbl: 0853.11032

[9] David Eisenbud Commutative algebra: with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer, 1995 | Zbl: 0819.13001

[10] Kimball Martin The Jacquet–Langlands correspondence, Eisenstein congruences, and integral $L$-values in weight 2, Math. Res. Lett., Volume 24 (2017) no. 6, pp. 1775-1795 | Article | MR: 3762695 | Zbl: 06900924

[11] Barry Mazur Modular curves and the Eisenstein ideal, Publ. Math., Inst. Hautes Étud. Sci., Volume 47 (1977), pp. 33-186 | Article | Zbl: 0394.14008

[12] Barry Mazur; Andrew Wiles Class fields of abelian extensions of $ℚ$, Invent. Math., Volume 76 (1984), pp. 179-330 | Article | MR: 742853 | Zbl: 0545.12005

[13] Toshitsune Miyake Modular forms, Springer Monographs in Mathematics, Springer, 2006 | MR: 2194815 | Zbl: 1159.11014

[14] Bartosz Naskręcki On higher congruences between cusp forms and Eisenstein series, Computations with modular forms (Heidelberg, 2011) (Contributions in Mathematical and Computational Sciences), Volume 6, Springer, 2014, pp. 257-277 | Article | MR: 3381456 | Zbl: 1355.11043

[15] Masami Ohta Eisenstein ideals and the rational torsion subgroups of modular Jacobian varieties II, Tokyo J. Math., Volume 37 (2014) no. 2, pp. 273-318 | Article | MR: 3304683 | Zbl: 1332.11061

[16] Christopher M. Skinner; Andrew Wiles Ordinary representations and modular forms, Proc. Natl. Acad. Sci. USA, Volume 94 (1997) no. 20, pp. 10520-10527 | Article | MR: 1471466 | Zbl: 0924.11044

[17] Preston Wake; Carl Wang-Erickson The rank of Mazur’s Eisenstein ideal (2017) (to appear in Duke Math. J.)

[18] Preston Wake; Carl Wang-Erickson The Eisenstein ideal with squarefree level (2018) (https://arxiv.org/abs/1804.06400)

[19] Hwajong Yoo The index of an Eisenstein ideal and multiplicity one, Math. Z., Volume 282 (2016) no. 3-4, pp. 1097-1116 | MR: 3473658 | Zbl: 1338.11057

[20] Hwajong Yoo Non-optimal levels of a reducible $\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}\ell$ modular representation, Trans. Am. Math. Soc., Volume 371 (2019) no. 6, pp. 3805-3830 | MR: 3917209 | Zbl: 07031935

Cited by Sources: