Higher congruences between newforms and Eisenstein series of squarefree level

Catherine M. Hsu

doi:10.5802/jtnb.1092

Catherine M. Hsu ¹

¹ School of Mathematics University of Bristol Bristol, BS8 1TH, UK and the Heilbronn Institute for Mathematical Research Bristol, UK

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 503-525.

Abstract
Résumé

Let $p \geq 5$ be prime. For elliptic modular forms of weight 2 and level $Γ_{0} (N)$ 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 $p \geq 5$ un nombre premier. Pour les formes modulaires elliptiques de poids $2$ et de niveau $Γ_{0} (N),$ où $N > 6$ 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.

Received: 2018-12-02
Accepted: 2019-07-14
Published online: 2019-10-29

DOI: 10.5802/jtnb.1092

Classification: 11F33
Keywords: Congruences between modular forms, Eisentein ideal

Author's affiliations:

Catherine M. Hsu ¹

¹ School of Mathematics University of Bristol Bristol, BS8 1TH, UK and the Heilbronn Institute for Mathematical Research Bristol, UK

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Higher congruences between newforms and {Eisenstein} series of squarefree level},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {503--525},
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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/

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