Dans cet article, nous donnons une nouvelle explication des idéaux de congruences des séries d’Eisenstein de niveau et de caractère . Notre approche est basée sur l’interprétation algébro-géométrique de Katz des congruences -adiques des séries d’Eisenstein normalisées de niveau 1. Une étape cruciale de notre approche consiste à reformuler une correspondance de Riemann–Hilbert dans l’approche de Katz en termes de la théorie de Dieudonné des -modules formels de hauteur et de leurs schémas de sous-groupes finis. Nous généralisons en outre cette correspondance de Riemann–Hilbert en termes de groupes formels de hauteur supérieure à .
In this paper, we give a new explanation of congruences of Eisenstein series of level and character . Our approach is based on Katz’s algebro-geometric explanation of -adic congruences of normalized Eisenstein series of level . One crucial step in our argument is to reformulate a Riemann–Hilbert correspondence in Katz’s explanation in terms of Dieudonné theory of height formal -modules and their finite subgroup schemes. We further generalize this Riemann–Hilbert correspondence in terms of formal groups of height greater than .
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Mots clés : Eisenstein series, Dieudonné modules, formal $A$-modules
@article{JTNB_2024__36_1_215_0, author = {Ningchuan Zhang}, title = {Congruences of {Eisenstein} series of level $\Gamma _1(N)$ via {Dieudonn\'e} theory of formal groups}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {215--249}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {36}, number = {1}, year = {2024}, doi = {10.5802/jtnb.1277}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1277/} }
TY - JOUR AU - Ningchuan Zhang TI - Congruences of Eisenstein series of level $\Gamma _1(N)$ via Dieudonné theory of formal groups JO - Journal de théorie des nombres de Bordeaux PY - 2024 SP - 215 EP - 249 VL - 36 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1277/ DO - 10.5802/jtnb.1277 LA - en ID - JTNB_2024__36_1_215_0 ER -
%0 Journal Article %A Ningchuan Zhang %T Congruences of Eisenstein series of level $\Gamma _1(N)$ via Dieudonné theory of formal groups %J Journal de théorie des nombres de Bordeaux %D 2024 %P 215-249 %V 36 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1277/ %R 10.5802/jtnb.1277 %G en %F JTNB_2024__36_1_215_0
Ningchuan Zhang. Congruences of Eisenstein series of level $\Gamma _1(N)$ via Dieudonné theory of formal groups. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 215-249. doi : 10.5802/jtnb.1277. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1277/
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