Congruences of Eisenstein series of level Γ 1 (N) via Dieudonné theory of formal groups
Ningchuan Zhang1
1 Department of Mathematics Indiana University Bloomington Bloomington, IN 47405 USA
Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 215-249.

Dans cet article, nous donnons une nouvelle explication des idéaux de congruences des séries d’Eisenstein de niveau Γ 1 (N) et de caractère χ. Notre approche est basée sur l’interprétation algébro-géométrique de Katz des congruences p-adiques des séries d’Eisenstein normalisées E 2k 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 A-modules formels de hauteur 1 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 1.

In this paper, we give a new explanation of congruences of Eisenstein series of level Γ 1 (N) and character χ. Our approach is based on Katz’s algebro-geometric explanation of p-adic congruences of normalized Eisenstein series E 2k of level 1. One crucial step in our argument is to reformulate a Riemann–Hilbert correspondence in Katz’s explanation in terms of Dieudonné theory of height 1 formal A-modules and their finite subgroup schemes. We further generalize this Riemann–Hilbert correspondence in terms of formal groups of height greater than 1.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1277
Classification : 11F33, 14L05
Mots clés : Eisenstein series, Dieudonné modules, formal $A$-modules
Ningchuan Zhang 1

1 Department of Mathematics Indiana University Bloomington Bloomington, IN 47405 USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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/

[1] Mark Behrens The construction of tmf, Topological modular forms (Christopher L. Douglas; John Francis; André G. Henriques; Michael A. Hill, eds.) (Mathematical Surveys and Monographs), Volume 201, American Mathematical Society, 2014, pp. 131-188 | DOI | Zbl

[2] David Burns; Cornelius Greither On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math., Volume 153 (2003) no. 2, pp. 303-359 | DOI | Zbl

[3] Leonard Carlitz Arithmetic properties of generalized Bernoulli numbers, J. Reine Angew. Math., Volume 202 (1959), pp. 174-182 | DOI

[4] Pierre Cartier Modules associés à un groupe formel commutatif. Courbes typiques, C. R. Math. Acad. Sci. Paris, Volume 265 (1967), p. A129-A132 | Zbl

[5] Brian Conrad Arithmetic moduli of generalized elliptic curves, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 209-278 | DOI | Zbl

[6] Michel Demazure Lectures on p-divisible groups, Lecture Notes in Mathematics, 302, Springer, 1972, v+98 pages | DOI

[7] Fred Diamond; Jerry Shurman A first course in modular forms, Graduate Texts in Mathematics, 228, Springer, 2005, xvi+436 pages | DOI

[8] Haruzo Hida Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts, 26, Cambridge University Press, 1993, xii+386 pages | DOI

[9] Annette Huber; Guido Kings Bloch-Kato conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J., Volume 119 (2003) no. 3, pp. 393-464 | DOI | Zbl

[10] Kenkichi Iwasawa Lectures on p-adic L-functions, Annals of Mathematics Studies, 74, Princeton University Press; University of Tokyo Press, 1972, vii+106 pages | DOI

[11] A. J. de Jong Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math., Inst. Hautes Étud. Sci., Volume 82 (1995), p. 5-96 (1996) | DOI | Numdam | MR | Zbl

[12] Nicholas M. Katz p-adic properties of modular schemes and modular forms, Modular functions of one variable III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) (Lecture Notes in Mathematics), Volume 350, Springer, 1973, pp. 69-190 | DOI | Zbl

[13] Nicholas M. Katz Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 409 (Lecture Notes in Mathematics), Volume 317, Springer, 1973, pp. 167-200 | DOI | Numdam | Zbl

[14] Nicholas M. Katz p-adic L-functions via moduli of elliptic curves, Algebraic geometry (Arcata 1974) (Proceedings of Symposia in Pure Mathematics), Volume 29, American Mathematical Society, 1975, pp. 479-506 | DOI | Zbl

[15] Nicholas M. Katz The Eisenstein measure and p-adic interpolation, Am. J. Math., Volume 99 (1977) no. 2, pp. 238-311 | DOI | Zbl

[16] Nicholas M. Katz Crystalline cohomology, Dieudonné modules, and Jacobi sums, Automorphic forms, representation theory and arithmetic (Bombay, 1979) (Tata Institute of Fundamental Research Studies in Mathematics), Volume 10, Tata Institute of Fundamental Research, 1981, pp. 165-246 | DOI | Zbl

[17] Nicholas M. Katz; Barry Mazur Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, 108, Princeton University Press, 1985, xiv+514 pages | DOI

[18] Michel Lazard Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. Fr., Volume 83 (1955), pp. 251-274 | DOI | Numdam | Zbl

[19] Saunders MacLane Categories for the working mathematician, Graduate Texts in Mathematics, 5, Springer, 1971, ix+262 pages | DOI

[20] James S. Milne Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980, xiii+323 pages | DOI

[21] Douglas C. Ravenel Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, 2004 (latest version available at https://people.math.rochester.edu/faculty/doug/mybooks/ravenel.pdf)

[22] The Sage Developers SageMath, the Sage Mathematics Software System (Version 10.1), 2023 (http://www.sagemath.org/)

[23] Joseph H. Silverman The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, 2009, xx+513 pages | DOI

[24] Christophe Soulé K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math., Volume 55 (1979) no. 3, pp. 251-295 | DOI | Zbl

[25] William Stein Modular forms, a computational approach, Graduate Studies in Mathematics, 79, American Mathematical Society, 2007, xvi+268 pages (with an appendix by Paul E. Gunnells) | DOI

[26] Tamás Szamuely Galois groups and fundamental groups, Cambridge Studies in Advanced Mathematics, 117, Cambridge University Press, 2009, x+270 pages | DOI

[27] Ningchuan Zhang Analogs of Dirichlet L-functions in chromatic homotopy theory, Adv. Math., Volume 399 (2022), 108267, 84 pages | DOI | Zbl

Cité par Sources :