no. 10
Number Theory
A counterexample to the Gouvêa–Mazur conjecture
[Un contre-exemple à la conjecture de Gouvêa–Mazur]

Présenté par : Christophe Soulé

Buzzard, Kevin1 ; Calegari, Frank2
1 Department of Mathematics, Imperial College, 180, Queen's Gate, London SW7 2AZ, UK
2 1, Science Center, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA
Comptes Rendus. Mathématique, Tome 338 (2004) no. 10, pp. 751-753.

Gouvêa et Mazur ont proposé une conjecture précise au sujet des pentes des formes modulaires. Des versions plus faibles de cette conjecture ont été prouvées par Coleman et Wan. Dans cette Note, nous exhibons des exemples contredisant la conjecture.

Gouvêa and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this Note, we exhibit examples contradicting the full conjecture as it currently stands.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.03.016
Buzzard, Kevin 1 ; Calegari, Frank 2

1 Department of Mathematics, Imperial College, 180, Queen's Gate, London SW7 2AZ, UK
2 1, Science Center, Department of Mathematics, Harvard University, Cambridge, MA 02138, USA 
@article{CRMATH_2004__338_10_751_0,
     author = {Buzzard, Kevin and Calegari, Frank},
     title = {A counterexample to the {Gouv\^ea{\textendash}Mazur} conjecture},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {751--753},
     publisher = {Elsevier},
     volume = {338},
     number = {10},
     year = {2004},
     doi = {10.1016/j.crma.2004.03.016},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2004.03.016/}
}
TY  - JOUR
AU  - Buzzard, Kevin
AU  - Calegari, Frank
TI  - A counterexample to the Gouvêa–Mazur conjecture
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 751
EP  - 753
VL  - 338
IS  - 10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2004.03.016/
DO  - 10.1016/j.crma.2004.03.016
LA  - en
ID  - CRMATH_2004__338_10_751_0
ER  - 
%0 Journal Article
%A Buzzard, Kevin
%A Calegari, Frank
%T A counterexample to the Gouvêa–Mazur conjecture
%J Comptes Rendus. Mathématique
%D 2004
%P 751-753
%V 338
%N 10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2004.03.016/
%R 10.1016/j.crma.2004.03.016
%G en
%F CRMATH_2004__338_10_751_0
Buzzard, Kevin; Calegari, Frank. A counterexample to the Gouvêa–Mazur conjecture. Comptes Rendus. Mathématique, Tome 338 (2004) no. 10, pp. 751-753. doi : 10.1016/j.crma.2004.03.016. http://www.numdam.org/articles/10.1016/j.crma.2004.03.016/

[1] Bosma, W.; Cannon, J.; Playoust, C. The Magma algebra system I: the user language, J. Symbolic Comput., Volume 24 (1997) no. 3–4, pp. 235-265 http://www.maths.usyd.edu.au:8000/u/magma/

[2] Breuil, C. Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) II, J. Inst. Math. Jussieu, Volume 2 (2003) no. 1, pp. 23-58

[3] K. Buzzard, Questions about slopes of modular forms, Astérisque, in press

[4] Coleman, R. p-adic Banach spaces and families of modular forms, Invent. Math., Volume 127 (1997) no. 3, pp. 417-479

[5] Gouvêa, F.; Mazur, B. Families of modular eigenforms, Math. Comp., Volume 58 (1992) no. 198, pp. 793-805

[6] Lang, S. Introduction to Modular Forms, Grundlehren Math. Wiss., vol. 222, Springer-Verlag, 1976

[7] Wan, D. Dimension variation of classical and p-adic modular forms, Invent. Math., Volume 133 (1998) no. 2, pp. 449-463

[8] Zagier, D. Correction to “The Eichler–Selberg trace formula on SL 2 (Z)”, Modular Functions of One Variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math., vol. 627, Springer, Berlin, 1977, pp. 171-173

Cité par Sources :