Deformations and derived categories
[Déformations et catégories dérivées]
Bleher, Frauke M.1 ; Chinburg, Ted2
1 Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA
2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 97-100.

Nous généralisons la théorie de déformation des représentations des groupes profinis développée par Mazur et Schlessinger aux complexes de modules sur de tels groupes. Comme exemple nous déterminons l'anneau de déformation universelle de l'hypercohomologie étale compacte de μp sur certaines courbes elliptiques affines de type CM étudiées par Boston et Ullom.

We generalize the deformation theory of representations of profinite groups developed by Mazur and Schlessinger to complexes of modules for such groups. As an example, we determine the universal deformation ring of the compact étale hypercohomology of μp on certain affine CM elliptic curves studied by Boston and Ullom.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02237-9
Bleher, Frauke M. 1 ; Chinburg, Ted 2

1 Department of Mathematics, University of Iowa, Iowa City, IA 52242-1419, USA
2 Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA 
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Bleher, Frauke M.; Chinburg, Ted. Deformations and derived categories. Comptes Rendus. Mathématique, Tome 334 (2002) no. 2, pp. 97-100. doi : 10.1016/S1631-073X(02)02237-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)02237-9/

[1] Bleher, F.M.; Chinburg, T. Universal deformation rings and cyclic blocks, Math. Ann., Volume 318 (2000), pp. 805-836

[2] Bleher F.M., Chinburg T., Applications of universal deformations to Galois theory, Preprint, 2001

[3] Boston, N.; Ullom, S.V. Representations related to CM elliptic curves, Math. Proc. Cambridge Philos. Soc., Volume 113 (1993), pp. 71-85

[4] Breuil, C.; Conrad, B.; Diamond, F.; Taylor, R. On the modularity of elliptic curves over Q: Wild 3-adic exercises, J. Amer. Math. Soc., Volume 14 (2001), pp. 843-939

[5] Broué, M. Isométries parfaites, types de blocs, catégories dérivées, Astérisque, Volume 181–182 (1990), pp. 61-92

[6] Modular Forms and Fermat's Last Theorem (Boston, 1995) (Cornell, G.; Silverman, J.H.; Stevens, G., eds.), Springer-Verlag, Berlin, 1997

[7] Deninger, C.; Murre, J. Motivic decomposition of Abelian schemes and the Fourier transform, J. Reine Angew. Math., Volume 422 (1991), pp. 201-219

[8] Illusie, L. Complexe cotangent et déformations. I, Lecture Notes in Math., 239, Springer-Verlag, Berlin, 1971

[9] Illusie, L. Complexe cotangent et déformations. II, Lecture Notes in Math., 283, Springer-Verlag, Berlin, 1972

[10] Künnemann, K. On the Chow motive of an Abelian scheme, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 1, American Mathematical Society, Providence, RI, 1994, pp. 189-205

[11] Mazur, B. Deforming Galois representations, Galois Groups over $ \mathrm{Q}$Q (Berkeley, CA, 1987), Springer-Verlag, Berlin, 1989, pp. 385-437

[12] Mazur, B. Deformation theory of Galois representations, Modular Forms and Fermat's Last Theorem (Boston, 1995), Springer-Verlag, Berlin, 1997, pp. 243-311

[13] Milne, J.S. Étale cohomology, Princeton University Press, Princeton, 1980

[14] Ribes, L.; Zalesskii, P. Profinite Groups, Ergeb. Math. Grenzgeb., 40, Springer-Verlag, Berlin, 2000

[15] Rickard, J. The Abelian defect group conjecture, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., Extra, II, 1998, pp. 121-128

[16] Schlessinger, M. Functors of Artin rings, Trans. Amer. Math. Soc., Volume 130 (1968), pp. 208-222

[17] Taylor, R.; Wiles, A. Ring-theoretic properties of certain Hecke algebras, Ann. Math., Volume 141 (1995), pp. 553-572

[18] Wiles, A. Modular elliptic curves and Fermat's last theorem, Ann. Math., Volume 141 (1995), pp. 443-551

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☆ Supported (respectively) by NSA Young Investigator Grant MDA904-01-1-0050 and NSF Grant DMS00-70433.