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Table des matières de ce fascicule | Article précédent | Article suivant Adam Mohamed Weight reduction for cohomological mod $p$ modular forms over imaginary quadratic fields Publications mathématiques de Besançon no. 1 (2014), p. 45-71, doi: 10.5802/pmb.4 Article PDF Class. Math.: 11F75, 11F67, 11F25, 11F41 Mots clés: Modular forms modulo $p$, imaginary quadratic fields, Hecke operators, Serre weight Résumé Soient $F$ un corps quadratique imaginaire et $ \mathcal{O}$ son anneau d’entiers. Soient $\mathfrak{ n} \subset \mathcal{ O} $ un idéal non nul et $ p> 5$ un nombre premier inerte dans $F$ copremier avec $\mathfrak{ n}$. Soit $ V$ une représentation irréductible de dimension finie de $ \overline{\mathbb{F}}_{p}[{\rm GL}_2(\mathbb{F}_{ p^2})]$. Nous établissons qu’un système de valeurs propres de Hecke appartenant au groupe de cohomologie â ?¡ coefficients dans $ V$ appartient aussi au groupe de cohomologie â ?¡ coefficients dans $ \overline{\mathbb{F}}_{p}\otimes det^e$ pour $e \ge 0$ à l’exception, éventuellement, de quelques cas. Bibliographie [2] A. Ash and G. Stevens, Modular forms in characteristic l and special values of their L-function, Duke Math. J 53, no 3 849-868. MR 860675 | Zbl 0618.10026 [3] A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986), 192–220. MR 826158 | Zbl 0596.10026 [4] A. Ash, D. Doud, and D. Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Mathematical Journal, Vol. 112, No. 3, 2002. MR 1896473 | Zbl 1023.11025 [5] A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of ${\rm GL}(n;\mathbb{ Z}) $, Duke Math. J. 105 (2000), 1-24. MR 1788040 | Zbl 1015.11018 [6] J. S. Bygott, Modular forms and modular symbols over imaginary quadratic fields, PhD thesis, University of Exeter, 1998. [7] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, Springer-Verlag New-York, 1982. MR 672956 | Zbl 0584.20036 [8] F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups, Proceedings of the LMS Durham Symposium on L-functions and Galois Representations, 2004. Zbl 1230.11069 [9] B. Edixhoven, C. Khare, Hasse invariant and group cohomology, Documenta Math 8 (2003) 43-50. MR 2029159 | Zbl 1044.11030 [10] M. Emerton, $p$-Adic families of modular forms, Séminaire Bourbaki, 62ème anneé, 2009-2010, No. 1013, (2009). [11] L. M. Figueiredo, Serre’s conjecture for imaginary quadratic fields, Compositio Mathematica. 118 (1999), No. 1, 103-122. MR 1705978 | Zbl 1021.11018 [12] M. H. Sengün and S. Türkelli, Weight Reduction for mod l Bianchi Modular forms, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 2010-2019. MR 2522720 | Zbl 1245.11069 [13] R. Taylor, On congruences between modular forms, PhD Thesis, Princeton University 1988. MR 2636500 [14] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Mathematical Journals, Vol.45, №. 3, (1978), 637-679. MR 507462 | Zbl 0394.10015 [15] E. Urban, Formes automorphes cuspidales pour $\rm GL_2 $ sur un corps quadratique imaginaire. Valeurs spéciales de fonctions L et congruences, Compositio Mathematica, tome 99, No. 3 ( 1995), 283-324. Numdam | MR 1361742 | Zbl 0846.11029 [16] G. Wiese, On the faithfulness of parabolic cohomology as a Hecke module over a finite field, J. Reine Angew. Math. ( 2007), 79-103. MR 2337642 | Zbl 1126.11028 |
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