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Iwasawa modules and p-modular representations of GL2

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Abstract

Let F be a finite extension of Q p . We associate, to certain smooth pmodular representations π of GL 2(F), a module \(\mathfrak{S}(\pi )\) on the mod-p Iwasawa algebra of the standard Iwahori subgroup I of GL 2(F). When F is unramified, we obtain a module on a suitable formally smooth F q-algebra, endowed with an action of \(\mathcal{O}_F^ \times \) (the units in the ring of integers of F) and an \(\mathcal{O}_F^ \times \) equivariant, Frobenius semilinear endomorphism which turns out to be p-étale. We study the torsion properties of such a module, as well as its Iwahori-radical filtration.

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References

  1. K. Ardakov and K. A. Brown, Ring-Theoretic Properties of Iwasawa Algebras: A Survey, Documenta Mathematica, Extra Volume Coates (2006), 7–33.

    Google Scholar 

  2. L. Barthel and R. Livné, Irreducible modular representation of GL2 of a local field, Duke Mathematical Journal 75 (1994), 261–292.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Berger, Hanfeng Li and Hui June Zhu, Construction of some families of 2-dimensional crystalline representations Mathematische Annalen 329 (2004), 365–377.

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Berger, Limites des représentations cristallines Compositio Mathematica 140 (2004), 1473–1498.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Berger, La correspondance de langlands locale p-adique pour GL2(Qp), exposé 1017 du séminaire Bourbaki, Astérisque 339 (2011), 157–180.

    Google Scholar 

  6. C. Breuil, Sur quelques répresentations modulaires et p-adiques de GL2(Qp) I, Compositio Mathematica 138 (2003), 165–188.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Breuil, Sur quelques répresentations modulaires et p-adiques de GL2(Qp) II, Journal of the Institute of Mathematics of Jussieu 2 (2003), 1–36.

    Article  Google Scholar 

  8. C. Breuil, Diagrammes de diamonds and (φ, Γ)-modules Israel Journal of Mathematics 182 (2011), 349–382.

    Article  MathSciNet  Google Scholar 

  9. C. Breuil and V. Paskunas, Towards a modulo p Langlands correspondence for GL2 Memoirs of the American Mathematical Society Society 216 (2012).

  10. P. Colmez, Représentations de GL2(Qp) et (ϕ, Γ)-modules, Astérisque 330 (2010), 281–509.

    MathSciNet  Google Scholar 

  11. J. Dixon, M. Du Sautoy, A. Mann and D. Segal Analytic pro p-groups, Cambridge studies in advanced mathematics 61, Cambridge University Press, 2003.

    MATH  Google Scholar 

  12. G. Dousmains, On reduction of families of crystalline Galois representations Documenta Mathematica 15 (2010), 873–938.

    MathSciNet  Google Scholar 

  13. J-M. Fontaine Représentations p-adiques des corps locaux I, The Grothendieck Festschrift vol. II, Progress in Mathematics, Birkhäuser, Boston 87 1990, 249–309.

    MATH  Google Scholar 

  14. M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups I. Definitions and first properties, Astérisque 331 (2010), 335–381.

    MathSciNet  MATH  Google Scholar 

  15. T. Gee, On the weights of mod-p Hilbert modular forms, Inventiones mathematicae 184 (2011), 1–46.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Herzig, A Satake isomorphism in characteristic p. Computational mathematics 147 (2011), 263–283.

    Google Scholar 

  17. F. Herzig, The classification of irreducible admissible mod-p representations of a p-adic GLn, Inventiones mathematicae 186 (2011), 373–434.

    Article  MathSciNet  MATH  Google Scholar 

  18. Y. Hu, Sur quelques représentation supersingulières de GL2(Qpf), Journal of Algebra 324 (2010), 1577–1615.

    Article  MathSciNet  Google Scholar 

  19. Y. Hu, Diagrammes canoniques et représentations modulo p de GL2(F), Journal of the Institute of Mathematics of Jussieu 11 (2012), 67–118

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Hu, S. Morra and B. Schraen, Sur la fidelité de certaines représentations de GL2(F) sous une algèbre d’Iwasawa, The Rendiconti del Seminario Matematico della Universitá di Padova 131 (2014), 49–65.

    Article  MATH  Google Scholar 

  21. S. Morra, Invariant elements for p-modular representations of GL2(Qp), Transactions of the American Mathematical Society, 365 (2013), 6625–6667.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Morra On some representations of the Iwahori subgroup, Journal of Number Theory 132 (2012), 1074–1150.

    Article  MathSciNet  MATH  Google Scholar 

  23. L. Ribes and P. Zalesskii, Profinite groups, Ergebnisse der Mathematik Vol. 40, Springer.

  24. B. Schraen, Sur la présentation des représentations supersingulières de GL2(F), Journal für die reine und angewandte Mathematik, to appear

  25. J-P. Serre, Arbres, amalgames, SL2, Astérisque 46 (1983).

  26. O. Venjakob, A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory, Journal für die reine und angewandte Mathematik 559 (2003), 153–191.

    MathSciNet  MATH  Google Scholar 

  27. J. S. Wilson, Profinite groups, London Mathematical Society 19, Oxford University Press, 1998.

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Author notes

  1. Stefano Morra

    Present address: Department of Mathematics, University of Toronto, Bahen Centre, 40 St. George Street, Toronto, Ontario, M5S 2E4, Canada

Authors and Affiliations

  1. Université Montpellier 2, Place Eugène Bataillon, Case Courrier 51, 34090, Montpellier, France

    Stefano Morra

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  1. Stefano Morra
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Morra, S. Iwasawa modules and p-modular representations of GL2. Isr. J. Math. 219, 1–70 (2017). https://doi.org/10.1007/s11856-017-1473-3

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  • DOI: https://doi.org/10.1007/s11856-017-1473-3

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