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Dihedral universal deformations

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This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘\(R=\mathbb {T}\)’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

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References

  1. Allen, P.B., Calegari, F.: Finiteness of unramified deformation rings. Algebra Number Theory 8(9), 2263–2272 (2014). https://doi.org/10.2140/ant.2014.8.2263

    Article  MathSciNet  MATH  Google Scholar 

  2. Aschbacher, M.: Finite Group Theory, Cambridge Studies in Advanced Mathematics, vol. 10. Cambridge University Press, Cambridge (1993). Corrected reprint of the 1986 original

  3. Balasubramanyam, B., Ghate, E., Vatsal, V.: On local Galois representations associated to ordinary Hilbert modular forms. Manuscr. Math. 142(3–4), 513–524 (2013). https://doi.org/10.1007/s00229-013-0614-1

    Article  MathSciNet  MATH  Google Scholar 

  4. Bellaïche, J.: Image of pseudorepresentations and coefficients of modular forms modulo \(p\). Adv. Math. 353, 647–721 (2019)

    Article  MathSciNet  Google Scholar 

  5. Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Astérisque (324), xii+314 (2009)

  6. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997). https://doi.org/10.1006/jsco.1996.0125. Computational algebra and number theory (London, 1993)

  7. Boston, N.: Explicit deformation of Galois representations. Invent. Math. 103(1), 181–196 (1991). https://doi.org/10.1007/BF01239511

    Article  MathSciNet  MATH  Google Scholar 

  8. Boston, N.: Some cases of the Fontaine–Mazur conjecture. II. J. Number Theory 75(2), 161–169 (1999). https://doi.org/10.1006/jnth.1998.2337

    Article  MathSciNet  MATH  Google Scholar 

  9. Boston, N., Mazur, B.: Explicit universal deformations of Galois representations. In: Neukirch, J. (ed.) Algebraic number theory, Advanced Studies in Pure Mathematics, vol. 17, pp. 1–21. Academic Press, Boston, MA (1989)

    Google Scholar 

  10. Calegari, F.: Non-minimal modularity lifting in weight one. J. Reine Angew. Math. 740, 41–62 (2018). https://doi.org/10.1515/crelle-2015-0071

    Article  MathSciNet  MATH  Google Scholar 

  11. Calegari, F., Geraghty, D.: Modularity lifting beyond the Taylor–Wiles method. Invent. Math. 211(1), 297–433 (2018). https://doi.org/10.1007/s00222-017-0749-x

    Article  MathSciNet  MATH  Google Scholar 

  12. Carayol H (1991) Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet. In: Mazur, B. (eds) \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemporary Mathematics, vol. 165. American Mathematical Society, Providence, RI, pp. 213–237 (1994). https://doi.org/10.1090/conm/165/01601

  13. Castella, F., Wang-Erickson, C.: Class groups and local indecomposability for non-CM forms (with an appendix by Haruzo Hida). J. Eur. Math. Soc. (JEMS)

  14. Coleman, R.F.: Classical and overconvergent modular forms. Invent. Math. 124(13), 215–241 (1996)

    Article  MathSciNet  Google Scholar 

  15. Curtis, C.W., Reiner, I.: Methods of Representation Theory With Applications to Finite Groups and Orders, A Wiley-Interscience Publication Pure and Applied Mathematics, vol. I. Wiley, New York (1981)

    Google Scholar 

  16. Darmon, H., Diamond, F., Taylor, R.: Fermat’s Last Theorem Elliptic Curves Modular Forms & Fermat’s Last Theorem (Hong Kong, 1993), pp. 2–140. International Press, Cambridge, MA (1997)

    Google Scholar 

  17. Gelbart, S.: Three Lectures on the Modularity of \({\overline{\rho }}_{E,3}\) and the Langlands Reciprocity Conjecture Modular forms and Fermat’s Last Theorem (Boston. MA, 1995), pp. 155–207. Springer, New York (1997)

    Google Scholar 

  18. Ghate, E., Vatsal, V.: On the local bheaviour of ordinary \(\lambda \)-adic representations. Ann. Inst. Fourier (Greenoble) 54(7), 2143–2162 (2004)

    Article  Google Scholar 

  19. Ghate, E., Vatsal, V.: Locally indecomposable galois representations. Canad. J. Math. 63(2), 277–297 (2011)

    Article  MathSciNet  Google Scholar 

  20. Hida, H.: On abelian varieties with complex multiplication as factors of the abelian variety attached to Hilbert modular forms. Japan. J. Math. (N.S.) 5(1), 157–208 (1979)

    Article  MathSciNet  Google Scholar 

  21. Kisin, M.: Overconvergent modular forms and the fontaine-mazur conjecture. Invent. Math. 153, 373–454 (2003)

    Article  MathSciNet  Google Scholar 

  22. LMFDB Collaboration, T.: The L-functions and modular forms database. http://www.lmfdb.org

  23. Mazur, B.: Deforming Galois representations. In: Ihara, Y. (ed.) Galois Groups Over Q (Berkeley, CA, 1987), Publications of the Research Institute for Mathematical Sciences, vol. 16, pp. 385–437. Springer, New York (1989). https://doi.org/10.1007/978-1-4613-9649-9_7

    Chapter  Google Scholar 

  24. Mazur, B.: An Introduction to the Deformation Theory of Galois Representations. Modular Forms and Fermat’s Last Theorem (Boston MA, 1995), pp. 243–311. Springer, New York (1997)

    Google Scholar 

  25. Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 323, 2nd edn. Springer, Berlin (2008). https://doi.org/10.1007/978-3-540-37889-1

    Book  Google Scholar 

  26. Ozawa, T.: Classical weight one forms in Hida families: Hilbert modular case. Manuscr. Math. 153(3–4), 501–521 (2017). https://doi.org/10.1007/s00229-016-0898-z

    Article  MathSciNet  MATH  Google Scholar 

  27. Raghuram, A., Tanabe, N.: Notes on the arithmetic of Hilbert modular forms. J. Ramanujan Math. Soc. 26(3), 261–319 (2011)

    MathSciNet  MATH  Google Scholar 

  28. Rogawski, J.D.: Functoriality and the Artin conjecture. In: Representation theory and automorphic forms (Edinburgh, 1996), Proceedings of Symposia in Pure Mathematics, vol. 61, pp. 331–353. American Mathematical Society, Providence, RI (1997). https://doi.org/10.1090/pspum/061/1476504

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Acknowledgements

The authors would like to thank Carl Wang-Erickson for useful discussions and encouraging them to include the equi-characteristic case of Theorem 1.4 as well. They also thank Tarun Dalal for pointing out an error in a previous version of this article. They also thank the referee for insightful and helpful comments and suggestions.

The majority of this work was done when S. D. was a postdoc at the University of Luxembourg. G. W. acknowledges support by the IRP AMFOR at the University of Luxembourg. This research is also supported by the Luxembourg National Research Fund INTER/ANR/18/12589973 GALF.

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  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India

    Shaunak V. Deo

  2. Department of Mathematics, University of Luxembourg, Maison du nombre, 6, avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Gabor Wiese

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Deo, S.V., Wiese, G. Dihedral universal deformations. Res. number theory 6, 29 (2020). https://doi.org/10.1007/s40993-020-00197-y

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