Computing the number of certain Galois representations mod p

Tommaso Giorgio Centeleghe[1]

  • [1] Universität Heidelberg IWR, Im Neuenheimer Feld 368 69120 Heidelberg, Germany

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 3, page 603-627
  • ISSN: 1246-7405

Abstract

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Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime p ≤ 2593 , a lower bound for the number of isomorphism classes of Galois representation of Q on a two–dimensional vector space over F ¯ p which are irreducible, odd, and unramified outside p .

How to cite

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Centeleghe, Tommaso Giorgio. "Computing the number of certain Galois representations mod $p$." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 603-627. <http://eudml.org/doc/219850>.

@article{Centeleghe2011,
abstract = {Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime $p\le 2593$, a lower bound for the number of isomorphism classes of Galois representation of $\mathbf\{Q\}$ on a two–dimensional vector space over $\overline\{\mathbf\{F\}\}_p$ which are irreducible, odd, and unramified outside $p$.},
affiliation = {Universität Heidelberg IWR, Im Neuenheimer Feld 368 69120 Heidelberg, Germany},
author = {Centeleghe, Tommaso Giorgio},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {11},
number = {3},
pages = {603-627},
publisher = {Société Arithmétique de Bordeaux},
title = {Computing the number of certain Galois representations mod $p$},
url = {http://eudml.org/doc/219850},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Centeleghe, Tommaso Giorgio
TI - Computing the number of certain Galois representations mod $p$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 603
EP - 627
AB - Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime $p\le 2593$, a lower bound for the number of isomorphism classes of Galois representation of $\mathbf{Q}$ on a two–dimensional vector space over $\overline{\mathbf{F}}_p$ which are irreducible, odd, and unramified outside $p$.
LA - eng
UR - http://eudml.org/doc/219850
ER -

References

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  1. M. F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1969. Zbl0175.03601MR242802
  2. A. Ash, G. Stevens, Modular Forms in characteristic ℓ and special values of their L -functions. Duke Math. J. 53 (1986), no.3, 849–868. Zbl0618.10026MR860675
  3. W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. Zbl0898.68039MR1484478
  4. C. Citro, A. Ghitza, Enumerating Galois representations in Sage. Preprint available at http://arxiv.org/abs/1006.4084. Zbl1244.11099
  5. B. Edixhoven, The weight in Serre’s conjecture on modular forms. Invent. Math. 109 (1992), 563–594. Zbl0777.11013MR1176206
  6. B. Gross, A tameness criterion for Galois representations associated to modular forms (mod p ). Duke Math. J. 61 (1990), no. 2, 445–517. Zbl0743.11030MR1074305
  7. N. Jochnowitz, A study of the local components of the Hecke Algebra mod l . Trans. Amer. Math. Soc. 270 (1982), no.1, 253–267. Zbl0536.10021MR642340
  8. N. Jochnowitz, Congruences between systems of eigenvalues of modular forms. Trans. Amer. Math. Soc. 270 (1982), no.1, 269–285. Zbl0536.10022MR642341
  9. N. Katz, p-adic properties of modular schemes and modular forms. Modular Functions of One Variable III, Lecture Notes in Math. 350, 69–190. Springer–Verlag, 1973. Zbl0271.10033MR447119
  10. C. Khare, Modularity of Galois representations and motives with good reduction properties. J. Ramanujan Math. Soc. 22 (2007), No. 1, 1-26. Zbl1192.11036MR2312549
  11. C. Khare, Serre’s modularity conjecture: the level one case. Duke Math. J. 134 (2006), no.3, 557–589. Zbl1105.11013MR2254626
  12. S. Lang, Introduction to Modular Forms. Springer–Verlag, 1976. Zbl0344.10011MR429740
  13. D.A. Marcus, Number Fields. Springer–Verlag, 1977. Zbl0383.12001MR457396
  14. J.–P. Serre, Congruences et formes modulaires (d’après H.P.F. Swinnerton-Dyer). Sém. Bourbaki 1972/72, no. 416. Zbl0276.14013MR466020
  15. J.–P. Serre, Corps Locaux. Hermann, Quatrième édition, corrigée, 2004. MR354618
  16. J.–P. Serre, A Course in Arithmetiic. Springer-Verlag, 1973. Zbl0432.10001MR344216
  17. J.–P. Serre, Modular forms of weight one and Galois representations. Algebraic Number Fields, Edited by A. Fröhlich, 193–268. Acad. Press, 1977. Zbl0366.10022MR450201
  18. G. Wiese, Dihedral Galois representations and Katz modular forms. Documenta Math. 9 (2004), 123–133. Zbl1119.14019MR2054983

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