Computing the number of certain Galois representations mod $p$

Centeleghe, Tommaso Giorgio

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 603-627.

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 $p$.

En utilisant le lien entre représentations galoisiennes et formes modulaires provenant de la Conjecture de Serre, nous calculons, pour tout premier $p\le 2593$, une borne pour le nombre de classes d’isomorphismes des représentations galoisiennes de $\mathbf{Q}$ sur un ${\overline{\mathbf{F}}}_p$–espace vectoriel de dimension deux qui sont irréductibles, impaires, et non–ramifiées en dehors de $p$.

[1] M. F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1969. | MR | Zbl

[2] A. Ash, G. Stevens, Modular Forms in characteristic $\ell$ and special values of their $L$-functions. Duke Math. J. 53 (1986), no.3, 849–868. | MR | Zbl

[3] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. | MR | Zbl

[4] C. Citro, A. Ghitza, Enumerating Galois representations in Sage. Preprint available at http://arxiv.org/abs/1006.4084.

[5] B. Edixhoven, The weight in Serre’s conjecture on modular forms. Invent. Math. 109 (1992), 563–594. | EuDML | MR | Zbl

[6] B. Gross, A tameness criterion for Galois representations associated to modular forms (mod $p$). Duke Math. J. 61 (1990), no. 2, 445–517. | MR | Zbl

[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. | MR | Zbl

[8] N. Jochnowitz, Congruences between systems of eigenvalues of modular forms. Trans. Amer. Math. Soc. 270 (1982), no.1, 269–285. | MR | Zbl

[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. | MR | Zbl

[10] C. Khare, Modularity of Galois representations and motives with good reduction properties. J. Ramanujan Math. Soc. 22 (2007), No. 1, 1-26. | MR | Zbl

[11] C. Khare, Serre’s modularity conjecture: the level one case. Duke Math. J. 134 (2006), no.3, 557–589. | MR | Zbl

[12] S. Lang, Introduction to Modular Forms. Springer–Verlag, 1976. | MR | Zbl

[13] D.A. Marcus, Number Fields. Springer–Verlag, 1977. | MR | Zbl

[14] J.–P. Serre, Congruences et formes modulaires (d’après H.P.F. Swinnerton-Dyer). Sém. Bourbaki 1972/72, no. 416. | Numdam | MR | Zbl

[15] J.–P. Serre, Corps Locaux. Hermann, Quatrième édition, corrigée, 2004. | MR

[16] J.–P. Serre, A Course in Arithmetiic. Springer-Verlag, 1973. | MR | Zbl

[17] J.–P. Serre, Modular forms of weight one and Galois representations. Algebraic Number Fields, Edited by A. Fröhlich, 193–268. Acad. Press, 1977. | MR | Zbl

[18] G. Wiese, Dihedral Galois representations and Katz modular forms. Documenta Math. 9 (2004), 123–133. | MR | Zbl