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Projective linear groups as Galois groups over \({\mathbb Q}\) via modular representations. (English) Zbl 0999.12006

The authors present an algorithm to determine the primes for which the image of the Galois representations associated to a weight 2 newform is not “as large as possible”. By a result of K. A. Ribet [Glasg. Math. J. 27, 185–194 (1985; Zbl 0596.10027)] there are only finitely many such primes. By applying this algorithm to suitable newforms they obtain realizations for the groups \(\text{PSL}(2,\mathbb F_{l^2})\), \(\text{PGL}(2,\mathbb F_{l^3})\) and \(\text{PSL}(2,\mathbb F_{l^4})\) for many primes as Galois groups over the rationals.

MSC:

12F12 Inverse Galois theory
11F80 Galois representations
11F11 Holomorphic modular forms of integral weight

Citations:

Zbl 0596.10027
Full Text: DOI

References:

[1] Brumer, A., The rank of \(J_0(N)\), S.M.F. Astérisque, 228, 41-68 (1995) · Zbl 0851.11035
[2] Carayol, H., Sur les représentations galoisiennes modulo \(l\) attachées aux formes modulaires, Duke Math. J., 59, 785-801 (1989) · Zbl 0703.11027
[3] L. Dieulefait; L. Dieulefait
[4] Faltings, G.; Jordan, B., Crystalline cohomology and GL(2,\(Q\)), Isr. J. Math., 90, 1-66 (1995) · Zbl 0854.14010
[5] Livné, R., On the conductors of mod \(l\) Galois representations coming from modular forms, J. Number Theory, 31, 133-141 (1989) · Zbl 0674.10024
[6] Mazur, B., Modular curves and the Eisenstein ideal, Publ. Math. IHES, 47, 33-186 (1977) · Zbl 0394.14008
[7] Mestre, J. F., Courbes hyperelliptiques à multiplications réelles, C.R. Acad. Sci. Paris, 307, 721-724 (1988) · Zbl 0704.14026
[8] Momose, F., On the \(l\) -adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 28:1, 89-109 (1981) · Zbl 0482.10023
[9] Reverter, A.; Vila, N., Some projective linear groups over finite fields as Galois groups over\(Q\), Contemp. Math., 186, 51-63 (1995) · Zbl 0836.12003
[10] Ribet, K. A., On \(l\) -adic representations attached to modular forms, Invent. Math., 28, 245-275 (1975) · Zbl 0302.10027
[11] Ribet, K., Twists of modular forms and endomorphisms of abelian varieties, Math. Ann., 253, 43-62 (1980) · Zbl 0421.14008
[12] Ribet, K., On \(l\) -adic representations attached to modular forms II, Glasgow Math. J., 27, 185-194 (1985) · Zbl 0596.10027
[13] Ribet, K., Images of semistable Galois representations, Pac. J. Math., 181 (1997)
[14] Serre, J. P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math., 15, 259-331 (1972) · Zbl 0235.14012
[15] Serre, J. P., Sur les représentations modulaires de degré 2 deGal (__\(Q\)/\(Q\)), Duke Math. J., 54, 179-230 (1987) · Zbl 0641.10026
[16] W. Stein; W. Stein
[17] Swinnerton-Dyer, H., On \(ℓ\) -adic Representations and Congruences for Coefficients of Modular Forms (1973), Springer-Verlag: Springer-Verlag Berlin, p. 1-55 · Zbl 0267.10032
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