Abstract
In this article, we give a survey of my results on the non-existence and finiteness of certain Galois extensions of the rational number field ℚ with prescribed ramification. The detail has been (will be) published in [8], [9], [10], [11], [12].
The author was supported by the JSPS Postdoctoral Fellowship for Foreign Researchers.
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References
A. Ash and W. Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod-p cohomology of GL(n, ℤ), Duke Math. J. 105 (2000), 1–24.
S. Brueggeman, The nonexistence of certain Galois extensions unramified outside 5, J. Number Theory 75 (1999), 47–52.
L. E. Dickson, “Linear Groups,” Dover, New York, 1958.
B. Edixhoven, Serre’s Conjectures, in “Modular forms and Fermat’s Last Theorem,” 209–242, Springer-Verlag, 1997.
J.-M. Fontaine, Il n’y a pas de variété abélienne sur ℤ, Invent. Math. 81 (1985), 515–538.
C. Khare, Conjectures on finiteness of mod p Galois representations, J. Ramanujan Math. Soc. 15 (2000), 23–42.
M. J. Larsen and R. Pink, Finite subgroups of algebraic groups, preprint (1998).
H. Moon, Finiteness results on certain mod p Galois representations, J. Number Theory 84 (2000), 156–165.
H. Moon, The number of monomial mod p Galois representations with bounded conductor, Tohoku Math. J. 55 (2003), 89–98.
H. Moon, The non-existence of certain mod p Galois representations, to appear in Bulletin of Korean Math. Soc.
H. Moon and Y. Taguchi, Mod p Galois representations of solvable image, Proc. Amer. Math. Soc. 129 (2001), 2529–2534.
H. Moon and Y. Taguchi, Refinement of Tate’s discriminant bound and nonexistence theorems for mod p Galois representations, to appear.
G. Poitou, Sur les petits discriminants, in “Séminaire Delange-Pisot-Poitou, 18e année: (1976/77), Théorie des nombres, Fasc. 1,” Exp. No. 6, Secrétariat Math., Paris, 1977.
J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(ℚ/ℚ) Duke Math. J. 54 (1987), 179–230.
D. A. Suprunenko, “Matrix Groups,” American Mathematical Society, Providence, R.I., 1976.
J. Tate, The non-existence of certain Galois extensions of ℚunramified outside 2, Contemp. Math. 174 (1994), 153–156.
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© 2004 Kluwer Academic Publishers
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Moon, H. (2004). On the Non-Existence of Certain Galois Extensions. In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_11
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DOI: https://doi.org/10.1007/978-1-4613-0249-0_11
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