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Representations related to CM elliptic curves

Published online by Cambridge University Press:  24 October 2008

Nigel Boston  and
Stephen V. Ullom
Nigel Boston
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, U.S.A.
Stephen V. Ullom
Affiliation:
Department of Mathematics, 1409 W. Green Street, University of Illinois, Urbana, IL 61801, U.S.A.
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Extract

In [10], Mazur showed that the p-adic lifts of a given absolutely irreducible representation are parametrized by a universal deformation ξ:Gℚ, S → GL2() where has the form . (Here Gℚ, S is the Galois group over ℚ of a maximal algebraic extension unramified outside a finite set S of rational primes.) In [1, 3, 10], situations were investigated where the universal deformation ring turned out to be ℚp[[T1T2, T3]] (i.e. r = 3, I = (0)). In [2], the tame relation of algebraic number theory led to more complicated universal deformation rings, ones whose prime spectra consist essentially of four-dimensional sheets.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 71 - 85
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Boston, N.. Explicit deformation of Galois representations. Invent. Math. 103 (1991), 181–196.CrossRefGoogle Scholar
[2]Boston, N.. Families of Galois representations – increasing the ramification. Duke Math. J. 66 (1992), 357–367.CrossRefGoogle Scholar
[3]Boston, N. and Mazue, B.. Explicit universal deformations of Galois representations. Adv. Stud. Pure Math. 17 (1989), 1–21.Google Scholar
[4]Bourbaki, N.. Eléments de Mathématique. Algèbre Commutative (Hermann, 1983).Google Scholar
[5]Coates, J. and Flach, M.. Private communication (1988).Google Scholar
[6]Flach, M.. A finiteness theorem for the symmetric square of an elliptic curve. Preprint.Google Scholar
[7]Ireland, K. and Rosen, M.. A Classical Introduction to Modern Number Theory (Springer-Verlag, 1990).CrossRefGoogle Scholar
[8]Jannsen, U. and Wingberg, K.. Die Struktur der absoluten Galoisgruppe p-adischer Zahlkörper. Invent. Math. 70 (1982), 71–98.CrossRefGoogle Scholar
[9]Koch, H.. Galoissche Theorie der p-Erweilerungen (VEB Deutscher Verlag der Wissenschaften, 1970).CrossRefGoogle Scholar
[10]Mazur, B.. Deforming Galois representations. Proceedings of the March 1987 Workshop on “Galois groups of ℚ” held at MSRI, Berkeley, California.Google Scholar
[11]Neukirch, J.. The absolute Galois group of a p-adic number field, Astérisque 94 (1982), 153–164.Google Scholar
[12]Neumann, O.. On p-closed number fields and an analogue of Riemann's existence theorem. In Algebraic Number Fields (Academic Press, 1977), pp. 625–647.Google Scholar
[13]Ribet, K.. Raising the levels of modular representations. Séminaire de Théorie des Nombres, Paris (1987–1988).Google Scholar
[14]Schlessinger, M.. Functors of Artin rings. Trans. Avier. Math. Soc. 130 (1968), 208–222.CrossRefGoogle Scholar
[15]Serre, J.-P.. Propriétés galoisiennes des points d'ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[16]Serre, J.-P. and Tate, J.. Good reduction of abelian varieties. Ann. of Math. 88 (1968), 492–517.CrossRefGoogle Scholar