Abstract
Let G be a profinite group and let k be a field. By a k-representation of G we mean a finite dimensional vector space over k with the discrete topology, equipped with a continuous k-linear action of G. If V is a k-representation of G and A is a complete local ring with residue field k, then a deformation of V in A is an isomorphism class of continuous representations of G over A that reduce to V modulo the maximal ideal of A; precise definitions are given in Section 2. We denote by Def(V, A) the set of such deformations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969.
N. Bourbaki, Théorie des ensembles, Hermann, Paris, 1970.
H. Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, pp. 213–237 in: B. Mazur and G. Stevens (eds), p-adic monodromy and the Birch and Swinnerton-Dyer conjecture, Contemp. Math. 165, Amer. Math. Soc., Providence, 1994.
B. Conrad, The flat deformation functor, Chapter XIV in this volume.
H. Darmon, F. Diamond, and R. Taylor, Fermat’s Last Theorem, pp. 1–107 in: R. Bott, A. Jaffe, and S. T. Yau (eds), Current developments in mathematics, 1995 International Press, Cambridge, MA, 1995.
A. Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique, 77, Sém. Bourbaki 12 (1959/60), n° 195.
S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading, Mass., 1993.
B. Mazur, Deforming Galois representations, pp. 385–437 in: Y. Ihara, K. Ribet, and J-P. Serre (eds), Galois groups over Q, MSRI Publications 16, Springer-Verlag, New York, 1989.
C. Procesi, Rings with polynomial identities, Marcel Dekker, New York, 1973.
C. Procesi, Deformations of representations, preprint, December 1995.
R. Ramakrishna, On a variation of Mazur’s deformation functor,Cornpositio Math. 87 (1993), 269–286.
M. Schlessingcr, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222.
A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. (2) 141 (1995), 443–551.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
de Smit, B., Lenstra, H.W. (1997). Explicit Construction of Universal Deformation Rings. In: Cornell, G., Silverman, J.H., Stevens, G. (eds) Modular Forms and Fermat’s Last Theorem. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1974-3_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1974-3_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98998-3
Online ISBN: 978-1-4612-1974-3
eBook Packages: Springer Book Archive