×

zbMATH — the first resource for mathematics

On a variation of Mazur’s deformation functor. (English) Zbl 0910.11023
Let \(R\) be in \({\mathcal C}^0\) and \(m_R\) be the maximal ideal of \(R\). Let \(\Gamma_n (R)\) be the kernel of the reduction map \(GL_n (R)\to GL_n (\mathbb{F}_q)\). Let \(\rho: G\to GL_n(R)\) be a homomorphism such that \(\pi\circ \rho= \overline\rho\) where \(\pi\) is the canonical projection \(R\to R/m_R =\mathbb{F}_q\). We call \(\rho_1\) and \(\rho_2\) strictly equivalent if \(\rho_1=Y\rho_2 Y^{-1}\) for some \(Y\) in \(\Gamma_n(R)\). A strict equivalence class of lifts of \(\overline\rho\) to \(R\) is called a deformation of \(\overline \rho\) to \(R\).
Let \(\overline \rho\) be given. For \(R\) in \({\mathcal C}^0\), Mazur’s functor is defined to be \(F: {\mathcal C}^0 \to\text{Sets}\) by \(F(R)=\){the set of deformations of \(\overline\rho\) to \(R\}\). Note that \(F\) is a functor. Mazur has shown that \(F\) satisfies the first three Schlessinger criteria. He has also shown that when \(\overline\rho\) is absolutely irreducible, \(F\) satisfies the fourth of these criteria. In fact, one needs only that the endomorphism ring of the Galois module associated to \(\overline\rho\) be \(\mathbb{F}_q\) to ensure that \(F\) satisfies the fourth criterion. The argument used in [N. Boston, Deformation theory of Galois representations, Harvard Ph.D. thesis (1987)] works with this weaker hypothesis. Let \(C(\overline\rho)\) denote this endomorphism ring. M. Schlessinger showed that a functor satisfying these four criteria is pro-representable. Thus for such \(\overline\rho\) there exists a universal deformation ring \(R(\overline\rho)\) [Trans. Am. Math. Soc. 130, 208-222 (1968; Zbl 0167.49503)].
The author defines a modified version of Mazur’s functor. Attention is restricted to those elements of \(F(R)\) such that the Galois modules determined by the deformation to \(R\) are the generic fibers of finite flat group schemes over \(A\). The aim of this paper is to do this functorially and in some cases compute the (uni)versal flat deformation rings \(R_{fl} (\overline\rho)\). B. Mazur [Publ., Math. Sci. Res. Inst. 16, 385-437 (1989; Zbl 0714.11076)] has considered a restriction that is similar in the ordinary case. The results here apply in the supersingular case. The author shows that if \(K=\mathbb{Q}_p\), \(C(\overline \rho)=\mathbb{F}_q\) and \(\overline\rho\) comes from the generic fiber of a finite flat group scheme over \(\mathbb{Z}_p\), then \(R_{fl} (\overline\rho)= W(\mathbb{F}_q) [[T_1,T_2]]\).

MSC:
11F85 \(p\)-adic theory, local fields
11F80 Galois representations
14L05 Formal groups, \(p\)-divisible groups
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Boston, N. : Deformation Theory of Galois Representations , Harvard Ph.D. thesis, 1987. · Zbl 0698.12012 · doi:10.1007/BF01239511 · eudml:143855
[2] Faltings, G. : Crystalline Cohomology and p-adic Galois Representations . In J. I. Igusa (ed.) Algebraic Analysis, Geometry and Number Theory , John Hopkins University Press, 1989, 25-80. · Zbl 0805.14008
[3] Fontaine, J.M. and Lafaille, G. : Construction de représentations p-adiques , Ann. Sc. ENS 15 (1982), 179-207. · Zbl 0579.14037 · doi:10.24033/asens.1437 · numdam:ASENS_1982_4_15_4_547_0 · eudml:82106
[4] Mazur, B. : Deforming Galois representations . In Y. Ihara, K. Ribet and J.-P. Serre (eds.) Proceedings of the March 1987 Workshop on ”Galois Groups over Q ”. MSRI, Berkeley, California, · Zbl 0714.11076
[5] Mazur, B. : Two-dimensional p-adic representations unramified away from p , Compositio Math. 74 (1990), 114-134. · Zbl 0773.11036 · numdam:CM_1990__74_2_115_0 · eudml:90012
[6] Raynaud, M. : Schémas en Groupes de type (p, p ... p) . Bull. Soc. Math. France 102 (1974), 241-280. · Zbl 0325.14020 · doi:10.24033/bsmf.1779 · numdam:BSMF_1974__102__241_0 · eudml:87227
[7] Schlessinger, M. : Functors of Artin rings , Trans. Amer. Math. Soc. 130 (1968), 208-222. · Zbl 0167.49503 · doi:10.2307/1994967
[8] Serre, J. - P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques , Inv. Math. 15 (1972), 259-331. · Zbl 0235.14012 · doi:10.1007/BF01405086 · eudml:142133
[9] Serre, J.-P. : Sur les représentations modulaires de degré 2 de Gal(Q/Q) , Duke Math. J. 54 (1987), 179-230. · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
[10] Serre, J.-P. : Cohomologie Galoisienne , Lecture Notes in Mathematics No. 5, Springer-Verlag, 1964. · Zbl 0812.12002
[11] Shatz, S. : Group schemes, formal groups and p-divisible groups . In (Cornell and Silverman eds.) Arithmetic Geometry , Springer Verlag, 1987. · Zbl 0603.14033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.