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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
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##### References:
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