On a variation of Mazur’s deformation functor. (English) Zbl 0910.11023

Ramakrishna, Ravi

Compos. Math. 87, No. 3, 269-286 (1993).

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

Citations:

Zbl 0167.49503; Zbl 0714.11076

References:

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