Extensions of Raynaud Schemes with Trivial Generic Fibers

We calculate the number of extensions of Raynaud schemes over the ring of integers of an absolutely ramified p-adic field with the trivial generic fibers by counting the rational points of the moduli space of finite flat models. Introduction Let K be a totally ramified extension of degree e over Qp for p > 2, and F be a finite field of characteristic p. We consider the constant group scheme CF over SpecK of the two-dimensional vector space over F. Let M(CF,K) be the isomorphism class of the pairs (A,CF ∼ → AK) such that A is a finite flat group scheme over OK with a structure of an F-vector space. Here AK is the generic fiber of A. A finite flat group scheme over OK with a structure of one-dimensional vector space over a finite field is called a Raynaud scheme. So A is an extension of Raynaud schemes by [Tat, Proposition 4.2.1(a)]. If e < p − 1, then M(CF,K) is one-point set by [Ray, Theorem 3.3.3]. However, if the ramification is big, there are surprisingly many finite flat models. In this paper, we calculate |M(CF,K)|, that is, the number of finite flat models of CF. The main theorem is the following. Theorem. Let q be the cardinality of F. Then we have |M(CF,K)| = ∑ n≥0 (an + a ′ n)q . Here an and a ′ n are defined as in the following. We express e and n by e = (p− 1)e0 + e1, n = (p− 1)n0 + n1 = (p− 1)n ′ 0 + n ′ 1 + e1 such that e0, n0, n ′ 0 ∈ Z and 0 ≤ e1, n1, n ′ 1 ≤ p− 2. Then an =max { e0 − (p+ 1)n0 − n1 − 1, 0 } if n1 6= 0, 1, an =max { e0 − (p+ 1)n0 − n1 − 1, 0 } +max { e0 − (p+ 1)n0 − n1 + 1, 0 } if n1 = 0, 1, and a′n =max { e0 − e1 − (p+ 1)n ′ 0 − n ′ 1 − 2, 0 } if n′1 6= 0, 1, a′n =max { e0 − e1 − (p+ 1)n ′ 0 − n ′ 1 − 2, 0 } +max { e0 − e1 − (p+ 1)n ′ 0 − n ′ 1, 0 } if n′1 = 0, 1 except in the case where n = 0 and e1 = p− 2, in which case we put a0 = e0. In the above theorem, we can easily check that |M(CF,K)| = 1 if e < p− 1. 1 2 NAOKI IMAI Acknowledgment. The author is grateful to his advisor Takeshi Saito for his careful reading of an earlier version of this paper and for his helpful comments. Notation. Throughout this paper, we use the following notation. Let p > 2 be a prime number, and K be a totally ramified extension of Qp of degree e. The ring of integers of K is denoted by OK , and the absolute Galois group of K is denoted by GK . Let F be a finite field of characteristic p. The formal power series ring of u over F is denoted by F[[u]], and its quotient field is denoted by F((u)). Let vu be the valuation of F((u)) normalized by vu(u) = 1, and we put vu(0) = ∞. For x ∈ R, the greatest integer less than or equal to x is denoted by [x]. 1. Preliminaries To calculate the number of finite flat models of CF, we use the moduli spaces of finite flat models constructed by Kisin in [Kis]. Let VF be the two-dimensional trivial representation of GK over F. The moduli space of finite flat models of VF, which is denoted by GRVF,0, is a projective scheme over F. An important property of GRVF,0 is the following Proposition. Proposition 1.1. For any finite extension F of F, there is a natural bijection between the set of isomorphism classes of finite flat models of VF′ = VF ⊗F F and GRVF,0(F ). Proof. This is [Kis, Corollary 2.1.13]. By Proposition 1.1, to calculate the number of finite flat models, it suffices to count the number of the F-rational points of GRVF,0. Let S = Zp[[u]], and OE be the p-adic completion of S[1/u]. There is an action of φ on OE determined by identity on Zp and u 7→ u. We choose elements πm ∈ K such that π0 = π and π p m+1 = πm for m ≥ 0, and put K∞ = ⋃ m≥0 K(πm). Let ΦMOE ,F be the category of finite OE ⊗Zp F-modules M equipped with φ-semilinear map M → M such that the induced OE ⊗Zp F-linear map φ (M) → M is an isomorphism. We take the φ-module MF ∈ ΦMOE ,F that corresponds to the GK∞-representation VF(−1). Here (−1) denotes the inverse of Tate twist. The moduli space GRVF,0 is described via the Kisin modules as in the following. Proposition 1.2. For any finite extension F of F, the elements of GRVF,0(F ) naturally correspond to free F[[u]]-submodules MF′ ⊂ MF ⊗F F of rank 2 that satisfy uMF′ ⊂ (1⊗ φ) ( φ(MF′) ) ⊂ MF′ . Proof. This follows from the construction of GRVF,0 in [Kis, Corollary 2.1.13]. By Proposition 1.2, we often identify a point of GRVF,0(F ) with the corresponding finite free F[[u]]-module. For A ∈ GL2 ( F((u)) ) , we write MF ∼ A if there is a basis {e1, e2} of MF over F((u)) such that φ ( e1 e2 )