Abstract
Abstract: By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.
Highlights
0 → μμp,B → Gmm,B −→p Gmm,B → 0 is exact, and the Galois descent of μμp,B is isomorphic to Ga,b under the suitable choice of a and b, one can compute the torsors for this kinds of group schemes
The aim of this paper is to study some group schemes of order a power of a prime number
We denote by p a prime number, by ζ a primitive (p − 1)-st root of the unity, and by A a Λp-algebra, where
Summary
1 log p, f (x) is monotonic increase for x ≥ (log p)−1. Since f (3) ≥ 2 and p = 2, we have that k − ordp k ≥ 2 for k ≥ 2, and ordp. By comparing the orders of the groups, we see that φ is isomorphism. Β ∈ (Z/pnZ)× with β ≡ ζ (mod p), whose orders are pn−1 and p − 1 respectively. Setting yi = yi,, we can define elements bi,j,k ∈ B by q yik =. For 1 ≤ i ≤ pn−j, we have the equalities q yrk+i =. Setting Z = zp, we have Zpn−1 = 1, and we can identify yr+i,k = (p − 1)ej 1 − i − 1 Zpj−2 as yr′ ′+i,j, br+i,r+j,k = b′r′+i,r′+j,k for 1 ≤ i, j ≤ pn−j. By Lemma 4.4, it suffices to show that Mpn,k,1 ≡ k! 0≤n1 ,n2 ,...,nk ≤p−2 αj−1≡βn1 +βn2 +···+βnk (mod pn) the coefficient of zαj−1 of yqk j=1 pn−1
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