IJPAM: Volume 96, No. 4 (2014)

ON THE TORSOS FOR SOME GROUP SCHEMES
OF PRIME-POWER ORDER

Yohei Toda
Department of Mathematics
Faculty of Science and Engineering
Chuo University
1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, JAPAN


Abstract. By the classification theorem by F. Oort and J. Tate [#!o-t!#], any group scheme of prime order is isomorphic to a group scheme $G_{a,b}$ under the suitable choice of $a$ and $b$. We computed the torsors for some kinds of group schemes $G_{a,b}$ in [#!s-t!#], which is a joint work with T. Sekiguchi, as in the following way: denote by $p$ a prime number and by $m=\phi(p-1)$ the value of the Euler function $\phi$. Suppose $\mathfrak{p}$ is a prime ideal lying over $p$ (which splits completely in $\mathbb{Z}[\zeta]$), where $\zeta$ is a primitive $(p-1)$-st root of the unity. In case $\mathfrak{p}$ is principal, the sequence

\begin{displaymath} 0 \rightarrow \mu\!\!\!\mu_{p,B} \rightarrow \mathbb{G}_{... ... \xrightarrow{\mathfrak{p}} \mathbb{G}_{m,B}^m \rightarrow 0 \end{displaymath}

is exact, and the Galois descent of $\mu\!\!\!\mu_{p,B}$ is isomorphic to $G_{a,b}$ 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 [#!koide!#] 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 [*] to [*], we would like to review the main result of the papers [#!o-t!#] by F. Oort and J. Tate, [#!k-s!#] by Y. Koide and T. Sekiguchi, and [#!s-t!#] by T. Sekiguchi and Y. Toda. In section [*], we give our main result, namely, the torsor for the Galois descent of $\mu\!\!\!\mu_{p^n,B}$.

Received: October 10, 2013

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DOI: 10.12732/ijpam.v96i4.1 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 96
Issue: 4
Pages: 407 - 425


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