Number theory seminar in memory of F. Momose
On the cyclotomic twisted torus and some torsors.
Presenters
Tsutomu Sekiguchi
Abstract
This is joint work with my graduate students Y. Koide and Y. Toda. For a cyclic extension $K/k$ of degree $n$ with Galois group $G$, the subgroup scheme $\mathcal{T} (n)_k$ of the Weil restriction $\prod_{K/k} \mathbb{G}_{m,K}$ given by the intersection of whole kernels of norm maps is interesting for cryptography. On the other hand, let $\zeta$ be a primitive $n$-th root of unity, and $I \in GL_m (\mathbb{Z})$ be the representation matrix of the multiplication $\zeta$ on $\mathbb{Z}[\zeta]$, where $m = \varphi(n)$. Then by using the matrix $I$, we can define an action of $G$ on $\mathbb{G}^m_{m,K}$, and we can descent this group scheme to the one $\mathbb{G}(n)_{k}$ over $k$, which we call a cyclotomic twisted torus. Then we can prove that $\mathbb{G}(n)_{k}$ and $\mathcal{T} (n)_k$ are isomorphic canonically, and these have a resolution consisting of Weil restrictions of tori and norm maps. Moreover, we would like to discuss about torsors for some kind of finite group schemes by using the above cyclotomic twisted tori.
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