## 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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