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arXiv · 2011.07716

The moduli space of $G$-algebras

Andrew O'Desky·Julian Rosen
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Abstract

Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal X$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb P/G$, where $\mathbb P$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x) \in \mathcal X(\mathbb Q)$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb P/G$.

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Andrew O'Desky, Julian Rosen. 2020-11-16. The moduli space of $G$-algebras. https://arxiv.org/abs/2011.07716

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