# The moduli space of $G$-algebras.

@article{ODesky2020TheMS, title={The moduli space of \$G\$-algebras.}, author={Andrew O’Desky and Julian Rosen}, journal={arXiv: Number Theory}, year={2020} }

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 realized by a subscheme of ${\mathbb P_G}/{G}$, where ${\mathbb P_G}$ is the projective space of the regular representation of ${G}$. We compute a formula for the heights of pairs $(L,x) \in {\mathcal X}({\mathbb Q})$ in terms of algebraic invariants with respect to a natural adelic metric on the anticanonical divisor of ${\mathbb P_G}/{G}$.

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