论文标题
数字字段中的根晶格
Root lattices in number fields
论文作者
论文摘要
我们探讨了一个根晶格是否可能类似于Inner Product $（x，y）的晶格$ \ mathscr o $ n数字$ k $（x，y）：= {\ rm trace} _ {k/\ mathbb q}（x \cdotθ（y））$ the $θ$。我们对所有对$ k $，$θ$进行分类，以便$ \ mathscr o $类似于均匀的lattice或root lattice $ \ mathbb z^{[k：\ mathbb q]} $。我们还对所有对$ k $，$θ$进行了分类，以便$ \ mathscr o $是根晶格。除此之外，我们还表明，$ \ mathscr o $从来都不类似于正排名的正等级的晶格$ \ leqslant 48 $，尤其是$ \ mathscr o $与leech lattice并不相似。在附录中，我们为$ \ mathscr o $的判别组的主要组件提供了一般的环节标准。
We explore whether a root lattice may be similar to the lattice $\mathscr O$ of integers of a number field $K$ endowed with the inner product $(x, y):={\rm Trace}_{K/\mathbb Q}(x\cdotθ(y))$, where $θ$ is an involution of $K$. We classify all pairs $K$, $θ$ such that $\mathscr O$ is similar to either an even root lattice or the root lattice $\mathbb Z^{[K:\mathbb Q]}$. We also classify all pairs $K$, $θ$ such that $\mathscr O$ is a root lattice. In addition to this, we show that $\mathscr O$ is never similar to a positive-definite even unimodular lattice of rank $\leqslant 48$, in particular, $\mathscr O$ is not similar to the Leech lattice. In appendix, we give a general cyclicity criterion for the primary components of the discriminant group of $\mathscr O$.
