论文标题
在不确定的$ k $ - 宇宙积分二次表格上
On indefinite $k$-universal integral quadratic forms over number fields
论文作者
论文摘要
如果代表给定的正整数$ k $的所有积分二次晶格，则一个整体二次晶格被称为无限$ k $ - 宇宙。 对于$ k \ geq 3 $，我们证明无限期的$ k $ - 宇宙财产满足了数字字段的本地全球原则。 对于$ k = 2 $，我们表明一个数字字段$ f $允许当地$ 2 $ - 宇宙的积分二次晶格，而不是当$ f $的班级数量均匀时，而不是不确定的2-宇宙。此外，超过$ f $的此类晶格只有有限的类别。 对于$ k = 1 $，我们证明$ f $承认了一个经典的积分晶格，当时是本地经典的$ 1 $ - 宇宙，但不是经典的不定$ 1 $ - 宇宙，并且只有$ f $具有所有二次不明显的扩展，而所有二型二型二二氧化差异$ f $ of $ f $。在这种情况下，超过$ f $的晶格有无限的类别。确定所有具有此属性的二次字段。
An integral quadratic lattice is called indefinite $k$-universal if it represents all integral quadratic lattices of rank $k$ for a given positive integer $k$. For $k\geq 3$, we prove that the indefinite $k$-universal property satisfies the local-global principle over number fields. For $k=2$, we show that a number field $F$ admits an integral quadratic lattice which is locally $2$-universal but not indefinite 2-universal if and only if the class number of $F$ is even. Moreover, there are only finitely many classes of such lattices over $F$. For $k=1$, we prove that $F$ admits a classic integral lattice which is locally classic $1$-universal but not classic indefinite $1$-universal if and only if $F$ has a quadratic unramified extension where all dyadic primes of $F$ split completely. In this case, there are infinitely many classes of such lattices over $F$. All quadratic fields with this property are determined.
