Higher composition laws III: The parametrization of quartic rings
Higher composition laws III: The parametrization of quartic rings
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DOI：
10.4007/annals.2004.159.1329
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发表时间：
2004-05
影响因子：
4.9
通讯作者：
M. Bhargava
中科院分区：
文献类型：
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作者：
M. Bhargava
在本系列的前两篇文章中，我们研究了高斯复合的各种高级类似物，并展示了如何将涉及二次和三次域中的阶的几个代数对象显式参数化。特别是，在理论中的核心作用是由二次和三次环本身的参数化。这些参数化很漂亮，也很容易表述。在二次情况下，只需要注意，二次环-即，任何不含秩2的环作为Z-模-由其判别式唯一指定直到同构;相反，给定任何判别式D，即，与0或1（mod 4）全等的任何整数，存在唯一具有判别式D的二次环，即
In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. In the quadratic case, one need only note that a quadratic ring—i.e., any ring that is free of rank 2 as a Z-module—is uniquely specified up to isomorphism by its discriminant; and conversely, given any discriminant D, i.e., any integer congruent to 0 or 1 (mod 4), there is a unique quadratic ring having discriminant D, namely