论文标题
关于二进制形式的等效性
On the equivalence of binary cubic forms
论文作者
论文摘要
我们考虑确定特征不是$ 2 $或$ 3 $的两个任意字段$ k $的两个二进制立方体的问题，在GL $（2，K）$或SL $（2，K）$的动作下，在每种情况下都获得了两个必要的标准。其中之一涉及二元立方形式的代数不变性，我们称之为Cardano不变性，该形式与经典公式密切相关，并且也出现在Bhargava等人的工作中。第二个标准以基地本身的形式表示，并且还以Sl $（2，k）$或GL $（2，k）的明确矩阵，将一个立方体转化为另一立方体（如果存在的话），则根据双线性双线性因素的系数，这两个立方体的双线性因素的系数。我们还考虑了单个二元立方形式的自态，展示了如何利用我们的结果来测试二进制立方体形式的等效性，而不是〜$ \ mathbb {z} $等积分域，并简要回忆起二进制立方体形式与椭圆曲线的算术之间的某些连接。 所使用的方法是基本的，类似于我们早期与Fisher有关二进制四分之一形式之间等价的工作中使用的方法。
We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either GL$(2,K)$ or SL$(2,K)$, deriving two necessary and sufficient criteria for such equivalence in each case. One of these involves an algebraic invariant of binary cubic forms which we call the Cardano invariant, which is closely connected to classical formulas and also appears in the work of Bhargava et al. The second criterion is expressed in terms of the base field itself, and also gives explicit matrices in SL$(2,K)$ or GL$(2,K)$ transforming one cubic into the other, if any exist, in terms of the coefficients of bilinear factors of a bicovariant of the two cubics. We also consider automorphisms of a single binary cubic form, show how to use our results to test equivalence of binary cubic forms over an integral domain such as~$\mathbb{Z}$, and briefly recall some connections between binary cubic forms and the arithmetic of elliptic curves. The methods used are elementary, and similar to those used in our earlier work with Fisher concerning equivalences between binary quartic forms.
