论文标题
Q（SQRT（3））和Q（SQRT（17））中的三个正方形的总和
Sums of three squares in Q(sqrt(3)), and in Q(sqrt(17))
论文作者
论文摘要
用$ \ mathbf {q}（\ sqrt {3}）$和$ \ Mathbf {q}（\ sqrt {17}）中的三个整数正方形的表示形式的数量，使用shimura lift top of Hilbert模态形式。我们显示以下结果。如果$ \ mathbf {q}（\ sqrt {3}）$，一个完全正的整数$ a+b \ sqrt {3} $表示为三个整数正方形的总和。如果$ \ Mathbf {q}（\ sqrt {17}）$，当且仅当它不以形式$π_{2}^{2e}^{2e}π_{2}π{2}'^{2e'^{2e'^{2E'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^{2e'^'^'^{2} Qunther and时，完全积极的整数表示为三个整数平方的总和。 $μ\ equiv7 \ pmod {π_{2}^{3}} $或$μ\ equiv7 \ pmod {π_{π_{2}'^{3}} $其中$π_{2}，π_{2}，π_{2}'$是$ 2} $ 2 =π_2} $ 2} 2}。在两种情况下，在$ \ mathbf {q}（\ sqrt {3}）$和$ \ mathbf {q}（\ sqrt {17}）$的两种情况下，类似的结果与高斯的三个平方定理相似，并且给出了其完全想象的扩展名的表格。
The numbers of representations of totally positive integers as sums of three integer squares in $\mathbf{Q}(\sqrt{3})$ and in $\mathbf{Q}(\sqrt{17})$, are studied by using Shimura lifting map of Hilbert modular forms. We show the following results. In case of $\mathbf{Q}(\sqrt{3})$, a totally positive integer $a+b\sqrt{3}$ is represented as a sum of three integer squares if and only if $b$ is even. In case of $\mathbf{Q}(\sqrt{17})$, a totally positive integer is represented as a sum of three integer squares if and only if it is not in the form $π_{2}^{2e}π_{2}'^{2e'}μ$ with $μ\equiv7\pmod{π_{2}^{3}}$ or $μ\equiv7\pmod{π_{2}'^{3}}$ where $π_{2},π_{2}'$ are prime elements with $2=π_{2}π_{2}'$. A similar result as Gauss's three squares theorem in both cases of $\mathbf{Q}(\sqrt{3})$ and $\mathbf{Q}(\sqrt{17})$, and as its application, tables of class numbers of their totally imaginary extensions are given.
