AI 中文总结
本文证明了模13三次剩余陪集上归一化互反乘积空间的显式Fricke变换，其矩阵为三次高斯周期差分的标量倍数，并给出了任意素数处的Fricke像及其正弦常数的域归属。
AI 中文摘要
我们证明了由模13的三次剩余陪集上的归一化互反乘积生成的二维空间的一个显式Fricke变换。该变换矩阵是三次高斯周期差分矩阵的标量倍数。其射影作用定义在循环三次域上，而归一化矩阵定义在实分圆域上。证明使用了广义eta函数、$X_1(13)$上的一个二次模单位以及一个五系数恒等式。一个独立的除子论证将射影变换提升为所断言的线性变换。我们还记录了在任意素数$p\equiv1\pmod3$处的Fricke像，并证明其归一化正弦常数属于相关的循环三次子域。
英文摘要
We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on $X_1(13)$, and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime $p\equiv1\pmod3$ and prove that their normalized sine constants belong to the associated cyclic cubic subfield.
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