博尔扎曲面与克莱因四次曲线的谱行列式
Spectral determinants of the Bolza surface and the Klein quartic
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中文总结 AI 辅助
该研究得到博尔扎曲面与克莱因四次曲线的谱行列式闭式公式，通过奇异商轨形行列式的乘积关系计算，还证明这类曲面是泰希米勒空间上谱行列式的临界点。
中文摘要 AI 辅助
我们得到了光滑双曲博尔扎曲面（Bolza surface）和克莱因四次曲线（Klein quartic）谱行列式的闭式显式公式。对于每一种情况，存在一个乘积关系，将曲面的行列式表示为亏格0和亏格1的奇异商轨形的行列式。通过将奇异Polyakov反常公式应用于判别式为-8和-7的复乘（CM）椭圆曲线上的显式贝里映射（Belyi maps）来计算椭圆因子，而亏格0因子则通过具有锥形奇点的常曲率球面的显式行列式公式计算。相同的乘积关系在对应的等对称形变层上逐纤维成立，并导出行列式和一阶变分恒等式。我们还证明，每一个紧拟柏拉图型双曲曲面都是其泰希米勒（Teichmüller）空间上谱行列式的临界点，博尔扎曲面和克莱因四次曲线尤其如此。
英文摘要
We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichmüller space; in particular, this applies to the Bolza surface and the Klein quartic.
