AI 中文总结
本文推导了双椭圆亏格二轨迹上平坦锥度量的谱行列式闭式公式，结合退化情形分析确定了相关常数，还解决了行列式公式的绝对归一化问题。
AI 中文摘要
我们得到了双椭圆亏格二轨迹上平坦锥度量的谱行列式的显式闭式公式。这类度量由带有两个简单零点的全纯一次形式生成。精确的克莱因四元谱恒等式将对称参考行列式简化为球面和环面上的标量行列式；随后的奇异反常公式给出了一般情形。所得公式涉及一次形式系数中的初等二元六次式，以及商球面上四锥度量的两个显式超几何面积。该公式还给出了三种退化情形的完整渐近行为：在可分情形中，曲面分裂为两个平坦环面；在单节点不可分情形中，同一条节点曲线有两个度量极限：光滑平坦环面或带有两个柱形端点的完备平坦环面；在同时双节点情形中，极限为带有四个柱形端点的完备平坦球面。对于柱形实现，与Bismut--Bost渐近行为的比较明确确定了对应的常数；在可分情形中，这还计算了Müller--Müller公式中出现的相对行列式。作为副产品，可分渐近行为确定了早期一般锥度量和变分行列式公式的绝对归一化。
英文摘要
We obtain a closed explicit formula for the spectral determinant of flat conical metrics on the bielliptic genus-two locus. The metrics are generated by holomorphic one-forms with two simple zeros. For a symmetric reference metric, an exact Klein-four spectral identity reduces the spectral determinant to scalar determinants on spheres and tori; the singular anomaly formula then yields the general case. The resulting formula involves an elementary binary sextic in the coefficients of the one-form and two explicit hypergeometric areas of four-cone metrics on a quotient sphere. We apply this formula to the separating, one-node nonseparating, and simultaneous two-node degenerations of the curve and obtain complete asymptotics of the spectral determinant in all cases. The one-node degeneration has two distinct metric limits, according to whether the limiting one-form is holomorphic or meromorphic. Comparison of the cylindrical cases with the Bismut--Bost asymptotics determines the corresponding constants explicitly; in the separating case it also evaluates the relative determinant appearing in the Müller--Müller formula. As a by-product, the separating asymptotics evaluate the multiplicative constants left undetermined in earlier general conical and variational determinant formulas.
