论文标题
Picard模块化形式和PICARD模块化表面上本地系统的共同体学
Picard modular forms and the cohomology of local systems on a Picard modular surface
论文作者
论文摘要
我们为与单一相似的PICARD模块化表面上的局部系统的共同体制定了详细的猜想Eichler-Shimura类型公式，以$ \ mathrm {gu}（2,1，\ Mathbb {q}（Q}（\ sqrt {-3}}）））$。该公式基于有限磁场上的计数点，这是三个属的曲线，这些曲线是射影线的循环三重覆盖物。假设我们能够在PICARD模块化形式的空间上计算Hecke运算符的痕迹。我们为猜想的公式提供了充分的证据。 在途中，我们证明了弗罗贝尼乌斯（Frobenius）的特征多项式作用于任何属的循环三重覆盖层的共同体学组，用于PICARD模块化形式的空间和公式的尺寸公式和局部系统的数值Euler特征的公式。
We formulate a detailed conjectural Eichler-Shimura type formula for the cohomology of local systems on a Picard modular surface associated to the group of unitary similitudes $\mathrm{GU}(2,1,\mathbb{Q}(\sqrt{-3}))$. The formula is based on counting points over finite fields on curves of genus three which are cyclic triple covers of the projective line. Assuming the conjecture we are able to calculate traces of Hecke operators on spaces of Picard modular forms. We provide ample evidence for the conjectural formula. Along the way we prove new results on characteristic polynomials of Frobenius acting on the first cohomology group of cyclic triple covers of any genus, dimension formulas for spaces of Picard modular forms and formulas for the numerical Euler characteristics of the local systems.
