论文标题
Bloch-Beilinson的Hecke角色和Eisenstein的同谋Picard表面的猜想
Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces
论文作者
论文摘要
我们考虑了某些hecke字符的家庭$ ϕ $，$ f $ f $。根据Bloch-Beilinson的猜想，中央点$ s = -1 $的$ l $ function $ l（ϕ，s）$消失的顺序应等于Tate Motive $ \ mathbb {q}（Q}（1）$的tate Motive $ \ Mathbb {Q}（1）$的动力。在本文中，我们假设$ l（ϕ，s）$的功能方程的符号为$ -1 $。这是通过与$ f $相关的PICARD模块化表面的变异和Harder的Eisenstein共同体理论相关的PICARD模块化表面的共同论来实现的。此外，我们证明了这些扩展是在某些Biextensions中自然实现的。我们概述了一个程序来计算Biextension高度，并利用它来确定这些扩展的非平凡性。
We consider certain families of Hecke characters $ϕ$ over a quadratic imaginary field $F$. According to the Bloch-Beilinson conjectures, the order of vanishing of the $L$-function $L(ϕ,s)$ at the central point $s=-1$ should be equal to the dimension of the space of extensions of the Tate motive $\mathbb{Q}(1)$ by the motive associated with $ϕ$. In this article, we construct candidates for the corresponding extensions of Hodge structures, assuming that the sign of the functional equation of $L(ϕ,s)$ is $-1$. This is accomplished through the cohomology of variations of Hodge structures over Picard modular surfaces associated with $F$ and Harder's theory of Eisenstein cohomology. Furthermore, we demonstrate that these extensions are naturally realized within certain biextensions. We outline a program to compute the biextension height and utilize it to establish the non-triviality of these extensions.
