Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces
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| Format: | Preprint |
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2022
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| _version_ | 1866914204636676096 |
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| author | Bajpai, Jitendra Cavicchi, Mattia |
| author_facet | Bajpai, Jitendra Cavicchi, Mattia |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2203_16435 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces Bajpai, Jitendra Cavicchi, Mattia Number Theory Algebraic Geometry Primary:11G40, 14G35, Secondary:11F75, 11F70, 14D07 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. |
| title | Bloch-Beilinson conjectures for Hecke characters and Eisenstein cohomology of Picard surfaces |
| topic | Number Theory Algebraic Geometry Primary:11G40, 14G35, Secondary:11F75, 11F70, 14D07 |
| url | https://arxiv.org/abs/2203.16435 |