The cone conjecture for vector-valued Siegel automorphic forms
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910380133974016 |
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| author | Koskivirta, Jean-Stefan |
| author_facet | Koskivirta, Jean-Stefan |
| contents | We prove that the effective cone of automorphic vector bundles on the Siegel modular variety of rank $n$ in characteristic $p$ at a place of good reduction is encoded by the stack of $G$-zips of Pink--Wedhorn--Ziegler. Specifically, we show that the degree zero cohomology groups of automorphic vector bundles always vanish outside of the zip cone. This result is a special case of a general conjecture formulated by the Goldring and the author for all Hodge-type Shimura varieties of good reduction. In the case $n=3$, we give explicit conditions for the vanishing of the $0$-th cohomology group. Finally, in the course of the proof we define the notion of automorphic forms of trivial-type and study their properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_16093 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The cone conjecture for vector-valued Siegel automorphic forms Koskivirta, Jean-Stefan Number Theory Algebraic Geometry 14G35, 11F46, 20G40 We prove that the effective cone of automorphic vector bundles on the Siegel modular variety of rank $n$ in characteristic $p$ at a place of good reduction is encoded by the stack of $G$-zips of Pink--Wedhorn--Ziegler. Specifically, we show that the degree zero cohomology groups of automorphic vector bundles always vanish outside of the zip cone. This result is a special case of a general conjecture formulated by the Goldring and the author for all Hodge-type Shimura varieties of good reduction. In the case $n=3$, we give explicit conditions for the vanishing of the $0$-th cohomology group. Finally, in the course of the proof we define the notion of automorphic forms of trivial-type and study their properties. |
| title | The cone conjecture for vector-valued Siegel automorphic forms |
| topic | Number Theory Algebraic Geometry 14G35, 11F46, 20G40 |
| url | https://arxiv.org/abs/2403.16093 |