Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912991538053120 |
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| author | Linden, Georg |
| author_facet | Linden, Georg |
| contents | We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial) valuation, and for $\mathrm{GL}_2(\mathcal{O}_F)$. Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat-Tits tree. Another tool we use is condensed group cohomology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane Linden, Georg Algebraic Geometry Number Theory 11F85 (Primary) 11G18, 14G22 (Secondary) We explicitly determine the group of isomorphism classes of equivariant line bundles on the non-archimedean Drinfeld upper half plane for $\mathrm{GL}_2(F)$, for its subgroups of matrices whose determinant has even (respectively trivial) valuation, and for $\mathrm{GL}_2(\mathcal{O}_F)$. Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat-Tits tree. Another tool we use is condensed group cohomology. |
| title | Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane |
| topic | Algebraic Geometry Number Theory 11F85 (Primary) 11G18, 14G22 (Secondary) |
| url | https://arxiv.org/abs/2510.10330 |