Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915775212683264 |
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| author | Li-Huerta, Siyan Daniel |
| author_facet | Li-Huerta, Siyan Daniel |
| contents | We study the global analogue of the Fargues-Fontaine curve over function fields $F$. We prove some foundational results about its moduli of $G$-bundles $\operatorname{Bun}_{G,F}$, which is a geometrization of the global Kottwitz set $B(F,G)$. For example, $\operatorname{Bun}_{G,F}$ plays the role of Igusa stacks over function fields. We use $\operatorname{Bun}_{G,F}$ to reformulate the global Langlands conjecture for $G$ over $F$ in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when $G$ is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_04978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale Li-Huerta, Siyan Daniel Number Theory Algebraic Geometry Representation Theory 11F70 (Primary) 11G18, 11G09 (Secondary) We study the global analogue of the Fargues-Fontaine curve over function fields $F$. We prove some foundational results about its moduli of $G$-bundles $\operatorname{Bun}_{G,F}$, which is a geometrization of the global Kottwitz set $B(F,G)$. For example, $\operatorname{Bun}_{G,F}$ plays the role of Igusa stacks over function fields. We use $\operatorname{Bun}_{G,F}$ to reformulate the global Langlands conjecture for $G$ over $F$ in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when $G$ is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest. |
| title | Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale |
| topic | Number Theory Algebraic Geometry Representation Theory 11F70 (Primary) 11G18, 11G09 (Secondary) |
| url | https://arxiv.org/abs/2602.04978 |