The completed Kirillov model and local-global compatibility for functions on Igusa varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909667494461440 |
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| author | Howe, Sean |
| author_facet | Howe, Sean |
| contents | We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tildeμ_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_24089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The completed Kirillov model and local-global compatibility for functions on Igusa varieties Howe, Sean Number Theory We describe the cuspidal functions $\mathbb{V}_b^{\mathrm{cusp}}$ on the ordinary Caraiani-Scholze Igusa variety for $\mathrm{GL}_2$ as a completion of the smooth Kirillov model for classical cuspidal modular forms, and identify a variant of Hida's ordinary $p$-adic modular forms with the coinvariants of an action of $\tildeμ_{p^\infty}$ on $\mathbb{V}_b^{\mathrm{cusp}}$. As a consequence of these results, we establish a weak local-global compatibility theorem for eigenspaces in $\mathbb{V}_b^{\mathrm{cusp}}$ associated to classical cuspidal modular forms. Based on these results, we conjecture an analog of Hida theory and an associated local-global compatibility for functions on more general Caraiani-Scholze Igusa varieties, which are natural spaces of $p$-adic automorphic forms. |
| title | The completed Kirillov model and local-global compatibility for functions on Igusa varieties |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.24089 |