The semi-stable Local Langlands Correspondence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909910975905792 |
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| author | Ghate, Eknath |
| author_facet | Ghate, Eknath |
| contents | We start with background that goes into an Iwahori-theoretic reformulation of the mod $p$ Local Langlands Correspondence (§2). We then explain some classical $p$-adic functional analytic results (§3) that go into defining the $p$-adic Banach space (§4) attached to a two-dimensional semi-stable representation $V_{k,{\mathcal L}}$ of the Galois group of ${\mathbb Q}_p$ of weight $k$ and ${\mathcal L}$-invariant ${\mathcal L}$ under the $p$-adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod $p$ LLC, allows one to completely determine the mod $p$ reduction of $V_{k,{\mathcal L}}$ for all weights $3 \leq k \leq p+1$ and all ${\mathcal L}$ for $p \geq 5$ (§5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_14382 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The semi-stable Local Langlands Correspondence Ghate, Eknath Number Theory Representation Theory We start with background that goes into an Iwahori-theoretic reformulation of the mod $p$ Local Langlands Correspondence (§2). We then explain some classical $p$-adic functional analytic results (§3) that go into defining the $p$-adic Banach space (§4) attached to a two-dimensional semi-stable representation $V_{k,{\mathcal L}}$ of the Galois group of ${\mathbb Q}_p$ of weight $k$ and ${\mathcal L}$-invariant ${\mathcal L}$ under the $p$-adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod $p$ LLC, allows one to completely determine the mod $p$ reduction of $V_{k,{\mathcal L}}$ for all weights $3 \leq k \leq p+1$ and all ${\mathcal L}$ for $p \geq 5$ (§5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness. |
| title | The semi-stable Local Langlands Correspondence |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2511.14382 |