P-adic L-functions for GL(3)
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911504696082432 |
|---|---|
| author | Loeffler, David Williams, Chris |
| author_facet | Loeffler, David Williams, Chris |
| contents | Let $Π$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $Π$ is $p$-nearly-ordinary for the maximal standard parabolic with Levi $\mathrm{GL}_1 \times \mathrm{GL}_2$, we construct a $p$-adic $L$-function for $Π$. More precisely, we construct a (single) bounded measure $L_p(Π)$ on $\mathbb{Z}_p^\times$ attached to $Π$, and show it interpolates all the critical values $L(Π\timesη,-j)$ at $p$ in the left-half of the critical strip for $Π$ (for varying $η$ and $j$). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic.
Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for $\mathrm{GL}_3$. We work in arbitrary cohomological weight, allow arbitrary ramification at $p$ along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of $p$-adic $L$-functions for RACARs of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ of 'general type' (i.e., those that do not arise as functorial lifts) for any $n > 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04535 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | P-adic L-functions for GL(3) Loeffler, David Williams, Chris Number Theory 11F67, 11R23 Let $Π$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $Π$ is $p$-nearly-ordinary for the maximal standard parabolic with Levi $\mathrm{GL}_1 \times \mathrm{GL}_2$, we construct a $p$-adic $L$-function for $Π$. More precisely, we construct a (single) bounded measure $L_p(Π)$ on $\mathbb{Z}_p^\times$ attached to $Π$, and show it interpolates all the critical values $L(Π\timesη,-j)$ at $p$ in the left-half of the critical strip for $Π$ (for varying $η$ and $j$). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for $\mathrm{GL}_3$. We work in arbitrary cohomological weight, allow arbitrary ramification at $p$ along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of $p$-adic $L$-functions for RACARs of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ of 'general type' (i.e., those that do not arise as functorial lifts) for any $n > 2$. |
| title | P-adic L-functions for GL(3) |
| topic | Number Theory 11F67, 11R23 |
| url | https://arxiv.org/abs/2111.04535 |