Local-global compatibility and the exceptional zero conjecture for GL(3)
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| Format: | Preprint |
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2025
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| author | Salazar, Daniel Barrera Graham, Andrew Williams, Chris |
| author_facet | Salazar, Daniel Barrera Graham, Andrew Williams, Chris |
| contents | We prove exceptional zero conjectures for $p$-ordinary regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3(\mathbb{A})$ which are Steinberg at $p$. We make no self-duality assumptions.
The paper has two parts. In Part 1, we use $p$-arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic $\mathcal{L}$-invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur $\mathcal{L}$-invariants, and thus deduce cases of the full Greenberg--Benois exceptional zero conjecture. As one of the key ingredients for this, we establish local-global compatibility at $\ell = p$ for Galois representations attached to $p$-ordinary torsion classes for $\mathrm{GL}_n$, confirming a conjecture of Hansen in this setting. We prove this for all $n$ following the strategy in the "10-author paper", and use the $n=3$ case to deduce the desired equality of $\mathcal{L}$-invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local-global compatibility and the exceptional zero conjecture for GL(3) Salazar, Daniel Barrera Graham, Andrew Williams, Chris Number Theory 11F80, 11F55, 11F75, 11F70, 11G18 We prove exceptional zero conjectures for $p$-ordinary regular algebraic cuspidal automorphic representations of $\mathrm{GL}_3(\mathbb{A})$ which are Steinberg at $p$. We make no self-duality assumptions. The paper has two parts. In Part 1, we use $p$-arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic $\mathcal{L}$-invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur $\mathcal{L}$-invariants, and thus deduce cases of the full Greenberg--Benois exceptional zero conjecture. As one of the key ingredients for this, we establish local-global compatibility at $\ell = p$ for Galois representations attached to $p$-ordinary torsion classes for $\mathrm{GL}_n$, confirming a conjecture of Hansen in this setting. We prove this for all $n$ following the strategy in the "10-author paper", and use the $n=3$ case to deduce the desired equality of $\mathcal{L}$-invariants. |
| title | Local-global compatibility and the exceptional zero conjecture for GL(3) |
| topic | Number Theory 11F80, 11F55, 11F75, 11F70, 11G18 |
| url | https://arxiv.org/abs/2508.10225 |