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Local-global compatibility and the exceptional zero conjecture for GL(3)

Daniel Barrera Salazar, Andrew Graham, Chris Williams

TL;DR

The paper studies exceptional zero phenomena for p-adic L-functions attached to regular algebraic cuspidal automorphic representations of GL3 that are p-ordinary with Steinberg at p and no self-duality assumption. It develops an automorphic L-invariant framework via p-arithmetic cohomology and extensions of generalized Steinberg representations, proving an abstract exceptional zero formula and an explicit automorphic derivative formula for L_p(π,s). Concurrently, it establishes local-global compatibility at p for Galois representations arising from GL3 eigenvarieties, and proves equality of automorphic and Fontaine–Mazur L-invariants in broad settings, leveraging trianguline deformation theory. In the symmetric-square (essentially self-dual) case, it connects to Rosso’s results and extends to non-self-dual cases through a trianguline/fundamental-lemma–type approach, providing new instances of the Greenberg–Benois exceptional zero conjectures for GL3 without self-duality. The work unifies automorphic constructions with p-adic Hodge-theoretic invariants, yielding precise derivative formulas for L_p^{(c)} at critical points and establishing a path to p-part of the Bloch–Kato conjecture in higher rank settings.

Abstract

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.

Local-global compatibility and the exceptional zero conjecture for GL(3)

TL;DR

The paper studies exceptional zero phenomena for p-adic L-functions attached to regular algebraic cuspidal automorphic representations of GL3 that are p-ordinary with Steinberg at p and no self-duality assumption. It develops an automorphic L-invariant framework via p-arithmetic cohomology and extensions of generalized Steinberg representations, proving an abstract exceptional zero formula and an explicit automorphic derivative formula for L_p(π,s). Concurrently, it establishes local-global compatibility at p for Galois representations arising from GL3 eigenvarieties, and proves equality of automorphic and Fontaine–Mazur L-invariants in broad settings, leveraging trianguline deformation theory. In the symmetric-square (essentially self-dual) case, it connects to Rosso’s results and extends to non-self-dual cases through a trianguline/fundamental-lemma–type approach, providing new instances of the Greenberg–Benois exceptional zero conjectures for GL3 without self-duality. The work unifies automorphic constructions with p-adic Hodge-theoretic invariants, yielding precise derivative formulas for L_p^{(c)} at critical points and establishing a path to p-part of the Bloch–Kato conjecture in higher rank settings.

Abstract

We prove exceptional zero conjectures for -ordinary regular algebraic cuspidal automorphic representations of which are Steinberg at . We make no self-duality assumptions. The paper has two parts. In Part 1, we use -arithmetic cohomology to unconditionally prove an automorphic exceptional zero conjecture in this setting, using Gehrmann's automorphic -invariant. In Part 2 we prove, under mild assumptions that are expected to always hold, the equality of automorphic and Fontaine--Mazur -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 for Galois representations attached to -ordinary torsion classes for , confirming a conjecture of Hansen in this setting. We prove this for all following the strategy in the "10-author paper", and use the case to deduce the desired equality of -invariants.

Paper Structure

This paper contains 80 sections, 82 theorems, 438 equations.

Table of Contents

  1. Introduction
  2. Set-up and statement of the main results
  3. Method of proof
  4. The automorphic side
  5. The Galois side: local-global compatibility for $\mathop{\mathrm{GL}}\nolimits_n$
  6. Structure of the paper
  7. Acknowledgements
  8. Notation
  9. Automorphic L-invariants
  10. Extensions of Steinberg representations
  11. Generalised Steinberg representations
  12. Extensions I
  13. Explicit computation of $c_{1,\lambda}^\bullet$
  14. Extensions II
  15. p-arithmetic cohomology
  16. ...and 65 more sections

Key Result

Theorem 1

Let $n=3$ and let $p \geqslant 7$. Suppose that the residual representation $\overline{\rho}_{\pi} \colon G_{\mathbb{Q}} \to \operatorname{GL}_3(\overline{\mathbb{F}}_p)$ is irreducible and decomposed generic. Then the exceptional zero conjecture holds for either $L_p^{(0)}$ or $L_p^{(1)}$. Furtherm

Theorems & Definitions (223)

  • Remark 1.1.3
  • Conjecture 1.1.4: The exceptional zero conjecture
  • Theorem 1
  • Remark 1.1.5
  • Remark 1.1.6
  • Remark 1.1.7
  • Remark 1.1.8
  • Theorem 2
  • Remark 1.2.2
  • Theorem 3
  • ...and 213 more