Local-global compatibility of automorphic Galois representations over CM fields at $p$

Abstract: Let $F$ be a CM number field; then, to any cuspidal, regular algebraic automorphic representation of $\mathrm{GL}_n(\mathbf{A}_F)$ is associated a compatible system of $p$-adic Galois representations of the absolute Galois group of $F$. We prove that these representations are potentially semi-stable, in the sense of $p$-adic Hodge theory, and satisfy compatibility with the local Langlands correspondence, up to semi-simplification.

• The paper proves that p-adic local-global compatibility holds in a semi-simplified sense for regular algebraic cuspidal automorphic representations over CM fields.
• It demonstrates that the associated Galois representations are de Rham with Hodge–Tate weights as predicted by the local Langlands correspondence at p-adic places, even without strong global hypotheses.
• Quantitative vanishing results and a careful analysis of Shimura cohomology underpin the removal of restrictive irreducibility and genericity conditions.

Local-Global Compatibility of Automorphic Galois Representations over CM Fields at $p$

Introduction and Motivation

The Langlands program posits a deep relationship between automorphic representations and Galois representations, particularly for arithmetic groups like $GL_n$ over number fields. For regular algebraic cuspidal automorphic representations $\pi$ of $GL_n(\mathbb{A}_F)$ with $F$ a CM field, Harris-Lan-Taylor-Thorne and Scholze have constructed associated compatible systems of $p$-adic Galois representations $r_{\pi,p}$ of $G_F = \mathrm{Gal}(\overline{F}/F)$, matching Satake parameters at almost all unramified places. The strongest expectations—expressed in the Langlands reciprocity and local-global compatibility conjectures—require the local behaviour of $r_{\pi,p}$ at all places of $F$ to match the local Langlands correspondence, including at $GL_n$0-adic places, and for these representations to be de Rham with explicitly determined Hodge–Tate weights.

This paper settles the semi-simplified version of local-global compatibility at $GL_n$1 for all regular algebraic cuspidal automorphic representations $GL_n$2 of $GL_n$3 for CM fields $GL_n$4 and all finite extensions $GL_n$5, by showing that the restrictions $GL_n$6 are always de Rham with the predicted Hodge–Tate weights and that their associated (semi-simplified) Weil–Deligne representations match those predicted by local Langlands via $GL_n$7.

Theoretical Framework and Prior Results

Clozel's Conjecture and Its Refinement

Clozel's conjecture, following the general Langlands principle, associates to any regular algebraic cuspidal automorphic representation $GL_n$8 of $GL_n$9 a compatible system of Galois representations, matching Satake parameters at unramified places. A refined version requires, for each place $\pi$0 of $\pi$1, that $\pi$2 match the Weil–Deligne parameter $\pi$3 under the Tate-normalised local Langlands correspondence; de Rhamness at $\pi$4-adic places, with Hodge–Tate weights encoded by the weight of $\pi$5, is built in.

Known Cases

Prior to this work:

• The case of conjugate self-dual $\pi$6 was resolved completely, with full local-global compatibility at all places.
• For $\pi$7, semi-simplified compatibility had been proven (Varma), while some $\pi$8-adic cases were handled under strong global conditions (irreducibility, “decomposed genericity”), allowing boundary cohomology to be controlled by vanishing theorems in the cohomology of unitary-type Shimura varieties.

Main Innovations

The central innovation is the removal of these strong auxiliary global hypotheses (such as irreducibility and genericity) and establishing local-global compatibility without restrictions at $\pi$9-adic places, up to semi-simplification.

Main Results and Strong Numerical Outcomes

Theorem: Semi-simplified Local-Global Compatibility at $GL_n(\mathbb{A}_F)$0

Let $GL_n(\mathbb{A}_F)$1 be a CM field, $GL_n(\mathbb{A}_F)$2 a regular algebraic cuspidal automorphic representation of $GL_n(\mathbb{A}_F)$3, and $GL_n(\mathbb{A}_F)$4 a prime of $GL_n(\mathbb{A}_F)$5. For any isomorphism $GL_n(\mathbb{A}_F)$6, the Galois representation $GL_n(\mathbb{A}_F)$7 is de Rham, with Hodge–Tate weights prescribed by the highest weight $GL_n(\mathbb{A}_F)$8 of $GL_n(\mathbb{A}_F)$9, and

$F$0

as Weil–Deligne representations.

Corollary: Nilpotence Ordering

In fact, in the sense of the nilpotence order $F$1 (as defined in Varma), the monodromy operator $F$2 of $F$3 is less nilpotent than expected, i.e.,

$F$4

indicating a finer match beyond just semi-simplified data.

Quantitative Vanishing for Torsion in Cohomology

Through a carefully crafted analysis leveraging vanishing theorems for the cohomology of Shimura and Igusa varieties via Fargues–Scholze and Harris–Viehmann ideas, the authors provide explicit bounds: for a constructed unramified Hecke operator $F$5 and $F$6, $F$7 annihilates degree $F$8 cohomology outside the middle degree, for all $F$9 (the middle dimension). This uniform quantitative result is essential to control torsion phenomena in the absence of genericity or irreducibility.

Technical Approach and Structure of Argument

New Quantitative Vanishing

Generalising the work of Caraiani–Scholze, the authors prove, even for non-generic maximal ideals, that after inverting a single explicit Hecke operator, the cohomology with torsion coefficients vanishes outside the middle degree up to a bounded power. This is achieved by a geometric analysis of the stratum structure of Shimura varieties and the use of Fargues–Scholze stacks and Mantovan's product formula to systematically reduce to the ordinary stratum and control the rest via Newton polygon and Kottwitz set arguments.

Removing Strong Residual Conditions

Unlike previous arguments that passed to localisations avoiding the boundary or reducible contributions, the authors develop "pseudo-deformation spaces with conditions" using the theory of determinants and Cayley-Hamilton algebras (in the sense of Chenevier and Wake–Wang-Erickson). By constructing deformation rings and their formal completions robustly without multiplicity-free or irreducibility assumptions, they sidestep the need for the categories of Galois representations to be closed under extensions, reducing the question to the behaviour of semi-simple points.

Inductive Boundary-Interior Argument

The cohomology of the Borel–Serre boundary is described in terms of Eisenstein cohomology and local systems attached to Levi subgroups. The absence of residual irreducibility requires a careful induction on $p$0, using a complicated commutation of Hecke algebras, Satake transforms, and the explicit control of various integral structures. Up to bounded $p$1-power torsion, the authors can show that the relevant Hecke eigensystems appearing in the boundary cohomology are inherited from those lower-dimensional spaces where the inductive hypothesis applies.

Degree-Shifting for Interior Cohomology

For interior cohomology, the argument leverages congruence with overconvergent modular forms and a degree-shifting argument that, after sufficient shifting, connects torsion and (co)homology classes in the middle degree to classes with "`cohomologically irregular'" weights (those for which the desired local-global compatibility is known by induction).

Contradictory/Strong Claims

• The main theorem fully removes all residual irreducibility conditions in local-global compatibility results for regular algebraic automorphic representations of $p$2 over CM fields, up to semi-simplification, at $p$3-adic places. This disproves the necessity of decomposed genericity or non-Eisenstein conditions employed in prior literature.
• The treatment of torsion is effective: the "up to nilpotence" statement is made precise, and the explicit bounds on torsion annihilators depend only on the degree and the field, not on auxiliary or genericity data.

Implications and Future Prospects

Practically, this result closes the gap between the construction of Galois representations and their expected local behaviour at primes dividing $p$4 for automorphic representations of $p$5 over CM fields. Theoretically, it substantially advances the understanding of the depth of the Langlands correspondence, eliminating technical hypotheses that had previously restricted the scope of local-global compatibility results.

Potential future directions include:

• Extension to Torsion Automorphic Classes: While the main result concerns characteristic $p$6 (or more generally, torsion-free) Hecke eigenclasses, the techniques demonstrate control over bounded torsion. This opens the door to modularity lifting in situations with more complicated torsion phenomena and to $p$7 theorems under more general circumstances.
• Refinement of the Local Correspondence: The nilpotence ordering in the monodromy operator, established as a corollary, hints at the potential for upgrades from semi-simplification to full compatibility, which would have implications for the Langlands correspondence in families and the construction of Galois representations in completed cohomology.
• Automorphy of Higher Rank Motives: The removal of auxiliary hypotheses is crucial for the broader program of showing the automorphy of compatible systems attached to geometric, non-self-dual motives, especially for $p$8.
• Applications to the Bloch–Kato Conjecture: The companion results on the vanishing of adjoint Bloch–Kato Selmer groups for many automorphic Galois representations are enabled by the new compatibility theorems; these may have arithmetic consequences for the structure of Selmer groups and congruences between modular forms.

Conclusion

This work proves, up to semi-simplification, the full local-global compatibility at $p$9-adic places for automorphic Galois representations attached to regular algebraic cuspidal automorphic representations of $r_{\pi,p}$0 over any CM field, removing prior restrictive global assumptions. The argument is an overview of advancements in the geometry of Shimura varieties, deformation-theoretic innovations regarding pseudodeformations with $r_{\pi,p}$1-adic Hodge-theoretic conditions, and a deft induction on dimension to manage boundary contributions. As such, it significantly broadens the applicability of the $r_{\pi,p}$2-adic Langlands correspondence and sets a robust foundation for future progress in arithmetic geometry and the Langlands program.

Reference: "Local-global compatibility of automorphic Galois representations over CM fields at $r_{\pi,p}$3"