[Paper Review] $p$-stabilization in higher dimension

Pascal Boyer|arXiv (Cornell University)|Aug 31, 2018

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper establishes new automorphic congruences between degenerate and tempered automorphic representations of the same weight on Shimura varieties of Kottwitz-Harris-Taylor type via $p$-stabilization in completed cohomology. By allowing level-raising at $l$, it constructs weakly congruent, tempered representations modulo $l$, extending Ribet-type congruences to higher rank groups and providing a cohomological mechanism for non-split Galois extensions in Selmer groups.

ABSTRACT

Using $l$-adic completed cohomology in the context of Shimura varieties of Kottwitz-Harris-Taylor type attached to some fixed similitude group $G$, we prove, allowing to increase the levet at $l$, some new automorphic congruences between any degenerate automorphic representation with a non degenerate one of the same weight.

Motivation & Objective

• To generalize Ribet's congruence method for modular forms to higher-rank reductive groups using completed cohomology.
• To construct weakly congruent automorphic representations of the same weight, one degenerate and one tempered, modulo $l$.
• To provide a cohomological mechanism for non-split Galois extensions in Selmer groups via $p$-stabilization.
• To extend the theory of torsion classes in cohomology to allow level-raising at $l$, analogous to ordinary $p$-stabilization.
• To establish a relative, flexible framework for constructing automorphic congruences in the context of similitude groups $G$ with signatures $(1,d-1), (0,d), \dots, (0,d)$.

Proposed method

• Utilizes $l$-adic completed cohomology of Kottwitz-Harris-Taylor Shimura varieties associated to a similitude group $G$.
• Applies the theory of perverse sheaves and the decomposition theorem to analyze cohomology groups $H^i_{I^l,\xi}(t)_{\mathfrak{m}}$ at various truncation levels $t$.
• Employs spectral sequences and long exact sequences from short exact sequences of sheaves to relate cohomology groups at different levels.
• Uses the structure of the local system $\mathcal{F}_\xi(\mathds{1}_v,t)$ and its intermediate extensions $^p j^{=t}_{!*}$ to analyze torsion and divisibility.
• Applies the principle of local-global compatibility via Emerton's completed cohomology to detect congruences between automorphic representations.
• Relies on the assumption that $H^i_{I^l,\xi}(h)_{\mathfrak{m}}$ is free for $h \geq s_0$, and derives a contradiction from torsion in $H^1_{I^l,\xi}(*,s_0-1)_{\mathfrak{m}}$ to prove non-triviality of the cohomology.

Experimental results

Research questions

• RQ1 Can automorphic congruences between degenerate and tempered representations of the same weight be constructed in higher-rank settings using completed cohomology?
• RQ2 How does level-raising at $l$ enable the construction of such congruences, analogous to $p$-stabilization in classical modular forms?
• RQ3 What is the role of torsion in the cohomology of Harris-Taylor perverse sheaves in realizing non-split Galois extensions?
• RQ4 Can the method produce weakly congruent representations with controlled degeneracy depth and level structure?
• RQ5 Is there a cohomological criterion ensuring that the level at $l$ remains unchanged ($I'_l = I_l$) while achieving congruence modulo $l$?

Key findings

• The paper proves the existence of a tempered automorphic representation $\Pi'$ of the same weight $\xi$ as a given degenerate $\Pi$, weakly congruent modulo $l$.
• $\Pi'$ is shown to have degeneracy depth exactly 1 and level $I'$ with $(I')^l = I^l$, meaning the level at $l$ is preserved.
• The construction relies on the non-vanishing of $H^0_{I^l,\xi}(s_0)_{\mathfrak{m}}$ and the non-trivial torsion in $H^1_{I^l,\xi}(*,s_0-1)_{\mathfrak{m}}$, leading to a contradiction if torsion were trivial.
• The key contradiction arises from assuming $H^i_{I^l,\xi}(h)_{\mathfrak{m}}$ is free for all $h \geq s_0$, which leads to non-trivial torsion in a cohomology group on an affine variety, violating cohomological vanishing.
• The result is interpreted as a higher-dimensional analogue of $p$-stabilization, where Eisenstein series become cuspidal after level-raising.
• The method provides a cohomological realization of non-split extensions of Galois modules, lying in appropriate Selmer groups, via the $L$-function special values of $\Pi$.