Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphic congruences between torsion cohomological classes

Boyer Pascal

TL;DR

The paper proves that two Harris-Taylor local systems on Newton strata, which are congruent modulo $l$, have isomorphic $l$-torsion in their $l$-adic cohomology with compact supports, equivalently making their free quotients congruent modulo $l$. This enables the construction of accurate non-tempered automorphic congruences for a CM-type similitude group with signature $(1,d-1)$ by isolating torsion and decomposing cohomology contributions according to automorphic data. The authors develop and leverage integral Harris-Taylor perverse sheaves, establish a torsion-control mechanism via a resolution and Newton stratification, and prove conjecture 5.10 of Boyer-AIF. They then derive quantitative multiplicity equalities for automorphic representations, yielding explicit congruence relations between automorphic forms with congruent local components. The results advance a geometric path to Langlands-compatible congruences and provide a precise description of how mod $l$ reductions govern cohomological and automorphic structures in this setting.

Abstract

For two representations of some local division algebra, congruent modulo $l$, giving rise to two Harris-Taylor local systems on the corresponding Newton strata of the special fiber of a KHT Shimura varieties, we prove that the $l$-torsion of each of their cohomology groups with compact supports are isomorphic, or equivalently the free quotients of each of the cohomology groups are congruent modulo $l$. We then deduce the construction of accurate non tempered automorphic congruences for a similitude group $G/\mathbb Q$ with signature $(1,d-1)$.

Automorphic congruences between torsion cohomological classes

TL;DR

The paper proves that two Harris-Taylor local systems on Newton strata, which are congruent modulo , have isomorphic -torsion in their -adic cohomology with compact supports, equivalently making their free quotients congruent modulo . This enables the construction of accurate non-tempered automorphic congruences for a CM-type similitude group with signature by isolating torsion and decomposing cohomology contributions according to automorphic data. The authors develop and leverage integral Harris-Taylor perverse sheaves, establish a torsion-control mechanism via a resolution and Newton stratification, and prove conjecture 5.10 of Boyer-AIF. They then derive quantitative multiplicity equalities for automorphic representations, yielding explicit congruence relations between automorphic forms with congruent local components. The results advance a geometric path to Langlands-compatible congruences and provide a precise description of how mod reductions govern cohomological and automorphic structures in this setting.

Abstract

For two representations of some local division algebra, congruent modulo , giving rise to two Harris-Taylor local systems on the corresponding Newton strata of the special fiber of a KHT Shimura varieties, we prove that the -torsion of each of their cohomology groups with compact supports are isomorphic, or equivalently the free quotients of each of the cohomology groups are congruent modulo . We then deduce the construction of accurate non tempered automorphic congruences for a similitude group with signature .
Paper Structure (8 sections, 7 theorems, 59 equations, 4 figures)

This paper contains 8 sections, 7 theorems, 59 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations about representations of $GL_n(K)$
  3. KHT-Shimura varieties and Harris-Taylor local systems
  4. $\overline {\mathbb Q}_l$-cohomology groups
  5. Integral Harris-Taylor perverse sheaves
  6. Torsion in the cohomology of a HT perverse sheaf
  7. Automorphic congruences
  8. Proof of conjecture 5.10 of boyer-aif

Key Result

Proposition 4.5

(proposition 3.6 of boyer-aif) Let $\pi_v$ be an irreducible cuspidal representation of $GL_g(F_v)$ and $1 \leq r \leq d/g$. Then we have where Concerning $R_{\pi_v}(s,t)(r,i)(\Pi_v)$ as a sum of representations of $GL_{d-rg}(F_v) \times {\mathbb Z}$, for $\Pi_v \simeq \mathop{\mathrm{Speh}}\nolimits_s({\mathop{\mathrm{St}}\nolimits}_{t_1}(\pi_{1,v})) \times \cdots \times \mathop{\mathrm{Speh}}

Figures (4)

  • Figure 1: The squares indicate the $(r,i)$ such that $m_{s,t}(r,i)=1$ for a Speh ($t=1$) at left and a Steinberg ($s=1$) on the right
  • Figure 2: $m_{s,t}(r,i)=1$ when $s \geq t$ at left and $t \geq s$ on the right
  • Figure 3: Superposition to compute $m(r,i)$ for $\Pi_v \simeq \mathop{\mathrm{Speh}}\nolimits_4(\pi_v) \times \mathop{\mathrm{Speh}}\nolimits_4({\mathop{\mathrm{St}}\nolimits}_3(\pi_v)) \times \mathop{\mathrm{Speh}}\nolimits_4({\mathop{\mathrm{St}}\nolimits}_5(\pi_v))$
  • Figure 4: Superposition to compute $n(r,i)$ for $\Pi_v \simeq \mathop{\mathrm{Speh}}\nolimits_4(\pi_v) \times \mathop{\mathrm{Speh}}\nolimits_4({\mathop{\mathrm{St}}\nolimits}_3(\pi_v)) \times \mathop{\mathrm{Speh}}\nolimits_4({\mathop{\mathrm{St}}\nolimits}_5(\pi_v))$

Theorems & Definitions (17)

  • Definition 2.2
  • Definition 2.3
  • Definition 3.3
  • Definition 4.2
  • Definition 4.3
  • Proposition 4.5
  • Definition 4.6
  • Proposition 4.7
  • Theorem 5.1
  • Lemma 6.1
  • ...and 7 more