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Cohomology of special unitary groups and congruence subgroups

Claudio Bravo

Abstract

We prove a homotopy invariance result for the first cohomology group of the special unitary group $\mathrm{SU}_3(F[t])$ with coefficients in irreducible representations of $\mathrm{PGL}_2(F)$. The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of $\mathrm{PGL}_2(F)$.

Cohomology of special unitary groups and congruence subgroups

Abstract

We prove a homotopy invariance result for the first cohomology group of the special unitary group with coefficients in irreducible representations of . The main theorem establishes that this cohomology is naturally isomorphic to the corresponding cohomology of .

Paper Structure

This paper contains 5 sections, 12 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Algebraic curves and definition of SU3
  3. Congruence subgroups and reduction to PGL2
  4. On the abelianization of congruence subgroups
  5. Cohomology over rational representations

Key Result

Theorem 1

Wendt If $F$ is an infinite field and $\mathcal{G}$ is a connected reductive smooth group defined over $F$, then the inclusion $F \hookrightarrow F[t]$ induces isomorphisms between the integral homology groups whenever the order of fundamental group of $\mathcal{G}$ is invertible in $F$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 1
  • Theorem 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 8 more