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Motives, cohomological invariants and Freudenthal magic square

Nikita Geldhauser, Alexander Henke, Maksim Zhykhovich

Abstract

We investigate cohomological invariants and motivic invariants of semisimple algebraic groups arising in the Freudenthal magic square. Besides, we show that if the Rost invariant of a strongly inner group of type $E_7$ is a sum of at most two symbols modulo 2, then it is isotropic over an odd degree field extension, and use this fact to give a different proof of a result of Petrov and Rigby. Moreover, we construct a cohomological invariant of degree 5 for certain groups of type $^2E_6$ which detects their isotropy.

Motives, cohomological invariants and Freudenthal magic square

Abstract

We investigate cohomological invariants and motivic invariants of semisimple algebraic groups arising in the Freudenthal magic square. Besides, we show that if the Rost invariant of a strongly inner group of type is a sum of at most two symbols modulo 2, then it is isotropic over an odd degree field extension, and use this fact to give a different proof of a result of Petrov and Rigby. Moreover, we construct a cohomological invariant of degree 5 for certain groups of type which detects their isotropy.
Paper Structure (7 sections, 12 theorems, 10 equations)

This paper contains 7 sections, 12 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on algebraic groups and motives
  3. Motives of $^2\mathrm{E}_6$-varieties and a cohomological invariant of degree $5$
  4. On the Rost invariant of groups of type $^2\mathrm{E}_6$ and $\mathrm{E}_7$
  5. Groups arising from the Allison--Faulkner construction
  6. On the $J$-invariant of groups of type $^2\mathrm{E}_6$ and $\mathrm{E}_7$
  7. Freudenthal magic square

Key Result

Theorem 3.1

Under above assumptions there exists a functorial cohomological invariant $u\in H^5(F,\mu_2)$ of $G$ such that for every field extension $L/F$ the variety $(X_{1,6})_L$ has a rational point if and only if $u_L=0\in H^5(L,\mu_2)$.

Theorems & Definitions (30)

  • Theorem 3.1
  • Proposition 3.2: Karpenko, Ka10a
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Remark 3.6
  • Proposition 3.8: De Clercq, DC13
  • Lemma 3.9
  • ...and 20 more