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The motivic Adams conjecture

Alexey Ananyevskiy, Elden Elmanto, Oliver Röndigs, Maria Yakerson

TL;DR

This work proves a motivic analogue of Adams' conjecture after inverting the base field's exponential characteristic $e$, showing that Thom spaces stabilize under Adams operations: $Th(k^N\otimes E) \simeq Th(k^N\otimes \psi^k E)$ in $SH(S)[1/e]$. The strategy combines a motivic mod $k$ Dold theorem away from $e$, a motivic Brown's trick to implement transfers, and Becker–Gottlieb transfer methods, together with reductions to line bundles and étale transfers to handle singular bases with Jouanolou devices. A finiteness/bounded-torsion analysis for higher motivic stable stems is established, highlighting torsion control away from the characteristic and suggesting directions for deeper realizations and J-homomorphism questions in the motivic setting. These results connect Adams-style phenomena to the motivic stable homotopy framework and pave the way for further study of images of motivic $J$-homomorphisms and transfer-based computations in algebraic geometry.

Abstract

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety of maximal tori in a general linear group, which turns out to be not stably A1-connected. We also show that the higher motivic stable stems are of bounded torsion.

The motivic Adams conjecture

TL;DR

This work proves a motivic analogue of Adams' conjecture after inverting the base field's exponential characteristic , showing that Thom spaces stabilize under Adams operations: in . The strategy combines a motivic mod Dold theorem away from , a motivic Brown's trick to implement transfers, and Becker–Gottlieb transfer methods, together with reductions to line bundles and étale transfers to handle singular bases with Jouanolou devices. A finiteness/bounded-torsion analysis for higher motivic stable stems is established, highlighting torsion control away from the characteristic and suggesting directions for deeper realizations and J-homomorphism questions in the motivic setting. These results connect Adams-style phenomena to the motivic stable homotopy framework and pave the way for further study of images of motivic -homomorphisms and transfer-based computations in algebraic geometry.

Abstract

We solve a motivic version of the Adams conjecture with the exponential characteristic of the base field inverted. In the way of the proof we obtain a motivic version of mod k Dold theorem and give a motivic version of Brown's trick studying the homogeneous variety of maximal tori in a general linear group, which turns out to be not stably A1-connected. We also show that the higher motivic stable stems are of bounded torsion.
Paper Structure (7 sections, 25 theorems, 124 equations)

This paper contains 7 sections, 25 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on cohomology theories
  3. A digression on the degree
  4. A motivic mod $k$ Dold theorem away from the characteristic
  5. Brown's trick
  6. Adams' conjecture
  7. Torsion bounds in motivic stable homotopy groups of spheres

Key Result

Theorem 1

Let $\mathcal{E}\xspace$ be a vector bundle over a regular scheme $S$ over a field $F$ of exponential characteristic $e$, and let $k\in \mathbb{Z}$ be an integer. Then for some $N\in \mathbb{N}_0$ one has $\operatorname{Th}(k^N\otimes \mathcal{E}\xspace) \cong \operatorname{Th}(k^N\otimes \psi^k \ma

Theorems & Definitions (67)

  • Theorem : Theorems \ref{['thm:Adams']} and \ref{['thm:Adams_at_char']}
  • Remark 1.1: Complex Betti realization and Adams conjecture
  • Remark 1.2: The real motivic Adams conjecture
  • Theorem : Theorem \ref{['thm:modkDold']}
  • Lemma : Lemma \ref{['lm:kernel_nilpotent']}
  • Theorem : Theorem \ref{['thm:bounded_exponent']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 57 more