Related papers: Some comments on motivic nilpotence

We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck…

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.

The term "motivic Moore spectrum" refers to a cone of an element in the motivic stable homotopy groups of spheres. This article discusses some properties of motivic Moore spectra, among them the question whether the ring structure on the…

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of…

Tits has defined Kac-Moody groups for all root systems, over all commutative rings with unit. A central concept is the idea of a prenilpotent pair of (real) roots. In particular, writing down his group presentation explicitly would require…

Let $K$ be a perfect field and let $E$ be a homotopy commutative ring spectrum in the Morel-Voevodsky stable motivic homotopy category $\mathcal{SH}(K)$. In this work we investigate the relation between the $E$-homology localization and…

A smooth projective scheme $X$ over a field $k$ is said to satisfy the Rost nilpotence principle if any endomorphism of $X$ in the category of Chow motives that vanishes on an extension of the base field $k$ is nilpotent. We show that an…

Let $G$ be a finite soluble group and $G^{(k)}$ the $k$th term of the derived series of $G$. We prove that $G^{(k)}$ is nilpotent if and only if $|ab|=|a||b|$ for any $\delta_k$-values $a,b\in G$ of coprime orders. In the course of the…

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric…

We prove that for a weakly exact magnetic system on a closed connected Riemannian manifold, almost all energy levels contain a closed orbit. More precisely, we prove the following stronger statements. Let $(M,g)$ denote a closed connected…

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers, and finite fields. We use this to extend Morel's identification of the endomorphism ring…

We identify the motivic $KGL/2$-local sphere as the fiber of $\psi^3-1$ on $(2,\eta)$-completed Hermitian $K$-theory, over any base scheme containing $1/2$. This is a motivic analogue of the classical resolution of the $K(1)$-local sphere,…

We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of…

We prove strong convergence results for the motivic Adams spectral sequence of the sphere spectrum over fields with finite virtual cohomological dimension at the prime 2, and over arbitrary fields at odd primes. We show that the motivic…

The motivic Mahowald invariant was introduced in \cite{Qui19a} and \cite{Qui19b} to study periodicity in the $\mathbb{C}$- and $\mathbb{R}$-motivic stable stems. In this paper, we define the motivic Mahowald invariant over any field $F$ of…

Motivated by potential theory on discrete spaces, we study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. These operators are discrete analogues of the classical…

We calculate the motivic stable homotopy groups of the two-complete sphere spectrum after inverting multiplication by the Hopf map eta over fields of cohomological dimension at most 2 with characteristic different from 2 (this includes the…

We construct an algebraic commutative ring T- spectrum BO which is stably fibrant and (8,4)- periodic and such that on SmOp/S the cohomology theory (X,U) -> BO^{p,q}(X_{+}/U_{+}) and Schlichting's hermitian K-theory functor (X,U) ->…

This paper delves into the study of Hilbert schemes of unibranch plane curves whose points have a fixed number of minimal generators. Building on the work of Oblomkov, Rasmussen and Shende we provide a formula for their motivic classes and…

We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of…