Related papers: Tate motives and the fundamental group

We construct the motivic t-structure on 1-motives with integral coefficients over a scheme of characteristic zero or a Dedekind scheme. When we invert the residue characteristic exponents of the base, this t-structure induces a t-structure…

This thesis is concerned with the mixed Tate property of reductive algebraic groups $G$, which in particular guarantees a Chow Kunneth property for the classifying space $BG$. Toward this goal, we first refine the construction of the…

In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we…

We study the multiplicities of pure motives modulo numerical equivalence, which are defined as scalars comparing the tannakian trace with the ring-theoretic trace. Our general set-up is that of a rigid semi-simple tensor category such that…

We prove a canonical Kuenneth decomposition of the relative motive with rational coefficients of a smooth commutative group scheme over a noetherian finite dimensional base. This paper is a follow-up of "On the motive of a commutative…

Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A_1,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of…

This paper investigates the structure of generic motives and their implications for the motivic cohomology of fields. Originating in Voevodsky's theory of motives and related to Beilinson's vision of a motivic $t$-structure, generic motives…

The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. As an…

We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including the field of algebraic numbers and the algebraic closure of a finite field, we arrive at a complete description…

We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf…

We define the height of a motive over a number field. We show that if we assume the finiteness of motives of bounded height, Tate conjecture for the $p$-adic Tate module can be proved for motives with good reduction at $p$.

In this paper we define the triangulated category of motives over a simplicial scheme. The morphisms between the Tate objects in this category compute the motivic cohomology of the underlying scheme. In the last section we consider the…

Mixed Tate motives are central objects in the study of cohomology groups of algebraic varieties and their arithmetic invariants. They also play a crucial role in a wide variety of questions related to multiple zeta values and…

We prove a few results concerning the notions of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology…

Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic…

This is the second installment in a sequence of articles devoted to "explicit Chabauty-Kim theory" for the thrice punctured line. Its ultimate goal is to construct an algorithmic solution to the unit equation whose halting will be…

In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all…

Over a perfect field k, let G be an extension of an abelian variety by the multiplicative group $\G_m$. We compute the motive of G in Voevodsky's category of etale motivic complexes with rational coefficients. The result is a decomposition…

We prove a canonical Kunneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the…

We classify the possible Mumford-Tate groups of polarizable rational Hodge structures. Along the way we deduce a polarized Hodge-theoretic analogue of a conjectural property of motivic Galois groups suggested by Serre.