Related papers: Mixed Tate motives over $\Z$
We give an evaluation for the stuffle-regularised $t^{\ast,V}(\{2\}^a,1,\{2\}^b)$ as a polynomial in single-zeta values, $\log(2)$ and $V$. We then apply this to establish some linear independence results of certain sets of motivic multiple…
The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. In this paper, the properties of this algebra are studied from the point of view of motivic periods.
For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of…
We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick…
The multiple zeta values (MZV) are a set of real numbers with a beautiful structure as an algebra over the rational numbers. They are related to maybe the most important conjecture on mathematics today, the Riemann hypothesis. In this paper…
Recent results of Zlobin and Cresson-Fischler-Rivoal allow one to decompose any suitable $p$-uple series of hypergeometric type into a linear combination (over the rationals) of multiple zeta values of depth at most $p$; in some cases, only…
Using some transformation formulas of the generalized hypergeometric series $\,_3F_2$, we give another proof of D. Zagier's evaluation formula of the multiple zeta values $\zeta(2,...,2,3,2,...,2)$.
In this paper we shall prove that every Witten multiple zeta value of weight w>3 attached to sl(4) at nonnegative integer arguments is a finite rational linear combinations of MZVs of the weight w and the depths three or less, except for…
The original article expressed the special values of the zeta function of a variety over a finite field in terms of the $\hat{Z}$-cohomology of the variety. As the article was being completed, Lichtenbaum conjectured the existence of…
For $k\leq n$, let $E(mn,k)$ be the sum of all multiple zeta values of depth $k$ and weight $mn$ with arguments are multiples of $m\geq 2$. More precisely, $E(mn,k)=\sum_{|\boldsymbol{\alpha}|=n}\zeta(m\alpha_1,m\alpha_2,\ldots,…
Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic…
We study the multiplicities of pure motives modulo numerical equivalence, which are defined as scalars comparing the tannakian trace with the ring-theoretic trace. Our general set-up is that of a rigid semi-simple tensor category such that…
In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. And from these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and…
We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements.…
We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group of the multiplicative group - {N-th roots of unity} and relate it to geometry of modular…
Multiple zeta values (MZVs, also called Euler sums or multiple harmonic series) are nested generalizations of the classical Riemann zeta function evaluated at integer values. The fact that an integral representation of MZVs obeys a shuffle…
We prove a few results concerning the notions of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology…
In 2015, Bachmann \cite{Ba3} conjectured that the~$\Q$-vector space~$\Zq$ of (formal)~$q$-analogues of Multiple Zeta Values (\qmzv s) is spanned by a very particular set compared to known spanning sets. In this work, we prove that this…
In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond…
Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial…