Related papers: Mixed Tate motives over $\Z$
We initiate a study of path spaces in the nascent context of "motivic dga's", under development in doctoral work by Gabriella Guzman. This enables us to reconstruct the unipotent fundamental group of a pointed scheme from the associated…
We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the…
The Hodge correlators ${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$ are functions of several complex variables, defined by Goncharov (arXiv:0803.0297) by an explicit integral formula. They satisfy some linear relations: dihedral symmetry…
Goncharov defined for each field $F$ and an integer $n$ greater than 1 a certain group $B_n(F)$. We consider the possibility of defining a linear map from $B_n(F)$ to the co-Lie algebra of the category of mixed Tate motives defined by Bloch…
In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the second as well as the third arguments. As an application of these formulas, we obtain results of…
We prove Zagier's conjecture on the value at s=4 of the Dedekind zeta-function of a number field F. For any field F, we define a map from the appropriate pieces of algebraic K-theory of F to the cohomology of the weight 4 polylogarithmic…
In this paper, we introduce the method of adding additional factors and a parameter to multiple zeta values and prove some generalizations of the duality theorem and several relations among multiple zeta values. In particular, we are able…
For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hat\mu}(T)$ denote the Denef-Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing…
Assuming the K\"unneth type standard conjecture, we propose a way to describe objects of mixed motives explicitly. We study their formal properties, and we associate mixed motives to schemes smooth and separated over a field. This serves as…
The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$…
In a recent preprint, Matherne, Morales and Selover conjectured that two different representations of unit interval posets are related by the famous zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently by G\'elinas,…
We show the compactly supported motive of the moduli stack of degree $n$ rational curves on the weighted projective stack $\mathcal{P}(a,b)$ is of mixed Tate type over any base field $K$ with $\text{char}(K) \nmid a,b$ and has class…
We prove that every profinite group in a certain class with a rational probabilistic zeta function has only finitely many maximal subgroups.
We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include…
K. Kato has recently defined and studied heights of mixed motives and proposed some interesting questions. In this paper, we relate the study of heights to the study of Tamagawa numbers of motives. We also partially answer one of Kato's…
In this paper we prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces $\mathfrak{M}_{0,n}$ of Riemann spheres with $n$ marked points are multiple zeta values. In order to do this, we introduce a…
We prove a new linear relation for multiple zeta values. This is a natural generalization of the restricted sum formula proved by Eie, Liaw and Ong. We also present an analogous result for finite multiple zeta values.
Multiples zeta values and alternating multiple zeta values in positive characteristic were introduced by Thakur and Harada as analogues of classical multiple zeta values of Euler and Euler sums. In this paper we determine all linear…
Multiple polylogarithms are equipped with rich algebraic structures including the motivic coaction and the single-valued map which both found fruitful applications in high-energy physics. In recent work arXiv:2312.00697, the current authors…
We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by…