相关论文: Mixed Tate Motives and Multiple Zeta Values

We give a natural construction of unramified over Z framed mixed Tate motives, whose periods are the multiple zeta values. Namely, for each convergent multiple zeta-value we define two boundary divisors A and B in the moduli space M_{0,n+3}…

We prove that the category of mixed Tate motives over $\Z$ is spanned by the motivic fundamental group of $\Pro^1$ minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a $\Q$-linear…

Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value…

For each field k, we define an abelian category of rationally decomposed mixed motives with integer coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near…

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\zeta(n)$. They naturally include the Beukers-Rhin-Viola integrals for $\zeta(2)$ and the Ball-Rivoal linear forms in odd zeta…

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras in positive characteristic introduced by the second author. If the number of variables involved is 'not…

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 that every multiple zeta value is a $\mathbb{Z}$-linear combination of $\zeta(k_1,\dots, k_r)$ where $k_i\geq 2$. Our proof also yields an explicit algorithm for such an expansion. The key ingredient is to introduce modified…

We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two…

We study a class of q-analogues of multiple zeta values given by certain formal q-series with rational coefficients. After introducing a notion of weight and depth for these q-analogues of multiple zeta values we present dimension…

Multiple zeta values (MZVs for short) can be represented as iterated integrals of $\mathbb{Q}$-rational algebraic differential forms on $\mathbb{P}^1(\mathbb{C})\setminus\{0, 1, \infty\}$. This interpretation allows us to consider MZVs…

Pellarin introduced the deformation of multiple zeta values of Thakur as elements over Tate algebras. In this paper, we relate these values to a certain coordinate of a higher dimensional Drinfeld module over Tate algebras which we will…

We consider multiple zeta values, which are periods of mixed Tate motives over $\mathbb Z$. For a given multiple zeta value $\zeta$, there exists a unique minimal motive $M(\zeta)$ such that $\zeta$ is a period of $M(\zeta)$. In general,…

We prove that the $\mathbb{Q}$-vector space generated by the multiple zeta values is generated by the finite real multiple zeta values introduced by Kaneko and Zagier.

For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $\zeta(i_1,...,i_k)$ with odd denominators. Like the…

Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other…

We study multiple zeta values and their generalizations from the point of view of Rota--Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota--Baxter algebras.

Beilinson and Deligne proved a weak version of Zagier's conjucture on special values of Dedekind zeta functions assuming the existence of a category of mixed Tate motives which has certain properties. We show that Bloch-Kriz category of…

This survey article is the written version of two talks given at the Journ\'ees X-UPS 2019 "P\'eriodes et transcendance" at \'Ecole polytechnique. We give a gentle introduction to the study of multiple zeta values, from Euler's solution to…

In this paper, we formally introduce the notion of Ap{\'e}ry-like sums and we show that every multiple zeta values can be expressed as a $\bf Z$-linear combination of them. We even describe a canonical way to do so. This allows us to put in…