Tomohide Terasoma
In this paper, we construct an object of the abelian category of mixed Tate motive associated to multiple zeta values. as a consequence, we prove the inequality of the dimension of the vector space generated by multiple zeta values, which is conjectured by Zagier. (Some errors was corrected from the first version. Namely, we changed the definition of the boundary $\bold B^0$.)
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This paper studies Artin-Tate motives over number rings. As a subcategory of geometric motives, the triangulated category of Artin-Tate motives DATM(S) is generated by motives of schemes that are finite over the base S. After establishing stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives is two, and there is an equivalence of the triangulated category…
On the Problem of Mixed-Tateness of the Motives of G-Varieties
本文研究了G-品种动机的混合塔特性质，证明在特定条件下这些动机属于混合塔特动机范畴，并探讨了其应用与限制。
For a number field, we have a Tannaka category of mixed Tate motives at our disposal. We construct p-adic points of the associated Tannaka group by using p-adic Hodge theory. Extensions of two Tate objects yield functions on the Tannaka group, and we show that evaluation at our p-adic points is essentially given by the inverse of the Bloch-Kato exponential map.
Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.
On André periods of mixed Tate motives
本文展示混合塔克动机中安德烈周期退化为经典概念，并通过连接安德烈周期与科利安积分，具体实现p进多项式对数的特殊值为安德烈周期。
Recently, the author defined multiple Dedekind zeta values \cite{MDZF} associated to a number $K$ field and a cone $C$. In this paper we construct explicitly non-trivial examples of mixed Tate motives over the ring of integers in $K$, for a real quadratic number field $K$ and a particular cone C. The period of such a motive is a multiple Dedekind zeta values at $(s_1,s_2)=(1,2)$, associated to the pair $(K;C)$, times a nonzero element of $K$.
Higher Tate traces of Chow motives
本文通过定义动机的Tate迹这一新不变量，并利用Chernousov-Gille-Merkurjev分解与Bialynicki-Birula-Hesselink-Iversen滤过，完成了p-内半单代数群射影齐性簇Chow动机的完整分类，同时推广了内半单群动机等价性的刻画。
On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives
研究通过混合塔克动机的co-Lie代数定义从Goncharov群B_n(F)到动机多项式对数的线性映射，支持Beilinson和Soulé关于字段K-群消失的猜想。
We establish a generalized Cassels-Tate dual exact sequence for 1-motives over global fields. We thereby extend the main theorem of [4] from abelian varieties to arbitrary 1-motives.
We define Euclidean scissor congruence groups for an arbitrary algebraically closed field F and propose their conjectural description. We suggest how they should be related to mixed Tate motives over dual numbers for F.
We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the property of having a rational zeta function satisfying a functional equation is preserved under products.
We construct derived fundamental group schemes for Tate motives over connected smooth schemes over fields. We show that there exists a pro affine derived group scheme over the rationals such that its category of perfect representations models the triangulated category of rational mixed Tate motives. Under a hypothesis which is weaker than an integral version of the Beilinson-Soule vanishing conjecture we show that there is an affine derived group scheme over the integers such that its perfect representations model Tate motives with integral coefficients. The hypothesis is for example fulfilled…
We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.
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 algebra.
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
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 combination of $ζ(n_1,..., n_r)$ where $n_i\in \{2,3\}$.
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 groups defined by means of Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ-ring structure on K_0(D).
We first study the weight structure on the triangulated category of Artin-Tate motives over a perfect base field k, building on results of Bondarko's. We then study the t-structure on the triangulated category of Artin-Tate motives, when k is algebraic over the rationals, generalizing a result of Levine's. We finally study the interaction of the weight structure and the t-structure. When k is a number field, this will give a useful criterion identifying the weight structure via realizations.
Using Beilinson's theory of f-categories, we prove that the triangulated category of Tate motives over a field k is equivalent to the bounded derived category of its heart, provided that k is algebraic over the rationals. This answers a question asked by Levine.
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
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 compactly supported motive of a quotient stack. In the first section, we construct the compactly supported motive $M^c(X)$ of an algebraic space $X$ and demonstrate that it satisfies expected properties, following closely Voevodsky's work in the case of schemes. In the second section, we construct a functorial version of Totaro's definition of the compactly suppor…
A natural place to study the Chow ring of the classifying space $BG$, for $G$ a linear algebraic group, is Voevodsky's triangulated category of motives, inside which Morel and Voevodsky, and Totaro have defined motives $M(BG)$ and $M^c(BG)$, respectively. We show that, for any group $G$ of order $p^3$ over a field of characteristic not $p$ which contains a primitive $p^2$-th root of unity, the motive $M(BG)$ is a mixed Tate motive. We also show that, for a finite group $G$ over a field of characteristic zero, $M(BG)$ is a mixed Tate motive if and only $M^c(BG)$ is a mixed Tate motive.
On Tate Milnor-Witt Motives
该研究针对带孤立有理不动点的光滑射影$\u0001D706_m$-簇，证明其具有Tate Milnor-Witt motives，并在欧几里得域下给出分裂公式，简化Chow-Witt群的计算。
We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with Galois group G.
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and c…
We show that the statement analogous to the Mumford-Tate conjecture for abelian varieties holds for 1-motives on unipotent parts. This is done by comparing the unipotent part of the associated Hodge group and the unipotent part of the image of the absolute Galois group with the unipotent part of the motivic fundamental group. Contains an appendix by P. Deligne. In this version: Minor corrections here and there. The Main result about the image of the Galois representations is now enriched by an adelic statement.
Let K be a field of characteristic 0 and A be a rigid tensor K-linear category. Let M be a finite-dimensional object of A in the sense of Kimura-O'Sullivan. We prove that the "motivic" zeta function of M with coefficients in K\_0(A) has a functional equation. When A is the category of Chow motives over a field, we thus recover and generalise previous work of Franziska Heinloth, who considered the case where M is the motive of an abelian variety. We also get a functional equation for the zeta function of any motive modulo homological equivalence over a finite field. Our functional equation invo…
An introduction to mixed Tate motives
本文是一篇综述，系统介绍混合Tate动机的基本概念及其在多重zeta值、代数K理论、双曲几何和粒子物理等领域的广泛应用。
In a number of recent works [6, 7] the authors have introduced and studied a functor $\mathcal{F}_k$ which associates to each loose graph $Γ$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that $\mathcal{F}_k(Γ)$ is largely controlled by the combinatorics of $Γ$. Here, $k$ is a field, and we allow $k$ to be $\mathbb{F}_1$, the field with one element. For each finite prime field $\mathbb{F}_p$, it is noted in [6] that any $\mathcal{F}_k(Γ)$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we s…
Let $G$ be a reductive algebraic group over a perfect field $k$ and $\cG$ a $G$-bundle over a scheme $X/k$. The main aim of this article is to study the motive associated with $\cG$, inside the Veovodsky Motivic categories. We consider the case that $\charakt k=0$ (resp. $\charakt k\geq 0$), the motive associated to $X$ is geometrically mixed Tate (resp. geometrically cellular) and $\cG$ is locally trivial for the Zariski (resp. étale) topology on $X$ and show that the motive of $\cG$ is geometrically mixed Tate. Moreover for a general $X$ we construct a nested filtration on the motive associa…
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 of the tensor triangular spectrum and a classification of thick tensor ideals.
Stratification of Derived Categories of Tate Motives
本研究对特定代数闭域上混合Tate动机导出范畴的局部化张量理想分类，证明其满足Barthel-Heard-Sanders意义下的分层，核心是提出借助Brown–Adams可表示性传递分层的新技术，具备独立研究价值。
Two different constructions of an invariant of an odd dimensional hyperbolic manifold in the K-group $K_{2n-1}(\bar \Bbb Q)\otimes \Bbb Q$ are given. The volume of the manifold is equal to the value of the Borel regulator on that element. The scissor congruence groups in non euclidian geometries are studied and their relationship with algebraic K-theory of the field of complex numbers is discussed.
We prove several duality theorems for the Galois and etale cohomology of 1-motives defined over local and global fields and establish a 12-term Poitou-Tate type exact sequence. The results give a common generalisation and sharpening of well-known theorems by Tate on abelian varieties as well as results by Tate/Nakayama and Kottwitz on algebraic tori.
In a previous work, the author have built two families of distinguished algebraic cycles in Bloch-Kriz cubical cycle complex over the projective line minus three points. The goal of this paper is to show how these cycles induce well-defined elements in the $\HH^0$ of the bar construction of the cycle complex and thus generated comodules over this $\HH^0$, that is a mixed Tate motives as in Bloch and Kriz construction. In addition, it is shown that out of the two families only ones is needed at the bar construction level. As a consequence, the author obtains that one of the family gives a basis…
Mumford-Tate groups of 1-motives and Weil pairing
本文研究1-动机的几何如何决定其动机Galois群的维数，通过周期矩阵描述Mumford-Tate群的作用，并在半椭圆情形下分类周期关系、计算群表示，为Grothendieck周期猜想提供新视角。
Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of positive characteristic $p$ and field of fractions $K$ of characteristic 0. In this paper we consider a $K$-1-motive $M_K$ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over $R$. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over $R$. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter $π$ of $R$, any logarithmic Barsotti-Tate group over $R$ is describe…
According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives.
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} of stable curves of genus zero. The corresponding multiple zeta-motive is the n-th cohomology of the pair (M_{0,n+3} -A,B).
The local Tamagawa number conjecure, first formulated by Fontaine and Perrin-Riou, expresses the compatibility of the (global) Tamagawa number conjecture on motivic $L$-functions with the functional equation. The local conjecture was proven for Tate motives over finite unramified extensions $K/\mathbb{Q}_p$ by Bloch and Kato. We use the theory of $(ϕ, Γ_K)$-modules and a reciprocity law due to Cherbonnier and Colmez to provide a new proof in the case of unramified extensions, and to prove the conjecture for the motive $\mathbb{Q}_p(2)$ over certain tamely ramified extensions.
We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational étale motives over a finite field or the homotopical stable algebraic derivator arising from the semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in suff…
In this paper we construct a Q-linear tannakian category MEM_1 of universal mixed elliptic motives over the moduli space M_{1,1} of elliptic curves. It contains MTM, the category of mixed Tate motives unramified over the integers. Each object of MEM_1 is an object of MTM endowed with an action of SL_2(Z) that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over M_{1,1} whose fiber over the tangential base point d/dq at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of MEM is determined and th…
Three real Artin-Tate motives
本文研究实数域上整系数 Artin-Tate 动机的张量三角谱，利用 Positselski 的模表示理论刻画素数 2 处的六点谱，并分类出十四类模 2 动机及三个生成性特殊动机。
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 integers to certain Ext groups. This is the submitted version, with minor corrections and additions from the first version.
This paper has been withdrawn by the author due to an error in the sufficient condition given for the proof of the Tate conjecture for Catanese surfaces.
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 over any number field K is a period of a mixed Tate motive. Moreover, if K is a totally real number field, then we can choose a cone C so that every multiple Dedekind zeta associated to the pair (K;C) is unramified over the ring of algebraic integers in K. In [7], the author proves similar statements in the special case of a real quadratic fields for a part…
The goal of this paper is to give an explicit description of the triangulated categories of Tate and Artin-Tate motives with finite coefficients Z/m over a field K containing a primitive m-root of unity as the derived categories of exact categories of filtered modules over the absolute Galois group of K with certain restrictions on the successive quotients. This description is conditional upon (and its validity is equivalent to) certain Koszulity hypotheses about the Milnor K-theory/Galois cohomology of K. This paper also purports to explain what it means for an arbitrary nonnegatively graded …
We prove: if the (étale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives comi…
We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $ζ(n)$. They naturally include the Beukers-Rhin-Viola integrals for $ζ(2)$ and the Ball-Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objec…
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 special case of "embedded" simplicial schemes, which correspond to the subsheaves of the constant sheaf and naturally appear in the descent problems for motivic cohomology such as the Bloch-Kato conjecture.
We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.
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 conditional on Goncharov's conjecture about exhaustion of mixed Tate motives by motivic iterated integrals (refined somewhat with respect to ramification), and on Kim's conjecture about the determination of integral points via $p$-adic iterated integrals. In this installment we explain what this means while developing basic tools for the construction of the algorithm…
Murmurations and Sato-Tate Conjectures for High Rank Zetas of Elliptic Curves
本研究将有理数域上椭圆曲线的Sato-Tate猜想与murmuration现象从阿贝尔Hasse-Weil Zeta函数推广至其p约化的非阿贝尔高秩Zeta函数，证明二者在该场景下同样成立。
As a higher genus version of universal mixed elliptic motives by Hain and Matsumoto, we consider mixed Teichmüller motives as certain motivic local systems on the moduli space of pointed curves. We show that the category of mixed Teichmüller motives is equivalent to a full subcategory of a certain product category of mixed Tate motives over Z and universal mixed elliptic motives. Furthermore, we show that unipotent fundamental torsors for the universal open curve give rise to a pro-object in the category of mixed Teichmüller motives. Our results can give a realization of motivic correlators pr…
We make explicit Serre's generalization of the Sato-Tate conjecture for motives, by expressing the construction in terms of fiber functors from the motivic category of absolute Hodge cycles into a suitable category of Hodge structures of odd weight. This extends the case of abelian varietes, which we treated in a previous paper. That description was used by Fite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abelian surfaces; the present description is used by Fite--Kedlaya--Sutherland to make a similar classification for certain motives of weight 3. We also give conditions under…
Isotropic localizations of mixed initial motives
本文定义初始动机三角范畴，通过Chow权结构证明各向同性局部化族保守且Pic-内射，推广了混合设定下高阶Tate迹的研究。
Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $π^{n[k:\Q]}ζ^*_k(1-n)$, where $ζ^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculat…
We study stacks of truncated Barsotti-Tate groups and the $G$-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group $G$ and its action on the root system. The main ingredients are the classification of $G$-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, …
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 triangulated monoidal subcategory generated by motives of quasi-projective curves in DM. Joint with a result of P.O'Sullivan it also gives an example of a variety whose motive is not finite-dimensional while the motivic zeta function is rational.
We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formula…
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