Shin-ya Koyama, Nobushige Kurokawa
We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.
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For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute automorphy.
Let $N$ be a power of $2$ or $3$, and $μ_{N}$ the set of $N$-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ is spanned by the motivic cyclotomic multiple zeta values of level $N$. This implies that the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[μ_{N},\frac{1}{N}]$ on the motivic fundamental group of $\mathbb{G}_{m}-μ_{N}$ is faithful. This is a generalization of the known results for $N\in\{1,2,3,4,8\}$ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight $2$ of oth…
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 $p$.
On zeta elements and functional equations for Tate motives over totally real fields
本文研究了完全实数域上塔特动机的伊瓦萨理论，构造了插值L函数在正整数处值的zeta元素，并建立了与L函数值相关的双 dual 上同调群中的canonical元素。
In a previous paper we constructed a new class of Iwasawa modules as $\ell$--adic realizations of what we called abstract $\ell$--adic $1$--motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the $\ell$--adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) $\ell$--adic Tate sequences for any Galois CM extension $K/k$ of an arbitrary totally real number field $k$. These explicit constructions are significant and useful in…
On the local equivariant Tamagawa number conjecture for Tate motives
本文在无分支条件下证明了塔特动机的局部等变Tamagawa数猜想，推广了Burns--Flach和Burns--Sano的先前工作。
We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…
In this paper we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of F_1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate F_zeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of F_zeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some qu…
Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.
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.
$v$-adic periods of Carlitz motives and Chowla-Selberg formula revisited
该研究在函数域数论框架下，利用带复乘的Carlitz动机的$v$-进晶体-德拉姆周期阐释$v$-进算术伽马值，建立Ogus型Chowla-Selberg公式，证明周期的代数无关性，并推导伽马值代数关系的来源。
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 automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis…
Survey on bounding Selmer groups for Rankin--Selberg motives
本文综述作者此前关于Rankin–Selberg动机的Beilinson–Bloch–Kato猜想研究，并结合X. Zhu的范畴局部Langlands纲领提出新观察与推测。
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions
本文研究了Rankin-Selberg L-函数的例外零点问题，并建立了关于Hilbert cuspidal形式的Sato-Tate分布的最强大收敛率。
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.
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 questions about the number of mixed motives of bounded heights in the case of mixed Tate motives with two graded quotients. Finally, we provide a concrete computation with the number of mixed Tate motives with three graded quotients.
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 $n_i\in \{2,3\}$.
An application of a Hodge realization of Bloch-Kriz mixed Tate motives
本文证明Bloch-Kriz混合Tate动机范畴及其构造的Hodge实现满足Beilinson和Deligne证明Zagier猜想弱版本所需的性质。
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).
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.
Iwasawa's main conjecture for Rankin-Selberg motives in the anticyclotomic case
本文在酉群的Gan--Gross--Prasad猜想框架下，研究了CM域上GL(n)×GL(n+1)尖点自守表示沿反循环方向的岩泽理论，并证明了相应岩泽主猜想的单侧整除性。
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.
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。
Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extension…
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.
$v$-adic Chowla-Selberg formula over function fields
本文在函数域上建立了 $v$-adic Chowla-Selberg 公式，通过引入 Weil 群的晶体作用，给出了 Thakur 几何伽马函数特殊值的周期解释，并基于 CM $t$-模的同源定理证明了 Ogus 型公式。
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.
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.
In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.
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…
Let $E/\mathbb{Q}$ be a totally real number field that is Galois over $\mathbb{Q}$, and let $π$ be a cuspidal, nondihedral automorphic representation of $\mathrm{GL}_2(\mathbb{A}_E)$ that is in the lowest weight discrete series at every real place of $E$. The representation $π$ cuts out a "motive" $M_\mathrm{et}(π^{\infty})$ from the $\ell$-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If $\ell$ is sufficiently large in a sense that depends on $π$ we compute the dimension of the space of Tate classes in $M_\mathrm{et}(π^{\infty})$. Moreover if the space …
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…
We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.
Mixed Tate motives over number fields
本文将域的代数K-理论与多重对数关联，对数域证明Goncharov李余代数到motivic李余代数的实现映射为同构，由此得到相关猜想与ζ函数结果，还构造了Borel类非零倍的多重对数上同调类。
Algebraic Relations among Special Gamma Values and the Chowla-Selberg Phenomenon over Function Fields
本文研究了函数域上特殊伽马值的代数关系，并证明了CM阿贝尔$t$-模准周期的丘-塞尔伯格型公式。
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…
Chern-Simons invariants of hyperbolic three-manifolds, mixed Tate motives, and motivic path torsor of augmented character varieties
针对有限体积完备双曲三维流形，构造了定义在不变迹场上的混合Tate motive并关联其陈-西蒙斯不变量与复体积，分析单环面边界情形下的霍奇实现，解释motivic意义并验证假设，理论与多重对数motivic理论平行。
Exponential motives on the affine Grassmannian
本文提出指数动机的概念，并通过Gaitsgory的Kirillov模型与Whittaker动机比较，建立仿射旗形猜想中的Tate动机和t-结构基础结果，证明了指数Tate动机与分类堆的诱导凝聚sheaves等价。
Over the past fifteen years or so, Minhyong Kim has developed a framework for making effective use of the fundamental group to bound (or even compute) integral points on hyperbolic curves. This is the third installment in a series whose goal is to realize the potential effectivity of Kim's approach in the case of the thrice punctured line. As envisioned in the last installment, we construct an algorithm whose output upon halting is provably the set of integral points, and whose halting would follow from conjectures. Our results go a long way towards achieving our goals over the rationals, whil…
In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors $A$ and $B$ in the moduli spaces $\mathcal{\bar{M}}_{0,n}$ of $n$-pointed stable curves of genus 0, and prove that the cohomology of the pair $(\mathcal{\bar{M}}_{0,n}-A,B-B\cap A)$ is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple zeta values. Then we apply our construction to the dilogarit…
We study the functional equation for the multi-signed Selmer groups for non-ordinary motives whose Hodge-Tate weights are $0$ and $1$, defined by Büyükboduk and the first named author. This generalizes simultaneously Greenberg's result for ordinary motives and Kim's result for supersingular elliptic curves.
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.
In the mirror symmetry of Calabi-Yau threefolds, the instanton expansion of the prepotential has a constant term that is a rational multiple of $ζ(3)/(2 πi)^3$, the motivic origin of which has been carefully studied in the author's paper with M. Kim. The Gamma conjecture claims that higher zeta values in fact appear in the theory of Calabi-Yau $n$-folds for $n \geq 4$. Moreover, a natural question is whether they also have a motivic origin. In this paper, we will study the limit mixed Hodge structure (MHS) of the Fermat pencil of Calabi-Yau $n$-folds at the large complex structure limit, which…
We provide a period interpretation for multizeta values (in the function field context) in terms of explicit iterated extensions of tensor powers of Carlitz motives (mixed Carlitz-Tate t-motives). We give examples of combinatorially involved relations that these multizeta values satisfy.
In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian $t$-modules and $t$-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable $t$-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analo…
In this paper we obtain a Poitou-Tate exact sequence for finite and flat group schemes over a global function field. We also extend the duality theorems for 1-motives over number fields obtained by D.Harari and T.Szamuely to the function field case.
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…
Crystalline Realizations of 1-Motives: Integral Structure and Classification via Linear Frobenius
本文在1-动机范围内证明了p进版本的塔猜想，通过Deligne 1-动机的全忠实性定理，将Barsotti-Tate晶体函子与滤过Dieudonné模的Hom群联系起来，从而得出Frobenius兼容的晶体类本质上是动机的，进而给出动机映射的线性代数描述。
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).
In this article we prove some period relations for the ratio of Deligne's periods for certain tensor product motives. These period relations give a motivic interpretation for certain algebraicity results for ratios of successive critical values for Rankin-Selberg L-functions for ${\rm GL}_n \times {\rm GL}_{n'}$ proved by Günter Harder and the second author.
The geometric Satake equivalence for integral motives
本文研究了整数动机的几何萨塔克等价性，通过引入新的几何结构和等价性，完善了之前对分裂简约群的等价性结果，并给出了德林德勒修改的对偶群和文林格单形的塔恩克构造。
This paper proposes a conjecture on special values of L-functions of geometric motives over Z. This includes L-functions of mixed motives over Q and Hasse-Weil zeta-functions of schemes over Z. We conjecture the following: the order of L(M, s) at s=0 is given by the negative Euler characteristic of motivic cohomology of $M^\vee(-1)$. Up to a nonzero rational factor, the L-value at s=0 is given by the determinant of the pairing of Arakelov motivic cohomology of M with the motivic homology of M. Under standard assumptions concerning mixed motives over Q, $F_p$, and Z, this conjecture is essentia…
This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. …
This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements $σ_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and completely solve the double shuffle equations for multiple zeta values in depths two and three.
We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_Δ$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_Δ^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by re…
We discuss the formalism of Iwasawa theory descent in the setting of the localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.
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