David Harari, Tamas Szamuely
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.
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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.
Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r-1$. There is a 1 -- 1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8…
We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.
In this paper, we propose a definition of Neron models of arbitrary Deligne 1-motives over Dedekind schemes, extending Neron models of semi-abelian varieties. The key property of our Neron models is that they satisfy a generalization of Grothendieck's duality conjecture in SGA 7 when the residue fields of the base scheme at closed points are perfect. The assumption on the residue fields is unnecessary for the class of 1-motives with semistable reduction everywhere. In general, this duality holds after inverting the residual characteristics. The definition of Neron models involves careful treat…
Duality for the condensed Weil-étale realisation of $1$-motives over $p$-adic fields
本文以凝聚 Weil 群上同调替代 Galois 上同调，引入 $1$-motives 的凝聚 Weil-étale 实现，从而将 Tate 对偶推广到 $p$-进数域上的 $1$-motives。
In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the later. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism al…
The Arithmetic of Reducible Rank 2 Hypergeometric Motives
本文研究数域上可约且不规则的${}_3F_2(1)$超几何动机的算术，利用动机技术证明新求值公式，并给出与模形式$L$-函数精确值相关的例子。
We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are t he homological 1-motive of Lichtenbaum and the motivic $H^1$ of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum's conjecture that the …
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。
This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in t…
We prove that the existence of an automorphism of finite order on a (defined over a number field) variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Gamma-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne. Our proof relies on the arithmetic fixed point formula (equivariant arithmetic Riemann-Roch theo…
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.
Local duality theorems for commutative algebraic groups
本研究构建任意域上含交换代数群对偶对象的k-1-motives范畴，建立任意特征局部域下关联群fppf上同调与对偶k-1-motive超上同调的Pontryagin对偶定理，还得到任意k-1-motive二阶上同调的对偶定理，可推广Lichtenbaum-van Hamel对偶。
Building upon our arithmetic duality theorems for 1-motives, we prove that the Manin obstruction related to a finite subquotient $\Be (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shaferevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate type dual exact se…
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
We give a generalization of Poitou-Tate duality to schemes of finite type over rings of integers of global fields.
Arithmetic duality for two-dimensional local rings with perfect residue field
本研究针对具有完美剩余域的二维局部环，通过为算术上同调群构建代数群结构，对Saito的算术对偶性理论作出了改进。
Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical values of automorphic L-functions for $GL_{n}\times GL_{1}$ can be interpreted in terms of these arithmetic automorphic periods. In the thesis, we generalize his results in two ways. Firstly, the arithmetic automorphic periods have been defined over general CM fields. We also prove that these periods factorize as products of local periods over infinity places. Secondly, we show that critical values of a…
Duality for Arithmetic $p$-adic Pro-étale Cohomology of Analytic Spaces
本研究验证了Colmez等人关于$p$-进数有限扩张上光滑部分紧合空间的算术$p$-进前平展上同调满足对偶性的猜想，借助Fontaine伽罗瓦下降技术由Fargues-Fontaine曲线的几何对偶性推导得出。
In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
In this paper, we prove that the arithmetic automorphic periods for $GL_{n}$ over a CM field factorize through the infinite places. This generalizes a conjecture of Shimura in 1983, and is predicted by the Langlands correspondence between automorphic representations and motives.
Arithmetic Duality
本文讨论了塔特在有限模和阿贝尔簇的伽罗瓦上同调中证明的对偶定理及其后续发展。
Arithmetic duality for $p$-adic pro-étale cohomology of analytic curves
该研究证明了$\textbf{Q}_p$有限扩张上光滑dagger曲线的算术$p$-进pro-étale上同调的庞加莱对偶性，推导结合了谱序列、几何比较定理与经典对偶，还验证了乘积相容性并得到相关上同调的泛函分析性质。
$v$-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。
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。
The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more re…
Arithmetic level raising for certain quaternionic unitary Shimura variety
本文证明了在分支情况下四维辛群的算术提升定理，为贝利森-布洛赫-卡托猜想中的某些兰金-塞尔伯格动机提供了关键步骤，同时可视为特殊纤维 Shimura 变种在分支特征下的伊哈拉引理或塔特猜想类比。
We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in \cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain $p$-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in \cite{tap}.
If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.
Introduction to Anderson t-motives: a survey
本文综述 Anderson t-模的基本理论，涵盖格、同调群、张量积、对偶定理、自同态与 L-函数，并引入仿射方程、T-可除模、完整序列等新概念，辅以显式计算实例与若干新结果。
Steenrod operations and symplectic arithmetic duality
本文基于2025年AMS代数几何夏季研讨会报告，介绍Tate提出的有限域上曲面Brauer群辛形式存在猜想，并梳理相关文献的证明思路。
Arithmetic volume of Shtukas and Langlands duality
本文扩展了Feng-Yun-Zhang的工作，将Shukas的算术体积与ζ函数的导数联系起来，允许任意共权重，并首次在L-函数导数的公式中赋予对偶群结构作用。
Class field theory, Hasse principles and Picard-Brauer duality for two-dimensional local rings
本文基于具有完美剩余域的二维局部环的算术对偶，通过分析算术上同调的ind-pro代数群结构并证明其有限性性质，推导出了类域论、Hasse原理及除子类群与Brauer群的对偶。
Weil-étale cohomology and duality for arithmetic schemes in negative weights
本文研究了算术方案在负权下的 Weil-étale 上同调和对偶性，推广了 Flach 和 Morin 的构造并去除了适当的和正则的假设。
We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Akio Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G_p(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G_p(M) may be…
Regulators in the Arithmetic of Function Fields
本文研究函数域算术中Anderson A-动机的调节器概念，证明了A-动机上同调的有限性，并在特定重量条件下显示调节器源和目标的维度相同，但发现调节器像可能不具有满秩，影响了Beilinson猜想的类比。
Duality for arithmetic Dijkgraaf-Witten theory
本文针对算术Dijkgraaf-Witten理论，在含n次单位根的全虚数域上建立类似等价性，通过二次域与四元数群Q_8的例子说明整数环上等价性失效情况，并给出其成立的充分条件。
Prismatic Steenrod operations and arithmetic duality on Brauer groups
本文构建并分析了
We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin's L-functions at negative integers. This generalizes conjectures by Colmez and Gross-Deligne and complements Beilinson's conjectures for the Artin motives. We announce several results in the direction of these statements.
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
On the Integral Part of A-Motivic Cohomology
本文研究了正特征全局域上的A-动机上同调的整数部分，提出模型版本和l-adic版本的定义，并证明模型版本包含l-adic版本，同时探讨了regulated extensions的子模块。
Arithmetic field theory via pro-p duality groups
本文通过pro-p对偶群理论定义适合算术拓扑的cobordism范畴，并利用Frobenius代数分类拓扑量子场论，推导出局部p进域上Galois扩展的计数公式。
Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(Γ_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\barρ_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\barρ_{f}$, which itself follows from the…
We discuss duality pairings on integral étale motivic cohomology groups of regular and proper schemes over algebraically closed fields, local fields, finite fields, and arithmetic schemes.
We improve the arithmetic duality formalism of the rational etale site. This improvement allows us to avoid some exotic approximation arguments on local fields with ind-rational base, thus simplifying the proofs of the previously established duality theorems in the rational etale site and making the formalism more user-friendly. In a subsequent paper, this new formulation will be used in a crucial way to study duality for two-dimensional local rings.
Arithmetic duality for finite Galois modules over two-dimensional local fields of mixed characteristic
研究更高局部域的对偶性，结合Suzuki和Kato的结果，在更弱条件下获得p- torsion对偶性陈述，涉及局部紧致Hausdorff拓扑下的étale上同调群。
Duality between prime factors and the Prime Number Theorem for Arithmetic Progressions -- II
本文利用涉及 Moebius 函数与不同素因子个数 ω(n) 的次大、最小素因子深层对偶恒等式，由算术级数素数定理推出相关加权和为零，并给出定量版本。
Finite Fourier Duality, Radial Values, and Arithmetic of a Mod Eight False Theta Quotient
研究模八假 theta 矩商式的径向行为，建立有限傅里叶闭包，证明周期狄利克雷值非零，并给出系数整除性、二进插值及欧拉变换正性等算术结果。
We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson's conjecture, to the adjoint $L$-function at $s=1$. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion.
Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we in…
Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities
本文研究了素因子与算术级数素数定理之间的对偶性，通过高阶对偶性揭示了μ(n)和ω(n)的性质，推导出新的求和公式和密度型定理。
Computations of Cohomology of Arithmetic Groups, Part 1
本文提出一种更简单、快速且结构化的方法，计算算术群$GL_4({\mathbb Z})$等在不同最高权表示系数下的边界与Eisenstein上同调，并提出了类似Serre对偶的猜想，经30个表示族验证。
K-theoretic Poitou-Tate duality in higher dimensions: proper case
该文将K-理论Poitou-Tate对偶推广至任意维数、在S-整数上光滑正常的算术概形，并利用局部紧模的K-理论处理紧支集侧。
We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.
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