Michael Dettweiler, Stefan Reiter
We classify rigid local systems of rank 7 whose monodromy group is dense in the simple algebraic group of type G2. This leads to motives with Galois group G2.
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We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.
In this paper, we prove that for suitably chosen Dwork motives, the local Galois representation arising from middle cohomology of fibers over a point with $p$-adic valuation $<0$ on the base is regular and ordinary. The result will be crucial in the forthcoming work Potential Automorphy for $GL_n$. The proof consists of the construction of a semistable blowup and a use of Hyodo-Kato's log crystalline cohomology theory.
In this mostly expository note we explain how Nori's theory of motives achieves the aim of establishing a Galois theory of periods, at least under the period conjecture. We explain and compare different notions periods, different versions of the period conjecture and view the evidence by explaining the examples of Artin motive, mixed Tate motives and 1-motives.
From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $π$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number theory? In which role? The aim of this text is to indicate what Grothendieck's theory of motives has to s…
A theorem of Grothendieck tells us that if the Galois action on the Tate module of an abelian variety factors through a smaller field, then the abelian variety, up to isogeny and finite extension of the base, is itself defined over the smaller field. Inspired by this, we give a Galois descent datum for a motive $H$ over a field by asking that the Galois action on an $\ell$-adic realisation factor through a smaller field. We conjecture that this descent datum is effective, that is if a motive $H$ satisfies the above criterion, then it must itself descend to the smaller field. We prove this conj…
Dimension of the motivic Galois group of a 1-motive
本文计算了复数域上1-动机M的motivic Galois群的维度，以乘法群的秩明确表达，并应用此结果重新表述了Grothendieck-André周期猜想。
Let T be a Tannakian category over a field k of characteristic 0 and π(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of π(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by in…
We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we generalize a well-known theorem of Nakajima on the Galois module structure of certain spaces of Kahler differentials associated to the top curve of the cover, assuming that the field of definition is of characteristic p. If the cover and the Picard 1-motive are defined over a finite field and if G is abelian, we compute the first Fitting ideal of each l-adic rea…
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.
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.
We prove an effective Hilbert Irreducibility result for residual realizations of a family of motives with motivic Galois group G2.
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.
The motivic Galois group for a double zeta value
本文研究了双重ζ值的最小动机及其动机伽罗华群，探讨了周期猜想与代数关系之间的联系。
Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.
A Weight Structure on Rigid Analytic Motives over a Field
本文通过伽罗瓦下降在局部域上刚性解析母题的稳定 ∞-范畴上构造幺半权结构，推广了 Binda-Gallauer-Vezzani 的好约化情形，并证明 Hyodo-Kato 实现经由权复形函子分解，从而得到收敛于 Hyodo-Kato 上同调的谱序列与权滤过。
Rigid analytic 1-motives and conjugate uniformization of abeloid varieties
本文针对p进域上的阿贝尔簇，研究刚性解析1-动机理论及阿贝尔簇的共轭单值化，建立了形式与刚性解析1-动机的等价性等结果，还构造了相关p可除群。
On the behavior of stringy motives under Galois quasi-étale covers
本文研究 Galois 拟平展覆盖下弦性动机的下降行为，证明其下降并在分歧时严格，进而将 KLT 奇点基本群有限性归结为 DCC 性质，并给出曲面及对数终端奇点的无特征证明。
This paper uses rigid Hecke eigensheaves, building on Yun's work on the construction of motives with exceptional Galois groups, to produce the first robust examples of `generalized Kuga-Satake theory' outside the Tannakian category of motives generated by abelian varieties. To strengthen our description of the `motivic' nature of Kuga-Satake lifts, we digress to establish a result that should be of independent interest: for any quasi-projective variety over a (finitely-generated) characteristic zero field, the associated weight-graded of its intersection cohomology arises from a motivated moti…
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.
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。
Rigidity for rigid analytic motives
本文证明了非阿基米德域上刚性解析簇的动机刚性定理，涵盖有/无转移情形，并应用于平展实现、动机比较及倾斜等价的升级。
In this article we further the study of noncommutative numerical motives. By exploring the change-of-coefficients mechanism, we start by improving some of our previous main results. Then, making use of the notion of Schur-finiteness, we prove that the category NNum(k)_F of noncommutative numerical motives is (neutral) super-Tannakian. As in the commutative world, NNum(k)_F is not Tannakian. In order to solve this problem we promote periodic cyclic homology to a well-defined symmetric monoidal functor HP on the category of noncommutative Chow motives. This allows us to introduce the correct non…
We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group of the multiplicative group - {N-th roots of unity} and relate it to geometry of modular varieties. In particular we get new information about the actionof the Galois group on the pro-l completion of the above fundamental group. This paper is the second part of "Multiple polylogarithms and mixed Tate motives" math.AG/0103059
Let $\mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $\mathbf{1}$, and equipped with a filtration $W_\cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $\mathbf{T}$, and $\underline{\mathfrak{u}}(M)\subset W_{-1}\underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $\underline{\mathfrak{u}}(M)$ in terms of the extensions $0\longrightarrow W_pM \longrightarrow M \longrightarr…
The six-functor formalism for rigid analytic motives
本文系统研究一般刚性解析空间上的刚性解析母题，建立其六函子形式体系，并通过归结为代数母题证明扩展真基变换定理，同时借助雷诺方法去除了诺特性质假设。
In the study of the arithmetic structure of elliptic modular groups which are the fundamental groups of compactified modular curves, these truncated group algebras and their direct sums are considered to construct elliptic modular motives. Our main result is a new theory of Hecke operators on these motives which gives a congruence relation to the Galois action, and their motivic decomposition. Using our Hecke theory, we show that elliptic modular motives are the direct sums of pure motives over certain number fields. This fact implies a kind of algebraicity on iterated Shimura integrals, i.e.,…
We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful.
In their paper Livné and Yui (math.AG/0304497) discuss several examples of non-rigid Calabi-Yau varieties which admit semi-stable K3-fibrations with 6 singular fibres over a base which is a rational modular curve. They also establish the modularity of the L-function of these examples. The purpose of this note is to point out that the examples which were listed in their paper, but which do not lead to semi-stable fibrations, are still modular in the sense that their L-function is associated to modular forms. We treat the case associated to the group Gamma_1(7) in detail, but our technique also …
Let $k$ be a field of characteristic zero containing all roots of unity and $K=k((t))$. We build a ring morphism from the Grothendieck group of semi-algebraic sets over $K$ to the Grothendieck group of motives of rigid analytic varieties over $K$. It extend the morphism sending the class of an algebraic variety over $K$ to its cohomological motive with compact support. We show that it fits inside a commutative diagram involving Hrushovski and Kazhdan's motivic integration and Ayoub's equivalence between motives of rigid analytic varieties over $K$ and quasi-unipotent motives over $k$ ; we also…
In this article we introduce the notion of a quasi-compatible system of Galois representations. The quasi-compatibility condition is a slight relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is the following: Let $M$ be an abelian motive, in the sense of Yves André. Then the $\ell$-adic realisations of $M$ form a quasi-compatible system of Galois representations. (In fact, we actually prove something stronger. See theorem 5.1.) As an application, we deduce that the absolute rank of the $\ell$-adic monodromy groups of $M$ does not depend …
Let M be a 1-motive defined over a field of characteristic 0. To M we can associate its motivic Galois group, G_mot(M), which is the geometrical interpretation of the Munford-Tate group of M. We prove that the unipotent radical of the Lie algebra of G_mot(M) is the semi-abelian variety defined by the adjoint action of the semi-simplified of the Lie algebra of G_mot(M) on itself.
We prove that the functor associating to a rigid analytic variety the singular complex of the underlying Berkovich topological space is motivic, and defines the maximal Artin quotient of a motive. We use this to generalize Berkovich's results on the weight-zero part of the étale cohomology of a variety defined over a non-archimedean valued field.
In this article we introduce the categories of noncommutative (mixed) Artin motives. In the pure world, we start by proving that the classical category AM(k) of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridge between pure motives and noncommutative pure motives we then show that the image of this full embedding, which we call the category NAM(k) of noncommutative Artin motives, is invariant under the different equivalence relations and modification of the sym…
Motivic local systems over a curve in finite characteristic form a countable set endowed with an action of the absolute Galois group of rational numbers commuting with the Frobenius map. I will discuss three series of conjectures about such sets, based on an analogy with algebraic dynamics, on a formalism of commutative algebras of motivic integral operators, and on an analogy with 2-dimensional lattice models.
In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level $Γ_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Grenié's results to compare thr…
Galois descent for motivic theories
本文给出了纤维化加性范畴成为stack的条件，适用于多种motive理论，并简化证明了motivic Galois群序列的正合性及Chow-Lefschetz motives的Galois群计算。
The modular properties of some higher dimensional varieties of special Fano type are analyzed by computing the L-function of their $Ω-$motives. It is shown that the emerging modular forms are string theoretic in origin, derived from the characters of the underlying rational conformal field theory. The definition of the class of Fano varieties of special type is motivated by the goal to find candidates for a geometric realization of the mirrors of rigid Calabi-Yau varieties. We consider explicitly the cubic sevenfold and the quartic fivefold, and show that their motivic L-functions agree with t…
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。
The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations
本文研究了特征二中 Dwork family 的剩余单值性表示及其在 2-adic 和模 2 Galois 表示中的应用，并结合已有成果证明了源自 Dwork 五次超曲面族的秩 4 辛动机在特定条件下的自守性。
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}.
Motivic p-adic periods of 1-motives
本文为1-动机显式构造了p-adic de Rham比较同构，并利用刚性解析动机构造了新的动机p-adic周期环，通过实例计算了周期配对。
Berkovich Motives
本文提出了一种统一处理 archimedean 和 nonarchimedean 设定的 Berkovich 动机理论，通过自包含的方法，实现了与 Voevodsky 理论的关联，并在基假设下保证了稳定 ∞-范畴的 rigid dualizable 性质。
We show that the André motive of a hyper-Kähler variety $X$ over a field $K \subset \mathbb{C}$ with $b_2(X)>6$ is governed by its component in degree $2$. More precisely, we prove that if $X_1$ and $X_2$ are deformation equivalent hyper-Kähler varieties with $b_2(X_i)>6$ and if there exists a Hodge isometry $f\colon H^2(X_1,\mathbb{Q})\to H^2(X_2,\mathbb{Q})$, then the André motives of $X_1$ and $X_2$ are isomorphic after a finite extension of $K$, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale coho…
Motivic Galois theory for one-loop Feynman integrals in momentum space
本文提出了一种动机构建一阶费曼积分在动量空间中的理论，通过关联通用动量学的动机局部系统来处理带有切口的图，并计算了这些系统的重量分量，这些分量是与最大切口商图相关的二次Artin动机的塔扭曲。
Artin motives in relative Nori and Voevodsky motives
本文证明特征零域上有限型概形的相对 Artin 动机在 Nori 与 Voevodsky 范畴中等价，并应用于对偶性及动机伽罗瓦群的正合序列类比。
Dwork Motives, Monodromy and Potential Automorphy
本文研究了作为Dwork族上同调直和项的motive族，通过计算找到几何monodromy稠密、实现多种unipotent算子且Hodge数不超过1的族，并借此构造potential automorphic表示及给出Matsumoto定理的新证明。
Non-abelian Rees construction and pure motives
本文基于非阿贝尔雷斯构造，建立准齐性空间与单调范畴之间的伽罗瓦对应关系，并应用于具有具体应用的动机单调范畴，推导了关于有限域上阿贝尔簇数值等价的新的证明与推广。
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 …
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…
This note explains an approach to producing examples of 'generalized Kuga-Satake theory' based on establishing special cases of Simpson's conjecture that rigid local systems are motivic. This strategy is then carried out, using work of Bogner and Reiter, in a low-dimensional example, producing the first non-trivial examples of generalized Kuga-Satake theory outside of the Tannakian category of motives generated by abelian varieties. For a more robust collection of examples, see the sequel "Generalized Kuga-Satake theory and rigid local systems, II: rigid Hecke eigensheaves."
We use a certain rigid local system in order to prove the potential automorphy of certain Galois representations with values in $G_2,$ found by N. Katz and the author.
In characteristic 0 there are essentially two approaches to the conjectural theory of mixed motives, one due to Nori and the other one due to, independently, Hanamura, Levine, and Voevodsky. Although these approaches are apriori quite different it is expected that ultimately they can be reduced to one another. In this article we provide some evidence for this belief by proving that their associated motivic Galois groups are canonically isomorphic.
We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.
We prove new instances of Simpson's rigidity conjecture which states that quasi-unipotent rigid local systems should be motivic. We construct new relative motives over the fourfold punctured Riemann sphere which give rise to $G_2$-rigid local systems which are not rigid in the group $\GL_7.$
We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of $L$-functions. The analogue in characteristic zero of the action of the Frobenius on l-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative …
The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a method due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.
Fix a base field F, a finite field K and consider a sequence of central simple F-algebras A_1,...,A_n. In this note we provide some results toward a classification of the indecomposable motives lying in the motivic decompositions of projective homogeneous varieties under the action of PGL(A_1)x...xPGL(A_n) with coefficients in K. We give a complete classification of those motives if n=1 and derive from it the motivic dichotomy of projective linear groups. We then provide several classification results as well as counterexamples for arbitrary n showing that the situation is less rigid. These re…
We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when $G$ …
We explain how to construct a cohomology theory on the category of separated quasi-compact smooth rigid spaces over $\mathbf{C}_p$ (or more general base fields), taking values in the category of vector bundles on the Fargues-Fontaine curve, which extends (in a suitable sense) Hyodo-Kato cohomology when the rigid space has a semi-stable proper formal model over the ring of integers of a finite extension of $\mathbf{Q}_p$. This cohomology theory factors through the category of rigid analytic motives of Ayoub.
We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O'Grady-10 situation. In the non-commutative setting, similar results are proved for …
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