Hodge-Tate Research Articles

Finiteness for crystalline representations of the absolute Galois group of a totally real field

Let K be a totally real field and GK:=Gal(K‾/K) its absolute Galois group, where K‾ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of Qℓ. Let S be a finite set of finite places of K not dividing ℓ. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if ℓ is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations r:GK→GLn(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|GK′≃⊕ri′ for some finite totally real field extension K′ of K unramified at all places of K over ℓ, where each representation ri′ over E is an 1-dimensional representation of GK′ or a totally odd irreducible 2-dimensional representation of GK′ with distinct Hodge-Tate numbers.

The Galois group of the category of mixed Hodge–Tate structures

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q. Since the category has homological dimension 1, the Hopf algebra H is isomorphic to the commutative graded Hopf algebra provided by the tensor algebra of the graded vector space given by the direct sum of the groups C/Q(n) over n>0. However this isomorphism is not natural in any sense, e.g. does not work in families. We give a natural explicit construction of the Hopf algebra H. Generalizing this, we define a Hopf dg-algebra related to any dg-algebra R over a field k, equipped with an invertible line k(1). When R is the sheaf of algebras given by the holomorphic de Rham complex of a complex manifold X, and the line is Q(1), the related Hopf dg-algebra describes a dg-model of the derived category of variations of Hodge-Tate structures on X. Its cobar complex provides a dg-model for the rational Deligne cohomology of X. We consider a variant of our construction which starts from Fontaine's crystalline / semi-stable period rings and produces graded / dg Hopf algebras, which we relate to the p-adic Hodge theory.

Potentially semi-stable deformations of specified Hodge–Tate type and Galois type

Let k be a perfect field of characteristic p>2, and let K be a finite totally ramified extension of W(k)[1p]. We prove that the locus of potentially semi-stable Gal(K¯/K)-representations of a given Hodge–Tate type and Galois type is a closed subspace of the universal deformation ring, generalizing the result of Kisin (2007) [Kis07] where k is assumed to be finite.

The category of flat Hodge–Tate structures

The category of flat Hodge–Tate structures

A simple construction of Grassmannian polylogarithms

The classical n-logarithm is a multivalued analytic function defined inductively: Lin(z):=∫0zLin−1(t)dlogt,Li1(z)=−log(1−z). In this paper we give a simple explicit construction of the Grassmannian n-logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of n-dimensional subspaces in C2n in generic position to the coordinate hyperplanes by the natural action of the torus (C∗)2n. The classical n-logarithm appears at a certain one dimensional boundary stratum.We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms dlogfi where fi are rational functions. We give a simple explicit formula for the Tate iterated integral which describes the Grassmannian n-logarithm.Another example is the Tate iterated integrals for the multiple polylogarithms on the moduli spaces M0,n, calculated in Section 4.4 of Goncharov (2005) [13] using the combinatorics of plane trivalent trees decorated by the arguments of the multiple polylogarithms.Variations of mixed Hodge–Tate structures on X are described by a Hopf algebra H•(X). We upgrade Tate iterated integrals on a (rational) complex variety X to elements of H•(X). The coproducts of these elements are very interesting invariants of the iterated integrals. In general their calculation is a non-trivial problem. We show however, that working modulo the ideal of H•(X) generated by constant variations, there is a simple way to calculate the coproduct.It is a pleasure to dedicate this paper to Andrey Suslin, whose works Suslin (1984) [21] and Suslin (1991) [22] played an essential role in the development of the story.

Closed form algebra on a disk is Koszul

We prove that the algebra of closed differential forms on an (algebraic, formal, or analytic) disk with logarithmic singularities along several coordinate hyperplanes is Koszul (both nontopologically and topologically). A relation to variations of mixed Hodge-Tate structures is discussed in the introduction.

Алгебра замкнутых форм на диске кошулева

We prove that the algebra of closed differential forms in an (algebraic, formal, or analytic) disk with logarithmic singularities along several coordinate hyperplanes is (both nontopologically and topologically) Koszul. The connection with variations of mixed Hodge-Tate structures, based on a preprint by Andrey Levin, is discussed in the introduction.

Hodge-Tate and de Rham representations in the imperfect residue field case

Soit K un corps local p-adique de corps residuel k tel que [k: k P ] = p e < +∞ et soit V une representation p-adique de Gal(K/K). Nous utilisons la theorie des modules differentiels p-adiques pour montrer que V est une representation de Hodge-Tate (resp. de Rham) de Gal(K/K) si et seulement si V est une representation de Hodge-Tate (resp. de Rham) de Gal(k pf /k pf ) ou K pf /K est un certain corps local p-adique de corps residuel le plus petit corps parfait k pf contenant k.

On reductions of families of crystalline Galois representations

Let K_f be the finite unramified extension of {Q}_p of degree f and E any finite large enough coefficient field containing K_f. We construct analytic families of étale (\varphi ,\Gamma ) -modules which give rise to families of crystalline E -representations of the absolute Galois group G_{K_f} of K_f. For any irreducible effective two-dimensional crystalline E -representation of G_{K_f} with labeled Hodge-Tate weights {0,-k_i}_{\tau _i} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E -representations of G_{K_f} of the same Hodge-Tate type which contains it. As an application, we compute the semisimplified mod p reductions of the members of each such family.

An update on semisimple quantum cohomology and F-manifolds

To the memory of V.A. Iskovskikh Abstract—In the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative OM -bilinear multiplication on its tangent sheaf TM is an F -manifold in the sense of Hertling- Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle T ∗ M , is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.

Variations of Mixed Hodge Structures of Multiple Polylogarithms

AbstractIt is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

A remark on potentially semi-stable representations of Hodge-Tate type (0,1)

In this note we complement a part of a theorem of Fontaine-Mazur. We show that if $(V,\rho)$ is an irreducible finite dimensional representation of the Galois group $Gal({\bar K}/K)$ of a finite extension of $K\Q_p$, of Hodge-Tate type $(0,1)$ then it is potentially semi-stable if and only if it is potentially crystalline. This was proved by Fontaine-Mazur for dimension two and $p\geq 5$ by their classfication theorem.

The Gauß–Manin connection on the Hodge–Tate structures

For each simplicial complex scheme X • the singular cohomology H ∗(X •) carries a canonical mixed Hodge structure. There exists a canonical homomorphism ∇ :H ∗(X •)→Ω 1 C/ Q ⊗H ∗(X •) , the Gauß–Manin connection. We show that there is a unique functorial connection on each mixed Hodge–Tate structure having certain properties of the Gauß–Manin connection. This connection is non-integrable in general, and therefore, its integrability is a non-trivial condition on the Hodge structure to be geometric.

Volumes of hyperbolic manifolds and mixed Tate motives

Two different constructions of an invariant of an odd-dimensional hyperbolic manifold with values in K 2 n − 1 ( Q ¯ ) ⊗ Q K_{2n-1}(\overline {\mathbb Q})\otimes \mathbb Q are given. We prove that the volume of the manifold equals the value of the Borel regulator on this invariant. The scissors congruence groups in noneuclidean geometries are studied and related to mixed Tate motives and algebraic K-theory of C \mathbb C . We contribute to the general theory of mixed Hodge structures by introducing for Hodge-Tate structures the big period map with values in C ⊗ C ∗ ( n − 2 ) \mathbb C \otimes \mathbb C^*(n-2) .

The sign of the functional equation of the L-function of an orthogonal motive

We prove that the sign of the functional equation of the L-function of an orthogonal motive of even weight is always positive, if we admit the conjecture of the product formula for the epsilon factor of Deligne [D3] (5.2). More precisely, we define the sign independently of any conjecture by using the product formula as the definition and by working with one chosen "good" prime f and we prove its positivity. The positivity has been established for the Artin motives [Fr-Q][D2], for the symmetric square of a modular elliptic curve [C-Sc] and in positive characteristic by Serre [Fr-Q]. As in the proof in [D2], we consider the second Stiefel-Whitney class in zBr(K) of an orthogonal representation of the absolute Galois group G~: of an algebraic number field K. The positivity of the sign is then a consequence of the reciprocity law of the square residue symbol. Fixing a "good" prime f, we work with an f-adic representation and compare the local epsilon factor and the local Stiefel-Whitney class. For a prime vXr it is a theorem of Deligne [D2]. For a prime vie , we compute the Stiefel-Whitney class assuming the representation is cristalline (Theorem 2) using a generalization of a theorem of Fr6hlich [Fr] to f-adic representation of Hodge-Tate type (Theorem 1). By combining them with an elementary computation for the Hodge structures at r iot , we conclude that the sign is positive (Theorem 3). The crucial step in the proof of Theorem 2, given in section 1, is the determination by Fontaine and Lafaille [Fo-L] of the action of the tame inertia on the semisimplification of reduction modulo p of a cristalline representation. The theory of orthogonal representations of Galois group including a proof of Fr6hlich's theorem is reviewed in a preliminary section 0.

On variation of Hodge-Tate structures

On variation of Hodge-Tate structures

Hodge-tate structures and modular forms

Hodge-tate structures and modular forms