The Study of the Homology of Kuga Varieties

Šokurov, V V

doi:10.1070/im1981v016n02abeh001314

Cited by 13 publications

(8 citation statements)

References 15 publications

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“…When the theory of modular symbols for the SL(2)-case had been conceived in the 70's (cf. [Ma1], [Ma2], [Sh1], [Sh2]), it was clear from the outset that it dealt with the Betti homology of some basic moduli spaces (modular curves, Kuga varieties, M 1,n , and alike), whereas the theory of modular forms involved the de Rham and Hodge cohomology of the same spaces.…”

Section: Introductionmentioning

confidence: 99%

Modular shadows and the Lévy—Mellin ∞—adic transform

Manin¹,

Marcolli²

2008

Modular Forms on Schiermonnikoog

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This paper continues the study of the structures induced on the "invisible boundary" of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo-measures generalizing the wellknown theory of modular symbols for SL(2). These pseudo-measures, and the related integral formula which we call the Lévy-Mellin transform, can be considered as an "∞-adic" version of Mazur's p-adic measures that have been introduced in the seventies in the theory of p-adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated Lévy-Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non-commutative geometry. Definition.A pseudo-measure on P 1 (R) with values in a commutative group (written additively) W is a function µ : P 1 (Q) × P 1 (Q) → W satisfying the following conditions: for any α, β, γ ∈ P 1 (Q), µ(α, α) = 0, µ(α, β) + µ(β, α) = 0 , (1.1) µ(α, β) + µ(β, γ) + µ(γ, α) = 0 .( 1.2)

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Section: Introductionmentioning

confidence: 99%

Modular shadows and the Lévy—Mellin ∞—adic transform

Manin¹,

Marcolli²

2008

Modular Forms on Schiermonnikoog

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“…Kuga fiber varieties are closely linked to the theory of automorphic forms (see e.g. [6], [7], [9], [10], [14], [15]). If x is an element of a symmetric space M, there is an isometry S x of M called a symmetry at x such that x is an isolated fixed point of S x and S 2 x = 1.…”

Section: Introductionmentioning

confidence: 99%

Rational equivariant holomorphic maps of symmetric domains

Lee

2002

Archiv der Mathematik

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An equivariant holomorphic map of symmetric domains associated to a homomorphism of semisimple algebraic groups defined over Q is rational if it carries a point belonging to a set determined by an arithmetic subgroup to a point in a similar set. We prove that an equivariant holomorphic map of symmetric domains is rational if the associated Kuga fiber variety does not have a nontrivial deformation. Introduction.In this paper we study the relation between the rationality of equivariant holomorphic maps of symmetric domains and the rigidity of Kuga fiber varieties which are families of polarized abelian varieties parametrized by a locally symmetric arithmetic variety.Let G be a semisimple algebraic group defined over Q that does not contain factors defined over Q and compact over R, and let G = G(R) be the associated real semisimple Lie group. Let K be a maximal compact subgroup of G, and assume that the corresponding symmetric space D = G/K has a G-invariant complex structure. Let V be a vector space over Q of dimension 2m, and let β be a nondegenerate alternating bilinear form on V . We set G = Sp(V, β) and denote by D the symmetric domain associated to G = G (R). Let ρ : G → G be a homomorphism of Lie groups induced by a morphism ρ : G → G of algebraic groups defined over Q. We define the space D ρ to be the set of all holomorphic maps τ : D → D satisfying τ(gz) = ρ(g)τ(z) for all g ∈ G and z ∈ D.Let L be a lattice in V R , and let Γ be a torsion-free subgroup of Sp(L, β) of finite index. We set

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“…A detailed analysis of singularities performed in [Sh2], [Sh3] shows that the map f → ω induces an isomorphism of the space of cusp forms of weight 2r with the space of holomorphic volume forms on an appropriate smooth projective KugaSato variety. (As I have already remarked in the Introduction, it would be useful to replace it by the base extension (M 1,2r−2 ) X Γ .…”

Section: Theorem (I) Ifmentioning

confidence: 99%

Iterated integrals of modular forms and noncommutative modular symbols

Manin

Algebraic Geometry and Number Theory

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Abstract. The main goal of this paper is to study properties of the iterated integrals of modular forms in the upper halfplane, eventually multiplied by z s−1 , along geodesics connecting two cusps. This setting generalizes simultaneously the theory of modular symbols and that of multiple zeta values. §0. Introduction and summary where the sequence of differential forms in the iterated integral consists of consecutive subsequences of the form dz z , . . . , dz z , dz 1 − z of lengths m k , m k−1 , . . . , m 1 . Easy combinatorial considerations allow one to express in two different ways products ζ(l 1 , . . . , l j ) · ζ(m 1 , . . . , m k ) as linear combinations of multiple zeta values.If one uses for this the integral representation (0.2), one gets a sum over shuffles which enumerate the simplices of highest dimension occurring in the natural simplicial decomposition of the product of two integration simplices. In fact, the formal generating series for (regularized) iterated integrals (0.2) appeared in the famous Drinfeld paper [Dr2], essentially as the Drinfeld associator, and more relations for multiple zeta values were implicitly deduced there. The question about interdependence of (double) shuffle and associator relations does not seem to be settled at the moment of writing this: cf.[Ra3]. The problem of completeness of these systems of relations is equivalent to some difficult transcendence questions. Multiple zeta values are interesting, because they and their generalizations appear in many different contexts involving mixed Tate motives ([DeGo], [T]), deformation quantization ([Kon]), knot invariants etc.0.2. Modular symbols and periods of modular forms. Let Γ be a congruence subgroup of the modular group acting upon the union H of the upper complex half-plane H and the set of cusps P 1 (Q).The quotient Γ \ H is the modular curve X Γ . Differentials of the first kind on X Γ lift to the cusp forms of weight 2 on H (multiplied by dz).The modular symbols {α, β} Γ ∈ H 1 (X Γ , Q), where α, β ∈ P 1 (Q), were introduced in [Ma1] as linear functionals on the space of differentials of the first kind obtained by lifting and integrating. The fact that one lands in H 1 (X Γ , Q) and not just H 1 (X Γ , R) is not obvious. It was proved in [Dr1] by refining a weaker argument given in [Ma1]. This is equivalent to the statement that difference of any two cusps in Γ has finite order in the Jacobian, or else that the mixed Hodge structure on H 1 (X Γ \ {cusps}, Q) is split (cf. [El]). One of the basic new insights of [Ma1] consisted in the realization that studying the action of Hecke operators on modular symbols one gets new arithmetic facts about periods and Fourier coefficients of cusp forms of weight two.The further generalizations of modular symbols proceeded, in particular, in the following directions. 3(a) In [Ma2] it was demonstrated that the same technique applies to the integrals of cusp forms of higher weight, eventually multiplied by polynomials in z, producing the similar information about their periods and...

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The Study of the Homology of Kuga Varieties

Cited by 13 publications

References 15 publications

Modular shadows and the Lévy—Mellin ∞—adic transform

Modular shadows and the Lévy—Mellin ∞—adic transform

Rational equivariant holomorphic maps of symmetric domains

Iterated integrals of modular forms and noncommutative modular symbols

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