Shimura Integrals of Cusp Forms

Šokurov, V V

doi:10.1070/im1981v016n03abeh001322

Cited by 27 publications

(23 citation statements)

References 10 publications

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“…In this way, modular symbols of higher weight can be interpreted as rational homology classes of middle dimension of Kuga-Sato varieties: cf. [Sh1]- [Sh3].…”

Section: Modular Symbols and Periods Of Modular Formsmentioning

confidence: 99%

“…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%

“…This paper was inspired by two sources: theory of multiple zeta values on the one hand (see [Za2]), and theory of modular symbols and periods of cusp forms, on the other ( [Ma1], [Ma2], [Sh1]- [Sh3], [Me]). Roughly speaking, it extends the theory of periods of modular forms replacing integration along geodesics in the upper complex half-plane by iterated integration.…”

Section: §0 Introduction and Summarymentioning

confidence: 99%

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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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“…In this way, modular symbols of higher weight can be interpreted as rational homology classes of middle dimension of Kuga-Sato varieties: cf. [Sh1]- [Sh3].…”

Section: Modular Symbols and Periods Of Modular Formsmentioning

confidence: 99%

Section: Theorem (I) Ifmentioning

confidence: 99%

Section: §0 Introduction and Summarymentioning

confidence: 99%

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Iterated integrals of modular forms and noncommutative modular symbols

Manin

Algebraic Geometry and Number Theory

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“…ad -bc = n, d -1 -c--0 mod N). For every n > 1 we define Hecke [9] and [17) proved (see [7]) in the more general setting of any subgroup r C SL2 (Z) of finite index (instead of rl (N)) and the setting of correspondences defined as double cosets (instead of the concrete Hecke or diamond operators).…”

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confidence: 99%

A combinatorial interpretation of Serre's conjecture on modular Galois representations

Herremans¹

2003

Annales De L’institut Fourier

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“…This includes an algebraic proof of the presentation of modular symbols in terms of Manin symbols. Thus, we avoid the use of the technically rather difficult paper [13], which is present in all the published treatments known to the author (e.g. [11], [14]).…”

Section: Introductionmentioning

confidence: 99%

On Modular Symbols and the Cohomology of Hecke Triangle Surfaces

Wiese

2009

Int. J. Number Theory

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The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalized from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers.MSC Classification: 11F67 (primary), 11F75, 11Y40, 20H10 (secondary).

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Shimura Integrals of Cusp Forms

Cited by 27 publications

References 10 publications

Iterated integrals of modular forms and noncommutative modular symbols

Iterated integrals of modular forms and noncommutative modular symbols

A combinatorial interpretation of Serre's conjecture on modular Galois representations

On Modular Symbols and the Cohomology of Hecke Triangle Surfaces

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