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1.
In this paper we express the multiple Hecke L-function in terms of a linear combination of iterated period integrals associated with elliptic cusp forms, which is introduced by Manin around 2004. This expression generalizes the classical formula of Hecke L-function obtained by the Mellin transformation of a cusp form. Also the expression gives a way of the analytic continuation of the multiple Hecke L-function.  相似文献   
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This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et?al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.  相似文献   
4.
The effects of the surface polarity of a glass substrate on the orientation of nematic liquid crystals (LCs) were studied using the polarised optical microscope and Fourier-transform infrared spectroscopy. On the surface of oxygen plasma treated glass, a homeotropic alignment of LCs was induced for LCs with negative dielectric anisotropy. This suggests that vertical orientation of LCs could be induced on a polar glass substrate without using an LC alignment layer. Upon cooling towards the isotropic–nematic transition, E7 with positive dielectric anisotropy changes its LC arrangement to isotropic, homeotropic, planar orientations in order. The nematic LC anchoring transition of E7 was interpreted by considering the competition between van der Waals forces and dipole interactions that control the alignment of LC molecules on a polar glass surface.  相似文献   
5.
Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.  相似文献   
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We determine a class of functions spanned by theta series of higher degree. We give two applications: A simple proof of the inversion formula of such theta series and a classification of skew-holomorphic Jacobi forms.

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Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.  相似文献   
8.
We investigate differential operators and their compatibility with subgroups of SL2n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.  相似文献   
9.
There is a known correspondence among modular forms, Jacobi forms and Siegel modular forms of genus 2. In this paper we show this correspondence can be extended to non-holomorphic Eisenstein series, in particular, among , E2,1(τ,z;δ;0), and .  相似文献   
10.
In this paper, we find a connection between the weight enumerator of self-dual codes and half-integral weight modular forms. We generalize in that way results of Broué-Enguehard, Hirzebruch, Ozeki, Rains-Sloane, Runge.

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