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Register a new userCohomological relation between Jacobi forms and skew-holomorphic Jacobi forms
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Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between holomorphic Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on $Gamma^J$ and the space of skew-holomorphic Jacobi cusp forms on $Gamma^J$ with the same half-integral weight to the Eichler cohomology group of $Gamma^J$ with a coefficient module coming from polynomials.
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We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi--Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skew-holomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.
We show that Hidas families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-ad
Families of quasimodular forms arise naturally in many situations such as curve counting on Abelian surfaces and counting ramified covers of orbifolds. In many cases the family of quasimodular forms naturally arises as the coefficients of a Taylor expansion of a Jacobi form. In this note we give examples of such expansions that arise in the study of partition statistics. The crank partition statistic has gathered much interest recently. For instance, Atkin and Garvan showed that the generating functions for the moments of the crank statistic are quasimodular forms. The two variable generating function for the crank partition statistic is a Jacobi form. Exploiting the structure inherent in the Jacobi theta function we construct explicit expressions for the functions of Atkin and Garvan. Furthermore, this perspective opens the door for further investigation including a study of the moments in arithmetic progressions. We conduct a thorough study of the crank statistic restricted to a residue class modulo 2.
Let $Gamma$ be a finitely generated Fuchsian group of the first kind which has at least one parabolic class. Eichler introduced a cohomology theory for Fuchsian groups, called as Eichler cohomology theory, and established the $CC$-linear isomorphism from the direct sum of two spaces of cusp forms on $Gamma$ with the same integral weight to the Eichler cohomology group of $Gamma$. After the results of Eichler, the Eichler cohomology theory was generalized in various ways. For example, these results were generalized by Knopp to the cases with arbitrary real weights. In this paper, we extend the Eichler cohomology theory to the context of Jacobi forms. We define the cohomology groups of Jacobi groups which are analogues of Eichler cohomology groups and prove an Eichler cohomology theorem for Jacobi forms of arbitrary real weights. Furthermore, we prove that every cocycle is parabolic and that for some special cases we have an isomorphism between the cohomology group and the space of Jacobi forms in terms of the critical values of partial $L$-functions of Jacobi cusp forms.
Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted to meromorphic Siegel paramodular forms. It was recently observed that the Fourier coefficients of such lifts are then either fast growing or slow growing. In this note we investigate the space of weak Jacobi forms that lead to slow growth. We provide analytic and numerical evidence for the conjecture that there are such slow growing forms for any index $m$.
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