Jacobi 적분에 관하여
- Jacobi 적분에 관하여
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- In his last letter to Hardy, Ramanujan defined mock theta functions and gave $17$ examples. Zwegers found that mock theta functions are essentially holomorphic parts of nonholomorphic modular forms. Zwegers' work is based on computations of Jacobi integrals and their period functions such as Lerch sums, real analytic $R$-functions and Mordell integrals. Although there was no definition of Jacobi integrals, Jacobi integrals are used by many researchers. Study of higher-level Appell functions and real analytic Jacobi Eisenstein series are related with Jacobi integrals.
This work is motivated by the theory of Eichler integrals. The concept of Eichler integrals was introduced by Eichler and more general automorphic integrals were studied further by many researchers. Period polynomials or period functions associated with automorphic integrals satisfy two functional equations and are explained in terms of parabolic cohomology. This cohomology group is called Eichler cohomology group and is isomorphic to the space of modular forms. The coefficients of period polynomials of Eichler integrals have played an important role to understand arithmetics of modular forms. The connection between periods and Maass wave forms was studied by Lewis and Zagier.
We introduce Jacobi integrals associated with Jacobi forms. We define cohomology group. We prove that we can find Jacobi integrals for given set of functions satisfying the cocycle condition. This is needed to prove the isomorphism theorem between the cohomology group and the space of Jacobi forms. We find a map between Jacobi integrals and Jacobi forms and period relations which characterizes period functions in Jacobi integrals. We explain various functions in terms of Jacobi integrals.
We define mock Jacobi forms as a special type of Jacobi integrals. Mock Jacobi forms are Jacobi integrals with dual forms. Lerch sum is given as an example of mock Jacobi forms.
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