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Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms. (English) Zbl 0702.11028

In this paper the kernel functions of the composite maps of the Shimura liftings \(S_{k,m}\to S_{2k-2}(m)\) and the Eichler-Shimura isomorphism \(S_{2k-2}(m)\to H^ 1_{par.}(\Gamma_ 0(m),C[x]_{2k-4})\) \((S_ k(m)\) \(=\) space of elliptic cusp forms of weight k on \(\Gamma_ 0(m)\), \(S_{k,m}\) \(=\) space of holomorphic and skew-holomorphic Jacobi forms of weight k and index m, \(H^ 1_{par.}(\Gamma_ 0(m),{\mathbb{C}}[x]_ k)\) \(=\) first cuspidal cohomology of \(\Gamma_ 0(m)\) acting in the natural way on \({\mathbb{C}}[x]_ k\) \(=\) space of complex polynomials of degree \(\leq k)\) are studied. These investigations lead to a set of Jacobi forms which degenerate the space \(J_{k,m}\), and whose Fourier expansion can be written down explicitly in closed form; in the simplest cases their Fourier coefficients C(D,r) count the number of integral binary quadratic forms of discriminant D which satisfy certain reduction conditions. Applying to these Jacobi forms the Shimura liftings yields a set of modular forms which generate \(S_{2k-2}(m)\) and whose Fourier coefficients are again given by interesting and effectively computable formulas.
This explicit construction of Jacobi and elliptic modular forms has immediate implications for the problem of tabulating automorphic forms and it can be used to deduce “Tunnell-like” theorems. There are also some interesting aspects on the technical side of this construction: e.g. one has to replace the abstract cohomology group \(H^ 1_{par.}({\mathbb{C}}[x]_{2k-4},\Gamma_ 0(m))\) by the more explicit “space of period polynomials” which itself may deserve further investigations.
This paper generalizes the results of a former paper by the same author (“Explicit formulas for the Fourier coefficients of Jacobi and elliptic modular forms”, to appear in Invent. Math.).
Reviewer: N.-P.Skoruppa

MSC:

11F30 Fourier coefficients of automorphic forms
11E16 General binary quadratic forms
11F11 Holomorphic modular forms of integral weight
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