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Heegner points, cycles and Maass forms. (English) Zbl 0787.11016
In the important paper [J. L. Waldspurger, J. Math. Pures Appl. 60, 375-484 (1981; Zbl 0431.10015)] it was proved that the value at the center of the critical strip of the $$L$$-series of a holomorphic newform $$f$$ of even integral weight on $$\Gamma_ 0(N)$$ twisted by a quadratic character of conductor $$t$$ is proportional to the square of the $$t$$-th Fourier coefficient of a form $$g$$ of half-integral weight corresponding to $$f$$ under the Shimura lift. This phenomenon was studied later by several authors including D. Zagier and the present reviewer. In particular, in [W. Kohnen, Math. Ann. 271, 237-268 (1985; Zbl 0542.10018)] for $$N$$ squarefree a generalization of Waldspurger’s theorem was given which relates the product of two Fourier coefficients of $$g$$ to the integral of $$f$$ over certain geodesic periods on the modular curve $$X_ 0(N)$$.
In the present paper the authors study similar questions in the context of a non-holomorphic Maass cusp form $$\phi$$ on $$SL_ 2(\mathbb{Z})$$, relating both special values at complex multiplication points of $$\phi$$ as well as integrals of $$\phi$$ over closed geodesics to products of Fourier coefficients of non-holomorphic cusp forms of weight 1/2 corresponding to $$\phi$$ by an analogous construction as the Shimura lift in the holomorphic case.

MSC:
 11F37 Forms of half-integer weight; nonholomorphic modular forms 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11F55 Other groups and their modular and automorphic forms (several variables)
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References:
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