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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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