Vector valued theta series and Waldspurger's theorem
In this paper the authors continue their investigation of theta series attached to norm forms of definite quaternion algebras \(D\) over \(\mathbb{Q}\). More precisely they study the properties of the Yoshida lifting \(Y^{(n)} (\varphi, \varphi)\) of an automorphic form \(\varphi\) defined on the adelization \(D^*_ \mathbb{A}\) to an Eisenstein series of Klingen type defined on the Siegel upper half space \(\mathbb{H}_ n\). If \(n=1\) one obtains Eichler's correspondence associating to \(\varphi\) an elliptic modular form \(f\). In a previous paper [Math. Ann. 288, 361-388 (1990; Zbl 0708.11026)], the authors, computing the Fourier coefficients of \(Y^{(2)} (\varphi, \varphi)\) with \(\varphi D^*\) right-invariant, gave a new proof for the case of weight 2 of a formula of Waldspurger relating the center of the critical strip of the \(L\)-series with squares of Fourier coefficients. In the present article they want to generalize this result to arbitrary weight, therefore they have to remove the right-invariant condition. This leads to consider vector valued modular forms. However, after some technical difficulties, as the proof that certain formulas for the Fourier coefficients of vector valued Eisenstein series of Klingen type are still valid outside the range of absolute convergence, they obtain a proof of Waldspurger's formula. More precisely they show that if \(g\) is in \(S^{3/2+ \beta/2} (N)^{\text{new}}\) and is an eigenform for all Hecke operators (\(N\) odd) and \(f\) in \(S^{2+\beta} (N)^{\text{new}}\) is the unique normalized eigenform corresponding to \(g\) via the Shimura correspondence, then the Waldspurger lifting \(W(\varphi_{N_ 1})\) is different from 0 for \(N_ 1/N\) iff \(L(f, 1+\beta/2)\) is different from 0.
scientific article; zbMATH DE number 706674
scientific article