Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular \(L\)-functions (Q2843990)

The main result asserts: Let \(k>1\) be an odd integer. Let \(N_1\), \(N_2\) be positive square free integers with \(M=(N_1,N_2)>1\). Let \(f\) (resp. \(g\)) be a holomorphic newform of weight \(2k\) on \(\Gamma_0(N_1)\) (resp. 2 on \(\Gamma_0(N_2)\)). Assume that for all primes \(p\) dividing \(M\) the Atkin-Lehner eigenvalues of \(f\) and \(g\) coincide. Then there exists an imaginary quadratic field \(K\) and a character \(\chi\) of the ideal class group of \(K\) with \(L(\frac{1}{2},\pi_f\times\theta_\chi) L(\frac{1}{2},\pi_g\times\theta_\chi) \not= 0\). Moreover, let \(D(f,g)\) be the set of odd, square free integers \(d>0\) such that \(-d\) is a fundamental discriminant, and there is a character \(\chi\) of the ideal class group of \(K=\mathbb{Q}(\sqrt{-d})\) such that \(L(\frac{1}{2},\pi_f\times\theta_\chi) L(\frac{1}{2},\pi_g\times\theta_\chi) \not= 0\). Then for each positive \(\delta<\frac{5}{8}\) the are \(C>0\), \(E>0\) with \(|\{0<d<X;\,d\in D(f,g)\}|>CX^\delta\) for all \(X>E\). A key ingredient in the proof is a translation to the language of modular forms of the endoscopic lifting of automorphic representations according to the embedding of \(\{(a,b)\in \mathrm{GL}(2, \mathbb{C})\times \mathrm{GL}(2, \mathbb{C}); \mathrm{det}(a)=\mathrm{det}(b)\}\) in \(\mathrm{GSp}(4, \mathbb{C})\) (as the centralizer of diag\((1,-1,1,-1)\) if \(\mathrm{GSp}(4)\) is defined using the form antidiag\((\mathrm{Id},-\mathrm{Id})\)).