On effective determination of symmetric-square lifts, level aspect. (English) Zbl 1390.11089
Summary: Let \(F\) be the symmetric-square lift with Laplace eigenvalue \(\lambda_F(\Delta) = 1 + 4\mu^2\). Suppose that \(|\mu| \leq\Lambda\). It is proved that \(F\) is uniquely determined by the central values of Rankin-Selberg \(L\)-functions \(L(s, F \otimes h)\), where \(h\) runs over the set of holomorphic cusp forms of weight 10 and level \(q \approx \Lambda^{\rho+\varepsilon}\) with \(\rho=\frac{64(\theta+2)}{5}\) for any \(\varepsilon > 0\). Here \(\theta\) is the exponent towards the Ramanujan conjecture for \(\mathrm{GL}_2\) Maass forms. We also prove an unconditional result in weight aspect.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
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