Abstract We establish explicit Ichino’s formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for GL 2 {\rm GL}_{2} , Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted L 2 {L^{2}} -norms of the L 2 {L^{2}} -normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 2013, 5, 1825–1852].

中文翻译：

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扭曲三重积 L 函数的特殊值公式及其在受限 L2 范数问题中的应用

摘要 我们建立了三重积L函数的中心值的显式Ichino公式，重点是对实数的计算。我们计算的关键要素是命题 8，它概括了 [P. Michel 和 A. Venkatesh，GL 2 {\rm GL}_{2} 的次凸性问题，Publ。数学。研究所 Hautes Études Sci. 111 2010，171-271]。作为一个应用，我们证明了 L 2 {L^{2}} -L 2 {L^{2}} - 归一化新形式的受限 L 2 {L^{2}} - 范数之和的最优上限按照[V. 布洛默，关于自守形式的 4 范数，J. Eur。数学。社会。(JEMS) 15 2013, 5, 1825–1852]。