AbstractWe 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 {\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}}-norms
of the {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].