Central values of degree six L-functions attached to two Hilbert modular newforms

Let f,g be two Hilbert modular newforms (functions on ’n-copies’ of the upper half-plane, satisfying properties similar to usual modular forms). Consider the L-function L(s,f \times \Sym^{2}g) (it is the degree six factor of the triple product L-function L(s,f \times g \times g)). In this talk we will give a formula for the central value of this L-function and work out its rationality in some special cases of relationships between weights of f and g. We will arrive at our formula via the refined Gan-Gross-Prasad formula for SL(2) \times \tilde(SL(2)). Our results on rationality are compatible with Deligne’s conjecture on the rationality of critical values of motivic L-functions.