Central Values of Degree Six L-Functions: The Case of Hilbert Modular Forms

Abstract

We will prove an explicit formula (Eq. 1.5 or Thm. 4.4.4) for the central value of the $L$-function $L(s,\Sym^{2} g\times f)$ when $f$ and $g$ are Hilbert newforms of level $\Gamma_{0}(1)$ by computing the local integrals appearing in the refined Gan-Gross-Prasad formula for $\SL_{2}\times\widetilde{\SL_{2}}$ for some suitable choice of vectors. We will also work out the rationality of this value in the two extreme cases, the \textit{purely balanced} and the \textit{purely unbalanced} (Thm. 1.3). Our results in these cases are compatible with Deligne's conjecture on rationality of critical values of motivic $L$-functions. We will also give an explicit conjecture on the rationality of \textit{all} critical values (cf. 1.2) of $L(s,\Sym^{2} g\times f)$ in the general case without any restrictions on the weights or the levels (Conj. 1.2).