On the global Gan-Gross-Prasad conjecture for general spin groups

We formulate a global Gan-Gross-Prasad conjecture for general spin groups. That is, we formulate a conjecture on a relation between periods of certain automorphic forms on $GSpin_{n+1} \times GSpin_n$ along the diagonal subgroup $GSpin_n$ and some $L$-values. To support the conjecture, we show that the conjecture holds for $n=2$ and $3$ and for certain cases for $n=4$.