I'm essentially looking for a reference for the following statement on the wikipedia:
It can be proved that if F is any local field other than $\mathbb{C}$, then the symplectic group $Sp_{2n}(F)$ admits a unique perfect central extension with the kernel $\mathbb{Z}/2\mathbb{Z}$, the cyclic group of order 2, which is called the metaplectic group over F.
Or any other reference that gives an explicit construction of $Mp_{2n}$ over arbitrary local fields and the adeles.