  We study weakly coupled $U(1)$ theories in $AdS_3$, their associated charged
BTZ solutions, and their charged spectra. We find that modular invariance of
the holographic dual two-dimensional CFT and compactness of the gauge group
together imply the existence of charged operators with conformal dimension
significantly below the black hole threshold. We regard this as a form of the
Weak Gravity Conjecture (WGC) in three dimensions. We also explore the
constraints posed by modular invariance on a particular discrete $\mathbb{Z}_N$
symmetry which arises in our discussion. In this case, modular invariance does
not guarantee the existence of light $\mathbb{Z}_N$-charged states. We also
highlight the differences between our discussion and the usual heuristic
arguments for the WGC based on black hole remnants.
