Let $G$ be a finite abelian group, and $C$ a compact Riemann surface (algebraic curve) of genus $g$. I am interested in topological Galois $G$-covers $X \to C$, aka \'etale $G$-principal bundles over $C$.
Let us write the abelian group as a direct sum of cyclic groups of order $d_1, \ldots, d_k$ with $d_1|d_2|\ldots|d_k$.
Is it true, and why, that if $k>2g$, then $X$ must be disconnected?