Galois covers of Riemann surfaces 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?

By the Galois correspondance for covering spaces, you will have a connected topological $G$-cover $X \to C$ precisely if you can find a surjective morphism $$\pi_1 C = \langle a_1, b_1, \ldots, a_n, b_n \, | \, [a_1,b_1]\cdots[a_g,b_g]\rangle \to G.$$ (The complex/algebraic structure is irrelevant here).
As $\pi_1 C$ is generated by $2g$ elements, any such $G$ (abelian or not) must also be generated by $2g$ elements. If $G$ is abelian, it is then a quotient of $\mathbb Z^{2g}$, and the classification of finite type abelian groups tell you that this can only happen if its canonical decomposition 
$G = \oplus_{i=1}^r \mathbb Z/d_i$, (canonical means $d_i|d_{i+1}$ and we know that such a decomposition is unique) has at most $2g$ factors.
So yes, it is true.
