Let $X$ be a smooth, complex projective variety, and $G$ a finite abelian group. We want to study $G$-principal bundles over $X$, or, in other words, étale $G$-covers over $X$.

Topologically, these objects are classified (up to isomorphism of $G$-covers over $X$) by $H^1(X, G)$, or equivalently, by homomorphisms $\phi:H_1(X, \mathbb{Z}) \to G$.

In algebraic geometry these objects are classified (up to isomorphism) by morphisms $L: G^{\vee} \to Pic^0(X)$. Here I write $G^{\vee}$ for the group of characters of $G$, or in other words the homomorphisms from $G$ to $\mathbb{C}^*$.

(For example, think of the well-known correspondence between $2$-torsion line bundles over $X$ and étale double covers on $X$).

There is then an association $L \to \phi$ given by forgetting the structure of algebraic variety and keeping only the topological structure. This is an homomorphism $$ \lambda: Hom(G^{\vee}, Pic^0(X)) \to Hom(H_1(X, \mathbb{Z}), G). $$

The question is: can one make sense of the latter homomorphism in a purely algebraic way? In other words, can you define $\lambda$ in a simple way that does not use geometry (as I did)?

(I apologize if this is too simple, I just cannot see it)

specifiedisomorphism $\mathcal O_X \equiv L^{\otimes 2}$. This is not the same as 2-torsion elements of $Pic^0$. For example, on $\mathbb A^1 - {0}$ there are no non-trivial line bundles, but there is a double cover. $\endgroup$