$$(G [1])^V = Hom(G, \mathbb{G}_m)$$ But there is a slight issue in the algebraic geometric side of the usual $\mathcal{B}$ side. Instead, let us do the class field theory in de Rham space. We can define $X_{dR}$ for an abelian and smooth variety $X$ as $$X_{dR} = X/\hat{\Delta}$$ where $\hat{\Delta}$ is the formal neighbourhood of the diagonal. It just imposes a relation $x \sim y$ if $x$ and $y$ are infinitesimally close to the identity. (A nice physics equivalent description is modding out the local gauge redundancy.) Though it is not technically a scheme, it is enough to do algebraic geometry. Actually, at many places, it is defined as a functor of points. So we will consider (quasic-coherent) sheaves on it. $$QC(X_{dR}) = QC(X)^\Delta$$ which is just another way of saying that $QC(X_{dR})$ contains flat-connections given by the identification $\mathcal{F}_x \xrightarrow{\sim} \mathcal{F}_y$ if $\mathcal{F}_x$ and $\mathcal{F}_y$ are $x$ and $y$ are infitemsally close. Actually, the following is true and interesting $$QC(X_{dR}) = \mathcal{D}_X-Mod$$ where $\mathcal{D}_X$ is the category of quasi-coherent sheaves. So the flat connections are exactly $\mathcal{D}$ modules. In physics, one can translate this by saying that the sheaves of $X_{dR}$ give the solutions to the Yang-Mills equation $F=dA + A \wedge A= 0$, so basically flat gauge field configurations.

All right. Let us take an abelian group $G$, then we can define the de Rham space of $G$ as $$G_{dR} = G/\hat{G}$$ where $\hat{G}$ is the formal neighbourhood of identity (same as before). A very good example is a vector space. The original Fourier-Mukai dual of $V$ has to do with the classifying space of the formal completion of the dual (we may discuss this in the following Quill(s)). But for $V_{dR}$ it is more interesting. Firstly, for $V$ (an abelian group) $$V_{dR}= V/ \hat{V}$$ Or equivalently, it is a chain complex (derived algebraic geometry) $V_{dR} = [ \hat{V} \to V]$. Now, the $1-$shifted Cartier duality is $$(V/\hat{V})^V [1]= (V^*/\hat{V}^*)$$ so $V_{dR}$ is canonically self-dual. Which means many things. But the best is that $\mathcal{D}(V) \simeq \mathcal{D}(V^*)$. This is better said as a Fourier transform of $\mathcal{D}$-modules. And it is nothing but the algebraic-geometric incarnation of electric-magnetic duality.

This is one nice symmetric way of doing abelian class field theory in algebraic geometry.

To learn more, see this paper by Ben-Zvi and Nadler.