For any hypergeometric differential equation of order $n$ on a Riemann sphere with three punctures $\mathbb{P}^1(\mathbb{C}) / \{0,1,\infty \}$, one has linearly $n$ independent solutions which are called hypergeometric functions. For the $\alpha, \beta$ parameters, we write the local monodromies using the (local) space of solutions through the monodromy representation $\rho: \pi_1 \to GL(V)$. The subgroup $\rho(\pi_1)$ of $GL(V)$ is called the monodromy group associated with our hypergeometric equation and parameters $\alpha, \beta$. Accoriding to Levelt's theorem (and later worked by Beukers–Heckman), there exists a (local) basis of solution space of hypergeometric equation such that the monodromy group associated to it (corresponding to $\alpha, \beta$) is generated by the companion matrices $A$ and $B$ of polynomials $f,g$ such that the polynomials are cyclotomic (then one can conjugate the monodromy groups) which means their roots are roots of unity, primitive, self-reciprocal, and thus have no common root. The monodromies are defined locally using these companion matrices. This monodromy group $\Gamma(f,g)$ is a subgroup of $GL_n(\mathbb{C})$.

Now, when we restrict to the polynomials $f,g$ with integer coefficients, then the monodromy group becomes a subgroup of $GL_n(\mathbb{Z})$. We are very much interested in this special representation. However, there is a very large gap between the monodromy group $\Gamma$ and $GL_n(\mathbb{C})$, hence we look for something in between, which is the Zariski closure of $\Gamma$. For the order $n=1$, we see that $\Gamma$ sits in $GL_1(\mathbb{Z})$; however, with order $2$ and higher, we have non-trivial things going on like finite and infinite index, and so on. For order $n=2$, the Zariski closure (here it is the smallest algebraic group that contains $\Gamma$ in $GL_n(\mathbb{Z})$) of $\Gamma$ is $SL_2(\mathbb{Z})$. Then a nice question is to ask if $\Gamma$ is of finite index in its Zariski closure, then it is called arithmetic, and if it infinite index, then it is called thin.

Let us denote for $\Gamma$ its Zariski closure $\Gamma^{Zar}$ as $G$, so $\Gamma \subset G(\mathbb{Z})$. Typically, one has $G=Sp_\Omega(\mathbb{Z})$ where $\Omega$ is a symplectic form on $\mathbb{Z}^n$ or $G=O_Q$ where $Q$ is a quadratic form. The details about when $G$ is symplectic and orthogonal are here. Anyway, the important question is to ask for any given order, $\Gamma(f,g)$ has a finite (arithmetic) or infinite (thin) index in $G(\mathbb{Z})$. For the number theoretic motivation behind the arithmeticity and thinness, see these notes by Sarnak. In order $n=2,3$, there are arithmetic monodromy groups. However, for $n=4$, the situation is more non-trivial and it contains both arithmetic and thin groups. There are $14$ cases in which the monodromy groups are associated with Calabi-Yau 3-folds. The periods of the three-form $\omega$ on the moduli of a CY 3-fold solve a fourth-order differential equation. (I do not understand much connection more than this for these 14 special cases connected to CY right now.) Anyway, again, one is asking how many of them are arithmetic and how many are thin. They were studied in this paper. As it turned out, 7 of the 14 cases are thin and the other 7 are arithmetic, and Kontsevich was also involved in the problem.

$$(G [1])^V = Hom(G, \mathbb{G}_m)$$ 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.