In 1974, Drinfeld published his great paper Elliptic Modules in which he introduced what we now call arbitrary rank Drinfeld modules. Let $k$ be the field of rational functions on a smooth, projective, geometrically connected curve $X$ defined over the finite field $\mathbb{F}_q$, and $K$ be a finite separable extension of $k$. Let $\infty$ be a closed point of $X$, and $A\subset k$ be the ring of functions regular away from $\infty$. A Drinfeld module defined over $K$ is an $\mathbb{F}_q$-algebra morphism from $A$ to a twisted polynomial ring with coefficients in $K$, satisfying some leading term condition. The notion of a Drinfeld module has been historically proven to be highly powerful. In 1986, Anderson generalized this notion to the higher-dimensional $t$-modules. If Drinfeld modules are analogous to elliptic curves, then $t$-modules are analogous to general abelian varieties.

On the other hand, given a Drinfeld module $E$ defined over $K$, Taelman defined an associated class module $H(E/A)$. He proved a class number formula which states that the special value at $s=0$ of the $L$-function of the Drinfeld module $E$ is equal to the product of the unique monic generator of the Fitting ideal $\mathrm{Fitt}_A^0(H(E/A))$ of $H(E/A)$ and a regulator term, in the case that $A=\mathbb{F}_q[t]$. Later, Popescu and his collaborators proved an equivariant Tamagawa number formula (ETNF) for the special value at $s=0$ of a Goss-type $L$-function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined over a finite, integral extension of $\mathbb{F}_q[t]$. Then Popescu and his collaborators generalize their result to the $t$-module case. In this case, if $K/F$ is a finite abelian extension of function fields of Galois group $G$ and $E$ is a $t$-module defined over the integral closure $O_F$ of $A=\mathbb{F}_q[t]$ in $F$, then the special value at $s=0$ of the $G$-equivariant $L$-function associated to $E$ is related to $\mathrm{Fitt}^0_{A[G]}(H(E/O_K))$.

The main theme of this talk is the theory of abelian $t$-modules of arbitrary dimension. In the first part, we talk about an Iwasawa theory for their Taelman class modules, extending work of Higgins from the 1-dimensional case to the arbitrary dimensional case. The second part deals with the theory of formal $t$-modules and extends the results of Michael Rosen from the 1-dimensional case to the case of arbitrary dimension.