  Motivated by Vafa's model, we study the $tt^{*}$ geometry of a degenerate
class of FQHE models with an abelian group of symmetry acting transitively on
the classical vacua. Despite it is not relevant for the phenomenology of the
FQHE, this class of theories has interesting mathematical properties. We find
that these models are parametrized by the family of modular curves $Y_{1}(N)=
\mathbb{H}/\Gamma_{1}(N)$, labelled by an integer $N\geq 2$. Each point of the
space of level $N$ is in correspondence with a one dimensional $\mathcal{N}=4$
Landau-Ginzburg theory, which is defined on an elliptic curve with $N$ vacua
and $N$ poles in the fundamental cell. The modular curve $Y(N)=
\mathbb{H}/\Gamma(N)$ is a cover of degree $N$ of $Y_{1}(N)$ and plays the role
of spectral cover for the space of models. The presence of an abelian symmetry
allows to diagonalize the Berry's connection of the vacuum bundle and the
$tt^{*}$ equations turn out to be the well known $\hat{A}_{N-1}$ Toda
equations. The underlying structure of the modular curves and the connection
between geometry and number theory emerge clearly when we study the modular
properties and classify the critical limits of these models.
