In this talk, I will discuss the problem of counting fractional ideals of a given order $R$ in a number field $E$ up to multiplication by invertible ideals, focusing on the cases where $R$ is a Bass order or $E$ is a cubic field.
I will introduce a local-global principle for fractional ideals, which reduces this counting problem to the enumeration of local overorders of $R$. I will then present an explicit parametrization of these local overorders, together with criteria describing their inclusion relations, highlighting the differences that arise in the two settings.
Finally, I will discuss several applications, including the global geometrization in the Hitchin fibration and orbit counting for Bhargava's $2\times3\times3$ cubes. An application to Beyond Endoscopy for $\mathrm{GL}_3(\mathbb{Q})$ will be presented in the subsequent talk by Yuchan Lee.
