I will propose a way to combine the theory of translation functors with the trace formula to study automorphic representations of connected semisimple anisotropic algebraic groups over the rational numbers whose Archimedean component is a limit of discrete series. I will explain the main ideas of the derivation of a trace formula which, modulo a conjecture on the decomposition of the tensor product of a limit of discrete series with a finite-dimensional representation into basic representations, allows to isolate the non-Archimedean parts of a finite family of C-algebraic automorphic representations containing the ones whose Archimedean component is a given limit of discrete series.

In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao established that, for a given number field $K$ with a degree $n\geq 5$, the size of the $2$-torsion is bounded by $h_{2}(K) \ll D^{\frac{1}{2}-\frac{1}{2n}}$, where $D_{K}$ is the discriminant of $K$ over $\mathbb{Q}$. In this presentation, we will introduce new bounds that take into account the geometry of the lattice underlying the ring of integers of $K$. This research is a joint project with Pierre Le Boudec.