Local, global, and relative problems in arithmetic intersection theory

This thesis consists of four independent articles. The overarching theme, arithmetic intersections, will present itself from very distinct angles.
Article [HM23] is an outlier in the sense that it is local in nature. It proves a conjectural identity between intersection numbers on Rapoport-Zink spaces and central derivatives of local orbital integrals, known as an arithmetic fundamental lemma(AFL), in the case of general linear groups over the quaternion division algebra. Articles [Hul23], [Hul24] and [DHS24] belong to the field of Arakelov geometry. Article [Hul23] studies arithmetic properties of algebraic extensions of Q through its Northcott number. We extend finiteness results from fields satisfying the Northcott property to fields with big enough Northcott number and provide examples of infinite extensions of Q with finitely many CM points.
In Article [Hul24], we study the Arakelov geometry of toric bundles in a systematic way. The purpose is two-fold. On one hand it provides us with a new class of examples that can be studied explicitly. On the other, toric bundles contain many varieties of interest such as semiabelian varieties and their compactifications. We compute the Okounkov body and Boucksom-Chen transform of toric line bundles on toric bundles in terms of information on the base. We prove a formula for intersection numbers in terms of convex geometry data, an arithmetic relative BKK theorem.
Lastly, we study arithmetic intersection numbers in families in Article [DHS24]. We associate to finite type schemes over globally valued fields topological spaces and prove continuity of arithmetic intersection numbers on them. We apply this to prove a conjecture of Gualdi and Sombra on the height of complete intersections in toric varieties.