Topics in Arithmetic Geometry: Diophantine Geometry and Modular Forms

We cover three topics: The geometric Bombieri-Lang conjecture and explain a proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties by J. Xie and X. Yuan; Faltings heights of abelian varieties and explain a proof of the Northcott property of Faltings heights in an isogeny class of abelian variety by M. Kisin and L. Mocz; and Kroneck limit formula for SL(3) and explain a proof of this formula and an application to the Stark conjecture for cubic fields by N. Bergeron, P. Charollois, and L. Garcia.