Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

Abstract: (Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties $A_g$ for any value of g. For $g<16$ the dimension is $g-1$, while for $g \geq 16$, it is determined by the larged dimensional compact Shimura subvariety, which we determine. Our methods rely on deforming the boundary using special varieties, and functional transcendence theory.