Workshop on Recent Developments in Hodge Theory and O-minimality

Abstract: Since Langlands's earliest paper on his now famous program, canonical integral models of Shimura varieties have occupied a central role in modern number theory. Steady progress has been made in the intervening 50 years toward the correct formulation of the notion of 'canonical', the study of the remarkable properties of such models, and ultimately of their construction. In this talk, I will discuss work with Madapusi (continuing previous work of Imai--Kato--Youcis in the abelian-type setting) on giving a new, and what we believe to be more robust, notion of 'canonical' using recent advances in $p$-adic Hodge theory due to Bhatt--Scholze, Drinfeld, Bhatt--Lurie, and others. In particular, we give constructions of models of pre-abelian type Shimura varieties for primes $p>2$, extending and more conceptually framing previous constructions of Kisin in the abelian-type setting. We moreover recapture the models for arbitrary Shimura varieties for $p\gg 0$ recently obtained by Bakker--Shankar--Tsimerman from this perspective. Finally, we discuss how this new notion of canonicity developed here provides the correct foundations to prove many strong results about such models, including Néron-type mapping properties and algebraization questions of the following kind: which $p$-divisible groups over $\overline{\mathbb{F}}_p$ arise as $A[p^\infty]$ for an abelian variety $A$?