Canonical extensions of $p$-adic shtukas on toroidal compactifications of Shimura varieties

Abstract: We construct canonical extensions of $p$-adic shtukas on integral models of toroidal compactifications of abelian-type Shimura varieties with quasi-parahoric levels at any prime number $p$. More precisely, we define the notion of a log diamond as a $v$-sheaf associated with a log scheme over $\mathbb{Z}_p$ and construct a $p$-adic log shtuka over the log diamond of an integral toroidal compactification of an abelian-type Shimura variety by studying the ``degeneration'' of the shtuka at the boundary. Moreover, we provide a definition of canonical integral models of toroidal and minimal compactifications in the sense of Pappas and Rapoport, and verify it in the same generality as above. Applications include the canonicity and functoriality of integral toroidal compactifications, as well as an axiomatic proof of the well-positionedness of all well-known stratifications on the special fiber.