Unitary Shimura varieties at ramified primes and arithmetic transfer

Abstract

We consider unitary Shimura varieties at places where the totally real field ramifies over $\mbQ$. Our first result constructs comparison isomorphisms between absolute and relative local models in this context, which relies on a reformulation of the Eisenstein condition of Rapoport--Zink and Rapoport--Smithling--Zhang. Related to that, we also provide a moduli description for the integral models of RSZ unitary Shimura varieties in new cases. Our second result lifts the comparison of local models to categories of $p$-divisible groups and, as a corollary, to various kinds of Rapoport--Zink spaces. Our third result is a proof of the arithmetic transfer conjecture of the third author in full generality. Using our statements about Rapoport--Zink spaces, we extend the previous proof from the unramified case to that of all $p$-adic local fields ($p$ odd).