Quasi-canonical AFL and arithmetic transfer conjectures at parahoric levels

Abstract

In the first part of the paper, we formulate several arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture in the presence of ramification. The ramification comes from the choice of non-hyperspecial parahoric level structure. We prove a graph version of these arithmetic transfer conjectures, by relating it to the quasi-canonical arithmetic fundamental lemma , which we also establish. We relate some of the arithmetic transfer conjectures to the arithmetic fundamental lemma conjecture for the whole Hecke algebra in our recent paper [C. Li, M. Rapoport and W. Zhang, Arithmetic fundamental lemma for the spherical Hecke algebra, Manuscripta Math. 175 (2024), 1–2, 1–51]. As a consequence, we prove these conjectures in some simple cases. In the second part of the paper, we elucidate the structure of an integral model of a certain member of the almost self-dual Rapoport–Zink tower, thereby proving conjectures in [S. S. Kudla and M. Rapoport, Notes on special cycles, unpublished notes (2012)] and [C. Li and W. Zhang, Kudla–Rapoport cycles and derivatives of local densities, J. Amer. Math. Soc. 35 (2022), 3, 705–797]. This result allows us to verify the hypotheses of the graph version of the arithmetic transfer conjectures in a particular case.