A Kudla-Rapoport Formula for Exotic Smooth Models of Odd Dimension

TL;DR

This paper proves a conjecture relating intersection numbers of cycles on exotic smooth unitary Rapoport-Zink spaces of odd dimension to derivatives of local representation densities, extending known results to new geometric settings.

Contribution

It establishes a Kudla-Rapoport formula for exotic smooth models of odd dimension, connecting geometric intersection numbers with analytic representation densities.

Findings

Proved the Kudla-Rapoport conjecture for odd-dimensional exotic smooth RZ spaces.

Compared $\\mathcal{Z}$-cycles and $\mathcal{Y}$-cycles on these spaces.

Reduced the conjecture to existing results in even-dimensional cases.

Abstract

In this article, we prove a Kudla-Rapoport conjecture for $\mathcal{Y}$-cycles on exotic smooth unitary Rapoport-Zink spaces of odd arithmetic dimension, i.e. the arithmetic intersection numbers for $\mathcal{Y}$-cycles equals the derivatives of local representation density. We also compare $\mathcal{Z}$-cycles and $\mathcal{Y}$-cycles on these RZ spaces. The method is to relate both geometric and analytic sides to the even dimensional case and reduce the conjecture to the results in arXiv:2101.09485.