Special cycles on unitary Shimura curves at ramified primes.

In this paper, we study special cycles on the Kr\"amer model of $\mathrm{GU}(1,1)(F)$ Rapoport-Zink spaces where $F$ is a ramified extension of $\mathbb{Q}_p$ with the assumption that the underlying hermitian form on the Dieudonn\'e module of the framing object of the Rapoport-Zink space is aniostropic. We write down the decomposition of these special cycles and compute their intersection numbers. We then apply the local results to compute the intersection numbers of special cycles on unitary Shimura curves and relate these intersection numbers to Fourier coefficients of central derivatives of certain Eisenstein series.