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In this paper, we proved a local arithmetic Siegel-Weil formula for a $U(1,1)$-Shimura variety at a ramified prime, a.k.a. a Kudla-Rapoport conjecture ata ramified case. The formula needs to be modified from the originalKudla-Rapoport conjecture. In the process, we also gives an explicitdecomposition of the special divisors of the Rapoport-Zink space of unitarytype $(1, 1)$ (Kr\"amer model). A key ingredient is to relate the Rapoport-Zinkspace to the Drinfeld upper plane.

The Kudla-Rapoport conjecture at a ramified prime for $U(1, 1)$