On Modular Curves: Scholze’s Generalization of the Langlands-Kottwitz Method to Places of Bad Reduction
We describe the Langlands-Kottwitz method for counting points mod p of Shimura varieties at places of good reduction in the special case of the modular curve. We then explicit the calculation of the semisimple trace of Frobenius on the nearby cycles sheaves, following the recent work of Peter Scholze. This effectively extends the Langlands-Kottwitz approach to all places.