Beyond endoscopy for ${\mathsf{GL}}_2$ over $\mathbb{Q}$ with ramification 5: cancellation theory

Yuhao Cheng

We complete our work on ${\mathsf{GL}}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy} proposed by Langlands. We prove that the asymptotic formula for each t…