Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification: Poisson summation

Abstract: At the beginning of this century, Langlands introduced a strategy known as \emph{Beyond Endoscopy} to attack the principle of functoriality. Altu\u{g} studied $\mathsf{GL}_2$ over $\mathbb{Q}$ in the unramified setting. The first step involves isolating specific representations, especially the residual part of the spectral side, in the elliptic part of the geometric side of the trace formula. We generalize this step to the case with ramification at $S={\infty,q_1,\dots,q_r}$ with $2\in S$, thereby fully resolving the problem of isolating these representations over $\mathbb{Q}$ which remained unresolved for over a decade. Such a formula that isolates the specific representations is derived by modifying Altu\u{g}'s approach. We use the approximate functional equation to ensure the validity of the Poisson summation formula. Then, we compute the residues of specific functions to isolate the desired representations.