Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 3: contribution of the elliptic part

Abstract

We continue to work on \emph{Beyond Endoscopy} for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification at $S = \{\infty, q_1, \dots, q_r\}$ (where $2 \in S$), generalizing the final step of Altu\u{g}'s work in the unramified setting. We derive an explicit asymptotic formula for the elliptic part when summing over $n<X$ with arbitrary smooth test functions at places in $S$ for the standard representation. As a consequence, we obtain the desired limit for the simple trace formula which only occurs in the ramified case. Moreover, we prove an asymptotic formula for the traces of Hecke operators on cusp forms with arbitrary level and weight $>2$, directly generalizing Altu\u{g}'s final result. Our approach differs entirely from Altu\u{g}'s: We apply a second Poisson summation with respect to the determinant, obtaining a formula on the Hitchin-Steinberg base $\mathfrak{g}/\!/ \mathsf{G}$. By changing variables from $(T, N)$ to $(T, \Delta)$ on $\mathfrak{g}/\!/ \mathsf{G}$, we perform analysis in the new coordinates.