Beyond endoscopy for ${\mathsf{GL}}_2$ over $\mathbb{Q}$ with ramification 4: contribution of non-elliptic parts

Abstract

We continue our work on ${\mathsf{GL}}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy}. We establish asymptotic formulas for each term of the trace formula when summing over $n<X$, using arbitrary smooth test functions at the places in $S=\{\infty ,q_1,\dots ,q_r\}$ where $2\in S$, for the standard representation, up to an error of $o(X)$. This yields an identity depending on a parameter $X$, leading to certain identities that can be regarded as a limit form of the trace formula for ${\mathsf{GL}}_2$ over $\mathbb{Q}$. On the spectral side, we employ the contour shift method and the Riemann-Lebesgue lemma. On the geometric side, both the identity part and the unipotent part contribute $o(X)$. The elliptic part was reduced to the hyperbolic part in a previous paper. Finally, using hyperbolic Poisson summation, we relate the hyperbolic part back to the spectral side and determine its contribution.