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

TL;DR

This paper advances the Beyond Endoscopy approach for GL_2 over Q with ramification at 4, establishing asymptotic trace formulas and analyzing spectral and geometric contributions to understand automorphic representations.

Contribution

It provides new asymptotic formulas for trace formula terms in the ramified setting, connecting spectral and geometric sides via hyperbolic Poisson summation.

Findings

Established asymptotic formulas for trace formula terms with error o(X)

Connected hyperbolic and spectral contributions using Poisson summation

Reduced elliptic to hyperbolic parts in previous work

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.