Beyond endoscopy for $\mathsf{GL}_2$ over $\mathbb{Q}$ with ramification 1: Poisson summation

TL;DR

This paper advances the Beyond Endoscopy approach for $ ext{GL}_2$ over $ ext{Q}$ with ramification, employing Poisson summation and residue calculations to isolate specific automorphic representations, resolving a decade-long problem.

Contribution

It generalizes Altu's unramified method to ramified cases, providing a new formula for isolating representations using Poisson summation and residue analysis.

Findings

Derived a formula that isolates specific automorphic representations with ramification.

Applied Poisson summation via the approximate functional equation to validate the approach.

Resolved a longstanding problem in the spectral analysis of $ ext{GL}_2$ over $ ext{Q}$ with ramification.

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.