Jacquet-Zagier treatment of the beyond endoscopy trace formula for $\mathrm{GL}_2$

Abstract

We begin the study of the beyond endoscopic trace formula for $\mathrm{GL}_2$ over $\mathbb{Q}$ attached to any symmetric power representation $σ_k$ of the dual group $\mathrm{G}L_2(\mathbb{C})$. For an adelic function that incorporates the $L$-functions $L(s_B,π,σ_k)$ through the basic functions at all the finite places, we integrate the cuspidal kernel against a corresponding Eisenstein series $E(g,s)$ and realize the trace formula as a residue at $s=1$, replacing Arthur's truncation operation by a continuously deformed trace formula. We argue Poisson summation on the trace variable of the Hitchin-Steinberg base and show that the dominant term of the elliptic part admits meromorphic continuation to $\mathfrak{R}(s_B) \geq 0$ with a pole of order $k$ at $s_B =1$.