Jacquet-Zagier Treatment of the Beyond Endoscopy Trace Formula for GL_2

Abstract

This thesis begins the study of the beyond endoscopic trace formula for $\mathrm{GL}_2$ over $\mathbb{Q}$ attached to an arbitrary symmetric power representation $\sigma_k$ of the dual group $\mathrm{GL}_2(\mathbb{C})$.

For an adelic function that incorporates the $L$-functions $L(s_B,\pi,\sigma_k)$ through the basic functions at all 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 with a continuously deformed trace formula. We apply Poisson summation to 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 $s_B=1$.