Beyond endoscopy for ${\mathsf{G}\mathsf{L}}_2$ over $\mathbb{Q}$ with ramification 5: cancellation theory

We complete our work on ${\mathsf{G}\mathsf{L}}_2$ over $\mathbb{Q}$ in the ramified setting for \emph{Beyond Endoscopy} proposed by Langlands. We prove that the asymptotic formula for each term of the trace formula when summing over $n<X$ with arbitrary smooth test functions at places in $S=\{\mathrm{\infty },q_1,\dots q_r\}$ with $2\in S$, for the standard representation, is $o(X)$. We prove an identity with a variable $X$, called the \emph{limit form of the trace formula} for ${\mathsf{G}\mathsf{L}}_2$ over $\mathbb{Q}$, directly. The proof uses Arthur’s result on the Fourier transform of weighted orbital integrals to rewrite the term involving intertwining operators, and then compares the expansion with the results of the real case due to Arthur-Herb-Sally and Hoffmann, and the nonarchimedean case by direct computation using Arthur’s definition.