Beyond endoscopy for ${\mathsf{G}\mathsf{L}}_2$ over $\mathbb{Q}$ with ramification 2: bounds towards the Ramanujan conjecture

We continue generalizing Altuğ’s work on ${\mathsf{G}\mathsf{L}}_2$ over $\mathbb{Q}$ in the unramified setting for \emph{Beyond Endoscopy} to the ramified case where ramification occurs at $S=\{\mathrm{\infty },q_1,\dots ,q_r\}$ with $2\in S$, after generalizing the first step. We establish a new proof of the $1\mathrm{/}4$ bound towards the Ramanujan conjecture for the trace of the cuspidal part in the ramified case, which is also provided by adapting Altuğ’s original approach. The proof proceeds in three stages: First, we estimate the contributions from the non-elliptic parts of the trace formula. Then, we apply the main result from our the previous work to isolate the $1$-dimensional representations within the elliptic part. Finally, we employ technical analytic estimates to bound the remainder terms in the elliptic part.