Beyond Endoscopy: Poisson Summation Formula and Kuznetsov Trace Formula on ${\mathrm{G}\mathrm{L}}_2$

In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of ${\mathrm{G}\mathrm{L}}_2$, which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a Kuznetsov-type trace formula for incorporating non-standard test functions. This formulation reveals that the Poisson summation formula naturally yields the functional equation for the standard $L$-functions of ${\mathrm{G}\mathrm{L}}_2$.