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With analytic applications in mind, in particular Beyond Endoscopy ([13]), weinitiate the study of the elliptic part of the trace formula. Incorporating theapproximate functional equation to the elliptic part we control the analyticbehavior of the volumes of tori that appear in the elliptic part. Furthermoreby carefully choosing the truncation parameter in the approximate functionalequation we smooth-out the singularities of orbital integrals. Finally by anapplication of Poisson summation we rewrite the elliptic part so that it isready to be used in analytic applications, and in particular in BeyondEndoscopy. As a by product we also isolate the contributions of specialrepresentations as pointed out in [13].

Beyond Endoscopy via the Trace Formula-I: Poisson Summation and Contributions of Special Representations