### Treatment of the Cuspidal Elliptic (Regular Cuspidal) contribution to the trace

Elstrodt Mennike and Grunewald calculate the contribution to the trace from the Cuspidal Elliptic elements in Chapter 6, section 5 of their book. What I don't understand is the transition from equation (5.19) to (5.20) in their treatment.

Trying to match up the steps in their calculation with those of the Jo-La manuscript is a bit difficult, because Jo-La use a lot of facts that are particular to the special case of $\Gamma=SL(2,Z)$ in this section. However, it seems that this step roughly corresponds to the proof on the top of p. 385 in the manuscript. (This is in part 5, chapter 13, towards the end of section 3.) The trouble is that the details in that proof are not filled in, and we are referred back to the details of proofs in section 1. More on this later...

Trying to match up the steps in their calculation with those of the Jo-La manuscript is a bit difficult, because Jo-La use a lot of facts that are particular to the special case of $\Gamma=SL(2,Z)$ in this section. However, it seems that this step roughly corresponds to the proof on the top of p. 385 in the manuscript. (This is in part 5, chapter 13, towards the end of section 3.) The trouble is that the details in that proof are not filled in, and we are referred back to the details of proofs in section 1. More on this later...

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