The Eichler-Selberg Trace Formula for Level-One Hecke Operators

This paper explains the steps involved in the proof of the Eichler-Selberg Trace Formula for Hecke operators of level one. It is based on an appendix in Serge Lang’s Introduction to Modular Forms written by Don Zagier, though I also draw heavily from sections of Toshitsune Miyake’s Modular Forms and Xueli Wang’s and Dingyi Pei’sModular Forms with Integral and Half-Integral Weights. Section 2 summarizes the necessary background in the theory of modular forms. The material covered here is standard, so I’ve left out most of the details and proofs. Most of the section is based on Chapter VII of Jean-Pierre Serre’s A Course in Arithmetic. Section 3 covers the first major step in Zagier’s terse proof of the EichlerSelberg Formula. I’ve tried to explain where Zaiger’s proof comes from and why such a formula should even exist, since the original proof lacks this sort of motivation for the reader. Section 4 completes the proof of the E-S Formula. This last part requires several results from the theory of binary quadratic forms. There are several well-known number theory books that cover these facts in detail, so I assume them without proof.