Over the last 20 years, the modular function has become widely known through its miraculous intervention in two great mathematical achievements: the proof of Fermat's last theorem and the "moonshine" of the monster simple group. In both cases, the modular function appears where no one expected it, and it bridges a chasm between seemingly unrelated fields. It is probably fair to say that, in these two cases, we do not yet fully understand how the modular magic works.
However, it can at least be said that these are not the first modular miracles. Ever since its discovery, in the early 19th century, the modular function has been an engine for spectacular and unexpected results. Now that things modular are back in the news, it is a good time to recall some of the modular miracles of the 19th century. They help us see the recent results in some perspective, and encourage us to believe that there is a lot more to be learned.
John Stillwell. Modular Miracles. The American Mathematical Monthly. Vol. 108, No. 1 (Jan., 2001), pp. 70-76. DOI: 10.2307/2695682