Chennai Mathematical Institute


Venue: CMI, Lecture Hall 5
Time: 19 April (Tuesday), 3.30 pm-4.30 pm
Higher Ramanujan congruences modulo 691

Loic Merel
University of Paris.


The congruence, due to Ramanujan, q\prod_{n=1}^{\infty}(1-q^n)^24 = \sum_{n=1}^{\infty}(\sum_{d|n}d^11)q^n modulo 691 is known to beginners in modular forms. It is not often realized that it brings into existence another modular form defined modulo 691 which is quite mysterious.

In our explanation of this type of phenomenon, we will focus on the the better understood case of modular forms of weight two and prime level. The forms that arise from Ramanujan-style congruences are called (in Lecouturier's work) higher Eisenstein series. All along we will see how the subject appears, and connects to a host of other questions: in the classical work of Mazur on the Eisenstein ideal, and later the constructions and conjectures of yours truly, Goncharov, McCallum–Sharifi, Calegari–Emerton, Fukaya–Kato, Lecouturier, and, most recently, Sharifi–Venkatesh.