A different look at $\tau(p) \equiv p^{11} +1$ mod 691

For zoom details contact Pieter Moree (moree@mpim-bonn.mpg.de).

I will introduce the denominator of Eisenstein classes and i will explain how these denominators give rise to congruences, the example of Ramanujan's congruence for tau(p) will be discussed in some detail.

These denominators arise from the fact that certain modules for the Hecke algebra do not split and I will explains how this influences the structure of certain Galois modules. This enhances some results of  Swinnerton- Dyer and Ribet.

If time permits I want to discuss some general conjectures about denominators of Eisenstein classes, their experimental verification and their arithmetic implications.