My impression is that the specialists in the field use geometric modular forms when proving congruences of modular forms. While this is probably the right way, I don't think I will be able to get a working knowledge of this point of view fast enough, as this is for an undergraduate summer research project.

Do you know of any references that of anything that involves congruences of modular forms being proved by elementary means? Mostly, I'm looking for examples of methods and tools rather than a specific theorem.

More specifically, I'm interested in congruences between modular forms of different weight and the same level. The only fact I know in this case is that the Eisenstein series $E_{p-1}\equiv 1\pmod{p}$, and multiplying by this Eistenstein series give you equivalences between modular forms of different weight. Also (though much less trivial), the converse is also true. However, with only this fact, it seems the only hope of proving anything is to come up with very explicit formulas for what is happening.

(I know there is a short paper by Serre that determines the structure of the ring of modular forms, under the full SL_2(Z), reduced mod p http://math.bu.edu/people/potthars/writings/serre-1.pdf. However, this does not generalize to modular forms of a given level, since it uses the structure of the ring of modular forms under the full modular group.

A couple of other papers I've found are "Congruences between systems of eigenvalues of modular forms" and "A study of the local components of the Hecke algebra mod l" by Jochnowitz, but I've only just started reading.

This was originally posted on stackexchange: https://math.stackexchange.com/questions/420509/elementary-tools-for-proving-congruences-of-modular-forms.)

An introduction to congruences between modular formsmath.tifr.res.in/%7Eeghate/basics.dvi $\endgroup$ – François Brunault Jun 15 '13 at 11:11