Elementary tools for proving congruences of modular forms 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. 
Do you know of any references that of anything that involves congruences of modular forms being proved by elementary means?
(The only one I know of 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.)
I would also appreciate it if somebody would let me know if this is appropriate for overflow.
Edit: Posted on mathoverflow https://mathoverflow.net/questions/133798/elementary-tools-for-proving-congruences-of-modular-forms 
 A: Have you looked at Serre's paper on p-adic modular forms, in Lecture Notes 349?  This works just at full level, but uses almost all elementary means.
It is fairly hard to go much further in an elementary way.  For example, when I was working on my thesis, I studied congruences mod powers of 2 between modular forms of different weights on $\Gamma_0(2)$, and found an approach a bit like Serre's that worked in this case, and was purely elementary.  This never made it into my thesis (as I worked more on these elementary methods, they pretty naturally morphed into more sophisticated methods, as these things tend to do), but I do have the
original notes somewhere.  If you're interested, I could try to find them, scan them, and post them, just as an illustrative example of what one can do. 
As for papers, have you looked at Ribet's "Converse to Herbrand's criterion" paper? There he proves some congruences in an elementary way (he changes both the weight and the level --- because he wants to go from wt. $k$ to wt. $2$ while preserving the congruence class of the cuspform mod $p$, and when $k \not\equiv 2 \bmod p-1$, this necessitates adding $p$ to the level). 
Also, if you look in Mazur's "Eisteinstein ideal" paper, while there are many congruences established there explicitly or implicity by geometric means,
my memory is that he also studies some congruences by elementary means;
you could look through it to see what examples you can find.
There is also an elementary approach to the whole theory of Hida families
(which is all about finding $p$-adic families of modular forms --- so all
of them congruence mod $p$ --- of varying weight and some fixed level), worked out by Wiles.  You can find it in Wiles's paper "On ordinary $\lambda$-adic representations" (I think), as well as in Hida's book with the title someting like "Elementary theory of $L$-functions and Eisenstein series". (Roughly, you take an eigenform $f$ (assumed to be ordinary at $p$, i.e. its eigenvalue at $p$ is non-zero mod $p$), multiply it by $E_{(p-1)p^{n-1}}$ (which is $\equiv 1 \bmod p^n$), and then apply $U_p$ many times.)
To see some examples of moving between weight one and weight two in an elementary way, you couldlook at section 3.3, 3.4, and 3.5 of this paper of mine (joint with Frank Calegari), and the paper of Gabor Wiese that is referenced there (which I vaguely remember has a mixture of elementary and geometric arguments).
