Elementary tools for proving congruences of modular forms

Question

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.)

Answer 1

To understand congruences between modular forms, one needs first to understand the abstract notion of congruences between elements of modules with (Hecke) operators acting on them. For this, Ghate's note, as recommended by François, is a veri good introduction.

But then I assume you want to go further, and understand how people prove, or use, congruences between modular forms. The problem with the traditional definition of modular forms as holomorphic functions on the upper half plane is that obviously, this is an analytic, not algebraic definition, and that therefore some (hard) work is needed to reveal the arithmetic nature of modular forms, in particular to study congruences between them. This hard work generally involve some serious algebraic geometry, such as defining, constructing and studying moduli space of elliptic curves with various structures over an arithmetic basis, etc. This is likely to be overwhelming for an undegraduate student. Yet ti understand seriously the aspect you are mentioning (congruences between modular forms of same level but various weights)

An alternative is to start with a different kind of object, somehow related to modular forms, but with a more direct connection to arithmetic. I am thinking of either "modular symbols" or "modular forms over quaternion algebra".

Let me just mention the second here. Let $D$ be a quaternion algebra over $\mathbb Q$ which is ramified at infinity (that is $D \otimes \mathbb R = \mathbb H$) and $G=D^\ast$ the algebraic group over $\mathbb Q$ of its invertble element. Define a "level" K as a compact open subgroup of $G(\mathbb A_f)$, and a "modular form of weight 2 and level K" as simply a function form $G(\mathbb A_f)$ to $\mathbb C$ which is left-invariant by $K$ and right invariant of $G(\mathbb Q)$. You get this way a notion which has a lot of analogy with modular forms: they form a finite-dimensional vector space, which has a natural structure over $\mathbb Z$ (just take the same functions with values in $\mathbb Z$), a natural action of Hecke operators, etc. Actually this is just more than a simple analogy: a deep theorem of Jacquet-Langlands tells you that the space of "modular forms of weight 2 for $D$" defined as above is actually a big subspace of the space of traditional modular forms of weight 2, in a way which respects the Hecke opertaors. But you don't need to understand fully this theorem, less alone its difficult proof, to use it as a motivation to study modular forms over $D$, and their congruences.

Now it turns that certain theorem about congruences between modular forms are much simpler to prove (but still non-trivial) for their $D$-counterpart. One example of this is the famous "Ribet level-raising theorem" which is quite difficult to prove for classical modular forms but is relatively simple and very beautiful to prove for modular forms for $D$. Understanding this theorem may seem a worthwhile goal for a summer project of a very good and very motivated undergrad. One reference is section 1 of Taylor's early and very deep paper "on Galois rep. associated with Hilbert modular forms" at inventiones. Unfortunately it is not extremely reader-friendly, as the main ideas (basically some computations on the Bruhat-Tits tree) are not explicitly explained.