# Congruence Relations between Modular Forms

Recently I read papers of Ahlgren and Boylan's paper:

Ahlgren, S. and Boylan, M., 2003. Arithmetic properties of the partition function. Inventiones mathematicae, 153(3), pp.487-502.

In their paper, they give the definition of the space of weight k modular forms modulo $\ell$. Let $\ell\geq5$ be a prime,and $f\in M_k\cap \mathbb{Z}[[q]]$, then the space of weight k modular forms modulo $\ell$ is defined by \begin{align*} \widetilde{M}_k: =\left\{f\pmod \ell:f\in M_k\cap\mathbb{Z}[[q]]\right\}. \end{align*} I want to understand more about this conception. As mentioned by @mixedmath, we can use Sturm Bound to verdict whether two modular forms are congruent. And generalized problems are that

1. is there a method to decide when two modular forms with different weights are congruent with each other module $\ell$ ?

2. moreover, what about for the case of modular forms on congruent subgroup $\Gamma_0(N)$?

Any help will be appreciated. :)

Thanks a lot for @Mathmo123 and @mixedmath's reminder. I hope I have made my question clear.

• The answer to both your questions is the same: check the Fourier coefficients match mod $\ell$ up to a certain bound. If they match, then the forms are congruent. There are some explicit bounds in this paper. Jun 26, 2017 at 9:02

### Question 1

You mention different weights, but all modular forms in $\widetilde{M}_k$ come from modular forms in $M_k$, which are weight $k$ modular forms on $\mathrm{SL}(2, \mathbb{Z})$. Unlike the level of a modular form, the weight is well-defined, and in fact modular forms form a graded ring (graded by weight). So it does not make sense to ask when forms of different weights become identified in $\widetilde{M}_k$, as all such forms have weight $k$.

This may be a side channel towards understanding whether two modular forms are congruent modulo $\ell$. This is a bit more complicated. It is expected that the first several coefficients uniquely identify a modular form (where the number is dependent on the level of the modular form), and that something similar is true even modulo $\ell$. A good concept to look up here is the Sturm Bound.

### Question 2

It is possible to define congruences between modular forms of different levels. This works in essentially the same way as in level 1, and the definitions are one should expect.

For an explicit example, Congruences between modular forms: Raising the level and dropping Euler factors by Diamond shows a congruence between a form on $\Gamma_0(11)$ and $\Gamma_0(77)$. Congruences between forms of different levels are much more interesting in general than forms of the same level, as there are only finitely many forms of a given weight and level.

You may also be interested in Frank Calegari's notes from the Arizona Winter School some years ago, which delves into many introductory (and p-adic) aspects of congruences between modular forms.

• I'm a bit confused by your first paragraph. The weight is not well-defined modulo $\ell$, since a form of weight $k$ is also a form of weight $k+\ell-1$. It is certainly possible to have congruences between forms of different weights! Jun 24, 2017 at 23:04