# Details about the $\mod p$ reduction map

Let $$N$$ be a natural number and let $$\Gamma_1(N)$$ be the congruence subgroup of $$SL_2(\mathbb{Z})$$. Let $$M(N)$$ denote the space of all integer weight holomorphic modular forms for $$\Gamma_1(N)$$ whose Fourier coefficients lie in $$\mathbb{Z}$$. Let $$p$$ be a prime. Then we have the natural map (reducing the coefficients modulo $$p$$),

$$\varphi : M(N)\to (\mathbb{Z}/p\mathbb{Z}) [[q]],$$

and let $$M_p(N)$$ denote the image of $$\varphi$$.

I am interested in understanding what is known about $$\varphi$$. Whether it is surjective? Whether it is injective? In particular, will the lift (under $$\varphi$$) of a $$\mod p$$ cusp form be always a cusp form?

If not, what is the kernel? I did some googling and wasn't able to find a complete source on this subject.

Any help is appreciated.

Thank you

The map $$\varphi$$ cannot be surjective because $$M(N)$$ is a countably generated abelian group (it is the direct sum over the weight $$k \in \mathbb{N}$$ of free $$\mathbb{Z}$$-modules of finite rank), while the $$\mathbb{Z}/p\mathbb{Z}$$-vector space $$\mathbb{Z}/p\mathbb{Z}[[q]]$$ has dimension $$2^{\aleph_0}$$ by Erdös-Kaplansky (see here).
Cusp forms can be congruent to Eisenstein series. Maybe the most famous example, in level $$N=1$$, is $$\Delta \equiv E_{12} \bmod{691}$$ where $$\Delta$$ is the cusp form of weight 12, and $$E_{12}$$ is the Eisenstein series of weight 12. (Actually the constant term of $$E_{12}$$ has a denominator, but one can clear it to find a congruence in your sense.)
• Any idea about the kernel of $\varphi$? Clearly the Kernel is non trivial as you say. I also found a paper by Naomi Jochnowitz online which was also helpful. There is a particular question that I am intereted in. Is there any way to determine, given $N,q$, precisely when two forms would be congruent to each other? This connects back to my question about the Kernel. Thank you. Jun 26, 2021 at 4:46