Details about the $\mod p$ reduction map

Question

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

Answer 1

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.)

For an introduction to this theory, you could try E. Ghate's paper An introduction to congruences between modular forms (Current trends in number theory, Hindustan Book Agency (2002), 39-58).