Modular form with coefficients in a ring $A$ is defined as follows
$$ M_k(N,A)=M_k(N,\mathbb{Z})\otimes_\mathbb{Z}A $$
where $M_k(N,\mathbb{Z})$ denote a set of all modular forms with integer fourier coefficient at infinity
After few arguments, we can view $M_k(N,A)$ as A-submodule of A[[q]]
On the other hands, mod $l$ modular form is defined as follows($l$:prime number)
$\overline{\mathbb{Q}_l}$ denote algebraic closure of $\mathbb{Q}_l$ where $\mathbb{Q}_l$ is a completion of $\mathbb{Q}$ at prime number $l$
It is known that residue field of $\overline{\mathbb{Q}_l}$ is an algebraic closure of $\mathbb{F}_l$
So we have a natural reduction map $\overline{\mathbb{Z}_l}\rightarrow\overline{\mathbb{F}_l}$(Here, $\overline{\mathbb{Z}_l}$ is a ring of integer of $\overline{\mathbb{Q}_l}$)
Then we define $M_k(N,\overline{\mathbb{F}_l})$ as a set of all formal power series F($z$)=$\sum_{n\geq 1}A_ne^{2\pi inz}, A_n\in\overline{\mathbb{F}_l}$,
for which there exists a modualr form f($z$)=$\sum_{n\geq 1}a_ne^{2\pi inz}\in M_k(N),a_n\in\overline{\mathbb{Z}}$, such that $\overline{a_n}=A_n$ for all $n\geq1$
Here, $\overline{a_n}$ is a image of $a_n$ under the map $\overline{\mathbb{Z}}\hookrightarrow \overline{\mathbb{Z}_l}\rightarrow\overline{\mathbb{F}_l} $
So we have two definitions for $M_k(N,\overline{\mathbb{F}_l})$
I guess that two definitions must be equivalent but I cannot prove.
Also I saw that $\overline{\mathbb{Z}}\hookrightarrow \overline{\mathbb{Z}_l}\rightarrow\overline{\mathbb{F}_l} $ is surjective map. Is it true?(second map is obviously surjective but whole map is surjective?)
Anyone help?
Thank you.