Question

Here is my setting: Let $E\in\mathcal{M}_k(\Gamma_0(N))$ be an Eisenstein series (of trivial Nebentypus) that is a normalized eigenform for all the Hecke operators at level $\Gamma_0(N)$. Assume that the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$ where $\mathcal{O}_K$ is the integer ring of some number field $K$. For any $\gamma\in\mathrm{SL}(2,\mathbf{Z})$ let us denote by $a_0(\gamma)$ the constant term of the Fourier expansion of $E|_k\gamma$. Here are my questions.

1. Is it true that for some number field $L$ containing $K$ and the $N$-th roots of unity we have $a_0(\gamma)\in L$ and $E|_k\gamma\in a_0(\gamma)+q^{1/N}\mathcal{R}[[q^{1/N}]]$ where $S$ is a finite set of places in $L$ containing the finite places above $N$ and the archimedian ones and $\mathcal{R}$ denotes the ring of $S$-integers in $L$?

2. Let $\lambda$ be a prime in $\mathcal{O}_K$ of residue characteristic $\ell\nmid N$. Since by assumption the Fourier expansion of $E$ lies in $\mathcal{O}_K[[q]]$, one may reduce $E$ modulo $\lambda$. Assume that the corresponding modular form over $\overline{\mathbf{F}}_l$ is cuspidal and let $\gamma\in\mathrm{SL}(2,\mathbf{Z})$. Assume moreover that $a_0(\gamma)$ is $\mathcal{L}$-integral for all $\mathcal{L}\mid\ell$. Is it true that $\ell$ divides the numerator of the norm of $a_0(\gamma)$?

I believe the answer to both questions is yes, but I'd like to have a proof or reference. Thanks for your help!

Answer 1

Let $f$ be any modular form of weight $k$ whose coefficients at the cusp $\infty$ lie in $\mathcal{O}_K$. Then (using GAGA and the $q$-expansion principle - see for example Katz's article on $p$-adic forms) $f$ gives rise to a section of the sheaf $\omega^k$ on $X_0(N)$ (or $X_1(N)$ + fixed under the diamond operators if you want to keep you moduli spaces nice) that is defined over the ring $\mathcal{O}_K[1/N](\mu_N)$.

Using the geometric description of $q$-expansions at all cusps in terms of the value of the geometric modular form on the Tate curve with level structure, one sees that in fact all $q$-expansions have coefficients in this ring (and are, as you say, generally expansions in $q^{1/N}$). This is simply because both $f$ and the Tate curve with these level structures are defined over this ring. This should answer your first question in the affirmative.

As for your second, I'm a bit confused because your conclusion is roughly what I would take to be the definition of "the corresponding modular form over $\overline{\mathbb{F}}_\ell$ is cuspidal."

Do you only mean to assume that the $q$-expansion at $\infty$ (making no assumptions at other cusps) has no constant term modulo $\ell$?

EDIT based on clarifications in the comments

Addressing the clarified second question, the $q$-expansions one obtains from the geometric modular form by evaluating at the Tate curve with level structure really are the $q$-expansions of the original form (all this assumes one has embedded $K$ into $\mathbb{C}$ already, but I gather that the OP has done that from the phrasing of his question). In particular, their reductions modulo $\lambda$ coincide, so if the original form has the property that all of the constant terms in the $q$-expansions are divisible by $\lambda$, then the reduced mod $\lambda$ geometric modular form is cuspidal.

Answer 2

I think your answers are contained in theorems (0.1) and (0.3) of Deligne and Ribet's paper in Inventiones, 1980.