Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
    
You may need to fix your LaTeX... –  Simon Rose Sep 19 '12 at 13:57
    
Right. Thanks. I think it's OK now. I apparently had a problem with subscripts. –  Nicolas B. Sep 19 '12 at 14:26
    
Bienvenu, Nico! Great to see you here. –  Filippo Alberto Edoardo Sep 19 '12 at 17:46

2 Answers 2

up vote 2 down vote accepted

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.

share|improve this answer
    
Thanks Ramsey for your very quick reply! So I guess Katz' paper you mention is the appropriate reference for the first statement, right? For my second question I was actually wondering (more generally) whether the reduction of $E|_k\gamma$ is the Fourier expansion of the reduction of $E$ at the cusp $\gamma\infty$. –  Nicolas B. Sep 19 '12 at 19:42
    
I guess I'm still a little confused. Classically, one defines the $q$-expansions at other cusps by considering $f|_k\gamma$. Thus, "the reduction of $f|_k\gamma$" is what I would mean by "the reduction of $f$ at $\gamma\infty$." –  Ramsey Sep 19 '12 at 19:53
    
Maybe this is what you're getting at (but maybe not!): One can reduce the geometric modular form in my answer mod $\lambda$ and consider the $q$-expansions of that form via Tate curves and ask how those compare to the reductions of the $q$-expansions. Is that sort of what you mean to do? –  Ramsey Sep 19 '12 at 19:55
    
RIGHT! This is exactly what I mean! –  Nicolas B. Sep 19 '12 at 20:05
    
Oh and I forgot to comment on references. I learned this point of view from Katz's article, which I think is pretty clear and nicely concise. Hida also has a book on this stuff ("Geometric Modular Forms and Elliptic Curves" I think is the title) that has lots more background on the moduli problems and so-on. –  Ramsey Sep 19 '12 at 23:35

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

share|improve this answer
    
Hi Fil! Thank you for this reference. I think the theorems you mention don't directly answer my questions but the paper of Rapoport they refer to may contain the appropriate statements. I'll check this. –  Nicolas B. Sep 19 '12 at 19:46
    
Oh, you are right. I keep on thinking at these as being the "q"-expansion principle, from which they directly derive. This is (5.4) and (5.5). In particular, it tells you that if the $q$ expansion lies in one ring $R$ at one cusp, than the form is defined over that ring, so $q$-exp. lies there for all cusps. This implies (1). For (2), apply the above to the genuine $\ell$-adic modular form $a_0(\gamma)$. Beware that DR work assuming $F\neq \mathbb{Q}$ so it does not really apply to your case, when may be Katz' paper in Antwerp proceedings (which you find in many scan folders) is the place. –  Filippo Alberto Edoardo Sep 20 '12 at 0:01
    
Thank you Fil for your contribution! As pointed out by you and Ramsey, Katz' paper is the reference I should look at to get the results I was looking for. –  Nicolas B. Sep 20 '12 at 9:20

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.